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To this end Wolfram specified a simple model for financial \ market price fluctuations in Chapter 8 of NKS. While exhibiting the \ characteristic fractal property of market prices this model did not allow an \ analysis of actual financial instruments. The evaluation of such instruments, \ especially financial derivatives, constitutes the main task of financial \ quants employed by banks and financial institutions. What this paper aims to \ do is show that the appropriate modification of the continuous CA \ transformation rules makes it possible to implement efficient code for the \ evaluation of financial options by modelling the trinomial lattice structures \ which correspond to the partial differential equations which describe all \ option behaviour. We show that there a number of possible models using \ inbuilt functions NestList, ListCorrelate and CellularAutomaton which also \ evaluate options by mimicking the stochastic evolution of underlying stock \ prices as cellular automata on a trinomial grid.\ \>", "Text", CellMargins->{{60, Inherited}, {Inherited, Inherited}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], 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Amongst the simplest models used by \ financiers is a trinomial structure which hypothesises that the next value of \ the stock can be in one of three possible future states: either up, down or \ at the same level. This is the so called trinomial model which envisages \ three possible future scenarios for each current state. This can be \ represented diagramtically by Figure 1 which has the horizontal axis \ representing time and the vertical axis for the equity under consideration. \ This three state model can be thought of as a continuous cellular automaton \ moving horizontally through time, where each cell represents the current \ state of the stock, equity or option and future states are obtained in the \ next step from the previous state. The up, down and same levels can be \ obtained from the prior or initial state by shifting, that is rotating to the \ left the initial state either zero, one or two steps. 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0fLiOomoo`0_Ool00fLiOomoo`0`Ool00fLiOomoo`0_Ool00fLiOomoo`0NOol003Uoo`03001oogoo 0>Qoo`00=goo1@00j7oo000gOol5003XOol003Moo`@00>Uoo`00>7oo0`00jGoo000hOol3003YOol0 03Qoo`<00>Uoo`00>7oo0P00jWoo000iOol00`00Oomoo`3XOol003Uoo`03001oogoo0>Qoo`00ogoo 9Goo003oOolUOol00?moobEoo`00ogoo9Goo003oOolUOol0031oo`@00?1oo`00;Woo0P0017oo00<0 07ooOol0kGoo000^Ool00`00Oomoo`04Ool00`00Oomoo`3/Ool003Eoo`03001oogoo0>aoo`00=Goo 00<007oo0000k7oo000bOol30003Ool00`00Oomoo`3YOol002ioo`@000Ioo`03001oogoo0>Uoo`00 ;Woo00<007ooOol01goo00<007ooOol0jGoo000^Ool00`00Oomoo`04Ool01000Oomoo`00jgoo000_ Ool00`00Oomoo`02Ool01@00Oomoogoo0000jgoo000`Ool40004Ool00`00Oomoo`3YOol003Qoo`03 001oogoo0>Uoo`00\ \>"], ImageRangeCache->{{{201.375, 434.625}, {652.25, 419}} -> {-5.1234, 10.2365, \ 0.0247591, 0.0247591}}, CellTags->"Explizite FDM Bild"], Cell["\<\ The combined operation of left shifting and multiplying by \ transition probabilities constitute a function which describes the action of \ a cellular automaton. What is not explicit in such trinomial models is the \ amount by which the stock goes up, represented by the function up[.], the \ amount by which it goes down, represented by the function down[.] and finally \ the amount by which the stock remains at the same level, represented by the \ function stay[.]. All of these functions are dependent upon market parameters \ which have to be specified initially. These include the current value of the \ stock, represented by the symbol S, the strike or exercise price K, which is \ the amount for which the option writer agrees to buy or sell the stock at \ some future date, called the maturity, T. Other parameters include the \ current interest rate r, the stock dividend yield q and the annualised \ standard deviation of the stock or its volatility \[Sigma]. Finally since \ this is a discrete process we also have the number of times the time \ increment dt divides into the time interval T/dt = n, which constitute the \ number of time steps in the model.\ \>", "Text", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[CellGroupData[{ Cell["Neighbouring Symmetry Constraints", "Subsubsection", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "The problem with all such models is in the specification of the amounts \ up[.], down[.] and stay[.] by which the stochastic process representing the \ equity deviates at ech incremental time step. There are two generally \ accepted models for this named after the people who first specified them, \ namely the Jarrow-Rudd and Cox-Ross-Rubinstein models. Both models are \ determined by equating the values for the mean and variance of the \ continuous equity process with its corresponding first two moments in the \ discrete trinomial representation. The problem with such an approach is that \ there are three parameters to specify, but only two equations. This leaves \ one of the parameters as a variable, which accounts for the difference \ between the models. A better approach is described by William Shaw in his \ book \"Modelling Financial Derivatives with ", StyleBox["Mathematica", FontSlant->"Italic"], "\", chapters 18 to 20. Starting with the general power solution to the \ Black-Scholes model " }], "Text", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \(TraditionalForm\`V = Z\ \((S\/K)\)\^n\ \(\[ExponentialE]\^\(\(-r\)\ \((T - t)\)\)\) \[ExponentialE]\^\(\((\((r - q)\)\ n + \(\(n(n - \ 1)\)\ \[Sigma]\^2\)\/2)\)\ \((T - t)\)\)\)], "NumberedEquation", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell["\<\ It can be shown that for appropriate choices of the parameters a \ and b, the relation between the up (u), down (d) and same level (s) amounts, \ along with the up, down and remain the same probabilities (p, q and r) are \ linked with the number of time steps n in the following relation:\ \>", "Text",\ CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \(TraditionalForm\`p\ u^n\ + \ q\ d^n\ + \ r\ s^n\ = \ a^n\ b^\((\((n - 1)\) n/2)\)\)], "NumberedEquation", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell["\<\ We first need to define appropriate functions for all of the above \ parameters before checking that the neighbouring symmetries apply.\ \>", \ "Text", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell["\<\ Discount[n_, r_, T_] := N[Exp[-r T/n]]; afunc[n_, r_, q_, T_] := N[Exp[(r-q) T/n]]; bfunc[n_, \[Sigma]_, T_] := N[Exp[\[Sigma]^2 T/n]];\ \>", "Input", CellLabel->"In[3]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}, InitializationCell->True, AspectRatioFixed->True], Cell[BoxData[{ \(\(U[\ b_]\ := \ 1/2\ \((b*\((b + 1)\) + Sqrt[b^4\ + \ 2\ b^3\ + \ b^2\ - \ 4])\);\)\), "\n", \(up[a_, \ b_]\ := \ a\ U[b]\), "\n", \(\(down[a_, \ b_]\ := \ a\ /U[b];\)\), "\n", \(\(stay[a_, \ b_]\ := \ a;\)\), "\n", \(\(pfunc[ b_]\ := \ \(\((b - 1)\)/\((1 - U[b])\)\)/\((1/U[b]\ - \ U[b])\);\)\), "\n", \(\(qfunc[ b_]\ := \ \(\((b - 1)\)/\((1 - 1/U[b])\)\)/\((U[b]\ - \ 1/U[b])\);\)\), "\n", \(\(rfunc[b_]\ := \ 1\ - \ pfunc[b] - qfunc[b];\)\)}], "Input", CellLabel->"In[6]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}, InitializationCell->True], Cell["\<\ We will now define a function to check the supersymmetric \ structures defined by equation (2):\ \>", "Text", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \(CheckSymmetry[u_, \ d_, s_, \ \ p_, \ q_, \ r_, \ n_, \ a_, \ b_]\ := \ Factor[ Simplify[ p\ u^n\ + \ q\ d^n\ + \ r\ s^n - \ a^n\ b^\((\((n - 1)\) n/2)\)]]\)], "Input", CellLabel->"In[70]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[CellGroupData[{ Cell[BoxData[ \(Table[ CheckSymmetry[up[a, b], down[a, b], stay[a, b], pfunc[b], qfunc[b], rfunc[b], 3, a, b], {n, \(-10\), 10}]\)], "Input", CellLabel->"In[71]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \({0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}\)], "Output", CellLabel->"Out[71]="] }, Open ]], Cell["\<\ The above clearly shows that supersymmetric identity of equation \ (2) is satisfied for a wide range of values of n\ \>", 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oeaN]OmPHkKoK6jkofY/^_m/KkcoNG_0ogMi_omiNl3oJ6ZioemR]OmLG[D02OmMGkD02?mKGKCoKW2k ogeo`OmlO/7oLgFlofe__?mXJk[oGV2d3?mMGkD01OmPHKGoJffkof=U^?mTJ;SoGV6f00SoGEne00Wo G5jeof5T]omTJ;SoI6Nhog5c_OmiNl3oHfFhof5S]omOHKH02OmMGkD02?mMG[GoHVFgofAV^?maL[go NG[0ofe^^om]Kk_oHF:f2OmMG[D2oeeO]@07ofAW^?mQHkOoIFNhogeo`Om[K[_oGEneof=U]`3ooeeO ]I[oGEne0000\ \>"], "ConferenceGraphicCell", Evaluatable->False, ImageSize->{1200, 58}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell["Basic European Options", "Section", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell["\<\ Perhaps the simplest type of options are European options which can \ only be exercised at the maturity date T. This fixed date for the cashing in \ of the option allows a description of the probability distribution of the \ option from that of the underlying asset it is based upon. This was the \ essential insight of Black and Scholes and it can be shown that the trinomial \ models converge in value to that of the theoretical Black-Scholes mode for \ sufficiently large n. We will first give the trinomial code for the european call and put options \ and then show that we can also replicate these values from an appropriately \ defined continuous cellular automata. \ \>", "Text", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \(\(OneTimeStepEuroTri\ = \ Compile[{{initial, \ _Real, \ 1}, {p, \ _Real, \ 0}, \ {q, \ _Real, \ 0}, \ {r, \ _Real, \ 0}}, \n\t p*Drop[initial, \(-2\)]\ + \ q*Drop[RotateLeft[initial, 2], \ \(-2\)] + r*Drop[RotateLeft[initial], \ \(-2\)]];\)\)], "Input", CellLabel->"In[13]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}, InitializationCell->True], Cell[BoxData[ \(\(ManyTimeStepEuroTri\ = \ \n Compile[{{initial, \ _Real, \ 1}, \ {p, _Real, \ 0}, \ {q, \ _Real, \ 0}, \ {r, \ _Real, \ 0}, {n, \ _Integer, \ 0}}, \n Module[{new\ = \ initial, \ k = 1, \ old = initial}, \n For[k\ = \ 1, \ k <= n, \ \(k++\), \ \n\((old\ = \ new; \ new\ = \ OneTimeStepEuroTri[old, p, q, r])\)]; \n Join[new, old]]];\)\)], "Input", CellLabel->"In[14]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}, InitializationCell->True], Cell["\<\ EuroTrinomialCall[S_, K_, n_, r_, q_, \[Sigma]_, T_] := Module[{a = afunc[n,r,q,T],b = bfunc[n, \[Sigma], T], disc = Discount[n,r,T],u, d, s, P, Q, R, result,payoffin, value, delta, \ gamma, theta, rho, vega}, u = up[a,b]; d = down[a,b]; s = stay[a,b]; P = disc*pfunc[b]; Q = disc*qfunc[b]; R = disc*rfunc[b]; payoffin =Join[ \t\t Table[Max[S s^node u^(n-node)-K,0], {node, 0,n, 1}], \t\t Table[Max[S d^node s^(n-node)-K,0], {node, 1,n, 1}]]; result = ManyTimeStepEuroTri[payoffin, P, Q, R, n]; value = result[[1]]; delta = (result[[2]] - result[[4]])/((u-d)*S); gamma = ((result[[2]] - result[[3]])/(u - s) - (result[[3]] - result[[4]])/(s - d))*2/(S^2 (u-d)); theta = (result[[3]]-(s-1)*S*delta- result[[1]])*n/T; rho = -T*(value - S*delta); vega = T*S*S*gamma*\[Sigma]; {value, delta, gamma, theta, rho, vega}]\ \>", "Input", CellLabel->"In[15]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}, InitializationCell->True, AspectRatioFixed->True], Cell["\<\ EuroTrinomialPut[S_, K_, n_, r_, q_, \[Sigma]_, T_] := Module[{a = afunc[n,r,q,T],b = bfunc[n, \[Sigma], T], disc = Discount[n,r,T],u, d, s, P, Q, R, result,payoffin, value, delta, \ gamma, theta, rho, vega}, u = up[a,b]; d = down[a,b]; s = stay[a,b]; P = disc*pfunc[b]; Q = disc*qfunc[b]; R = disc*rfunc[b]; payoffin =Join[ \t\t Table[Max[K-S s^node u^(n-node),0], {node, 0,n, 1}], \t\t Table[Max[K-S d^node s^(n-node),0], {node, 1,n, 1}]]; result = ManyTimeStepEuroTri[payoffin, P, Q, R, n]; value = result[[1]]; delta = (result[[2]] - result[[4]])/((u-d)*S); gamma = ((result[[2]] - result[[3]])/(u - s) - (result[[3]] - result[[4]])/(s - d))*2/(S^2 (u-d)); theta = (result[[3]]-(s-1)*S*delta- result[[1]])*n/T; rho = -T*(value - S*delta); vega = T*S*S*gamma*\[Sigma]; {value, delta, gamma, theta, rho, vega}]\ \>", "Input", CellLabel->"In[16]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}, InitializationCell->True, AspectRatioFixed->True], Cell["\<\ The points to notice about the above code are that it is decidely \ procedural and that both option functions call the same compiled general \ procedure ManyTimeStepEuroTri which in turn repeatedly calls the single step \ function OneTimeStepEuroTri. If we first load in a package designed to \ calculate the traditional known theoretical values we can then see how well \ the trinomial tree schemes compare. \ \>", "Text", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \(\(Needs["\"];\)\)], "Input", CellLabel->"In[17]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}, InitializationCell->True], Cell["\<\ In the following calculations we determine the prices of european \ options and their sensitivity functions, known as the greeks, for the \ following market values: The stock price is S = $9, the strike price is K = \ $10, the risk free interst rate r = 10%, the dividend yield q = 3%, \ volatility \[Sigma] = 20% and the time remaining tothe expiry date is T = 1 \ year. Here we choose n = 500 time steps.\ \>", "Text", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[CellGroupData[{ Cell[BoxData[ \({BlackScholesCall[9, \ 10, \ 0.2, \ 0.1, \ 0.03, \ 1], \ BlackScholesCallDelta[9, \ 10, \ 0.2, \ 0.1, \ 0.03, \ 1], BlackScholesCallGamma[9, \ 10, \ 0.2, \ 0.1, \ 0.03, \ 1], \n\t BlackScholesCallTheta[9, \ 10, \ 0.2, \ 0.1, \ 0.03, \ 1], \n\t\tBlackScholesCallRho[9, \ 10, \ 0.2, \ 0.1, \ 0.03, \ 1], \n\t\tBlackScholesCallVega[9, \ 10, \ 0.2, \ 0.1, \ 0.03, \ 1]}\)], "Input", CellLabel->"In[58]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \({0.5620541710025266`, 0.45551771994214485`, 0.21445089037605863`, \(-0.5781811888725132`\), 3.537605308476773`, 3.4741044240921486`}\)], "Output", CellLabel->"Out[58]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(EuroTrinomialCall[9, 10, \ 500, \ 0.1, \ 0.03, \ 0.2, \ 1]\)], "Input", CellLabel->"In[59]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], 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oeaN]OmPHkKoK6jkofY/^_m/KkcoNG_0ogMi_omiNl3oJ6ZioemR]OmLG[D02OmMGkD02?mKGKCoKW2k ogeo`OmlO/7oLgFlofe__?mXJk[oGV2d3?mMGkD01OmPHKGoJffkof=U^?mTJ;SoGV6f00SoGEne00Wo G5jeof5T]omTJ;SoI6Nhog5c_OmiNl3oHfFhof5S]omOHKH02OmMGkD02?mMG[GoHVFgofAV^?maL[go NG[0ofe^^om]Kk_oHF:f2OmMG[D2oeeO]@07ofAW^?mQHkOoIFNhogeo`Om[K[_oGEneof=U]`3ooeeO ]I[oGEne0000\ \>"], "ConferenceGraphicCell", Evaluatable->False, ImageSize->{1200, 58}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell["Continuous CAs for European Options", "Section", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell["\<\ The typical structure of a continuous cellular automaton is \ described in chapter 3 of Wolfram's book \"A New Kind Of Science\" and \ consists of two main components: an evolutionary Step involving the mapping \ of some function onto the cells of an initial condition and another \ evolutionary List function which repeatedly applies the first function using \ the built-in command NestList. We shall also need a few supplementary \ functions padLeftRight and paddedPayoffs, whose main purpose is to fill out \ the tree structure into a rectangular array suitable for repeated application \ of the EvolveStep procedure.\ \>", "Text", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[BoxData[{ \(\(padLeftRight[list_List, n_Integer] := PadLeft[list, n, {0}, \((n - Length[list])\)/2];\)\), "\[IndentingNewLine]", \(paddedPayoffs[payoff_List, payoffs_List] := \(padLeftRight[#, Length[payoff]] &\)\ /@ \ payoffs\)}], "Input", CellLabel->"In[18]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}, InitializationCell->True], Cell["\<\ There are a number of ways we can construct the appropriate \ cellular automata. Either we can specify the structure of the Step and Evolve \ List functions ourselves, or we can utilise the inbuilt CellularAutomaton[.] \ function. In the first case we will replicate the behaviour of the \ OneTimeStepEuroTri function and build a compiled procedure that shifts the \ previous array of cells depending upon whether we have moved into the up, \ down states or remained in the same state. Suppose that we are in the stae \ called initial, then at the next step we will be at the same price level with \ probability r we will be at the level down with probability q or finally we \ will be at the upper level with probability p. If we are at the same price \ level then the prices remain unchanged, whereas if we go down one level then \ the second value now occupies the leading position and is therefore shifted \ to the left, but if we go one level up then the first element is forced into \ the second position. In this way the expected future value of the next step \ of the process is given by: p*RotateRight[initial] \ +r*initial+q*RotateLeft[initial]. In order that first and last values at \ either end do not get incorrectly transposed, we remove them successively \ from each step by applying the procedure Take[.., {2,-2}]. Finally we \ repeatedly apply the single step procedure using NestList \ \>", "Text", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[BoxData[{ \(\(CompCAEuropeanOptionEvolveStep\ = Compile[{{initial, \ _Real, \ 1}, {p, \ _Real, \ 0}, \ {q, \ _Real, 0}, \ {r, \ _Real, \ 0}}, \[IndentingNewLine]Take[ p*RotateRight[initial]\ + r*initial + q*RotateLeft[initial], {2, \(-2\)}]];\)\), "\n", \(\(CAEuropeanOptionCompEvolveList[initial_List, p_, q_, r_, t_Integer] := NestList[CompCAEuropeanOptionEvolveStep[#, p, q, r] &, initial, t];\)\)}], "Input", CellLabel->"In[20]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}, InitializationCell->True], Cell["\<\ Of course both procedures can be collapsed into one line of code:\ \ \>", "Text", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \(\(CompCAEuropeanOptionEvolveList\ = Compile[{{initial, \ _Real, \ 1}, {p, \ _Real, \ 0}, \ {q, \ _Real, 0}, \ {r, \ _Real, \ 0}, {t, _Integer, 0}}, \[IndentingNewLine]NestList[ p*RotateRight[#] + r*# + q*RotateLeft[#] &, initial, t]];\)\)], "Input", CellLabel->"In[22]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}, InitializationCell->True], Cell["\<\ We can also avail ourselves of the inbuilt function ListCorrelate \ (or ListConvolve) which carry out discrete convolution of the kernel of \ probabilities {p,q,r} with the state of the system. This has the same effect \ as one step of the previous procedure but can be more optimised for large \ arrays. Again NestList is applied for repeated application of the basic \ convolution as we move through time. \ \>", "Text", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[BoxData[{ \(\(CompCAEuroEvolveStep\ = Compile[{{initial, \ _Real, \ 1}, {p, \ _Real, \ 0}, \ {q, \ _Real, 0}, \ {r, \ _Real, \ 0}}, \[IndentingNewLine]ListCorrelate[{p, r, q}, initial]];\)\), "\n", \(\(CAEuroCompEvolveList[initial_List, p_, q_, r_, t_Integer] := NestList[CompCAEuroEvolveStep[#, p, q, r] &, initial, t];\)\)}], "Input", CellLabel->"In[23]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}, InitializationCell->True], Cell["\<\ As before both procedures can be collapsed into one line of code. \ But with the minor adjustment required by the fact that compiled functions \ only work on arrays of full rank and which therefore require the further \ application of the padLeftRight[.] function.\ \>", "Text", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \(\(CompCAEuroEvolveList\ = Compile[{{initial, \ _Real, \ 1}, {p, \ _Real, \ 0}, \ {q, \ _Real, 0}, \ {r, \ _Real, \ 0}, {t, _Integer, 0}}, \[IndentingNewLine]NestList[ padLeftRight[ListCorrelate[{p, r, q}, #], Length[#]] &, initial, t], {{padLeftRight[_], _Real, 1}}];\)\)], "Input", CellLabel->"In[25]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}, InitializationCell->True], Cell["\<\ Finally we consider the use of the inbuilt function \ CellularAutomaton[.]. This undoubtedly constitutes the simplest use of the CA \ methodology and is also highly optimised. Again we are using the \ specification that requires a function in the first argument that convolves \ the probability vector {p,q,r} with the current state of the system.\ \>", \ "Text", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \(\(EuroCAEvolveStep[initial_List, p_, q_, r_] := CellularAutomaton[{# . {p, r, q} &, {}, 1}, initial, 1];\)\)], "Input",\ CellLabel->"In[26]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}, InitializationCell->True], Cell["\<\ The use of the function CellularAutomaton[.] removes the need for \ further use of the NestList operator. However we do not need to carry all the \ results as we move to the final step of the automaton and we therefore \ specify the inclusion of only the last two time steps and cells ranging from \ positions t-1 to t+1. \ \>", "Text", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \(\(EuroCAEvolveList[initial_List, p_, q_, r_, t_Integer] := CellularAutomaton[{# . {p, r, q} &, {}, 1}, initial, t, {\(-2\), {t - 1, t + 1}}];\)\)], "Input", CellLabel->"In[27]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}, InitializationCell->True], Cell["\<\ In order to create the european option programs we merely have to \ take the existing trinomial function pattern and replace the function \ ManyTimeStepEuroTri[.] wherever we see it with the new CA functions \ CompCAEuropeanOptionEvolveList, CompCAEuroEvolveList or EuroCAEvolveList. The \ following functions not only determine the european option values but with \ very little further calculation, all of the sensitivity functions or greeks \ also.\ \>", "Text", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell["\<\ CompCAEuropeanOptionCall[S_, K_, n_, r_, q_, \[Sigma]_, T_] := Module[{a = afunc[n,r,q,T],b = bfunc[n, \[Sigma], T], disc = Discount[n,r,T],u, d, s, P, Q, R, result,payoffin, value, delta, \ gamma, theta, rho, vega}, u = up[a,b]; d = down[a,b]; s = stay[a,b]; P = disc*pfunc[b]; Q = disc*qfunc[b]; R = disc*rfunc[b]; payoffin =Join[ \t\t Table[Max[S s^node u^(n-node)-K,0], {node, 0,n, 1}], \t\t Table[Max[S d^node s^(n-node)-K,0], {node, 1,n, 1}]]; result = CompCAEuropeanOptionEvolveList[payoffin, P, Q, R, \ n][[{-2,-1},{n,n+1,n+2}]]; value = result[[-1,2]]; delta = (result[[1,1]] - result[[1,3]])/((u-d)*S); gamma = ((result[[1,1]] - result[[1,2]])/(u - s) - (result[[1,2]] - result[[1,3]])/(s - d))*2/(S^2 (u-d)); theta = (result[[1,2]]-(s-1)*S*delta- result[[-1,2]])*n/T; rho = -T*(value - S*delta); vega = T*S*S*gamma*\[Sigma]; {value, delta, gamma, theta, rho, vega}]\ \>", "Input", CellLabel->"In[28]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}, InitializationCell->True, AspectRatioFixed->True], Cell["\<\ CompCAEuropeanOptionPut[S_, K_, n_, r_, q_, \[Sigma]_, T_] := Module[{a = afunc[n,r,q,T],b = bfunc[n, \[Sigma], T], disc = Discount[n,r,T],u, d, s, P, Q, R, result,payoffin, value, delta, \ gamma, theta, rho, vega}, u = up[a,b]; d = down[a,b]; s = stay[a,b]; P = disc*pfunc[b]; Q = disc*qfunc[b]; R = disc*rfunc[b]; payoffin =Join[ \t\t Table[Max[K-S s^node u^(n-node),0], {node, 0,n, 1}], \t\t Table[Max[K-S d^node s^(n-node),0], {node, 1,n, 1}]]; result = CompCAEuropeanOptionEvolveList[payoffin, P, Q, R, \ n][[{-2,-1},{n,n+1,n+2}]]; value = result[[-1,2]]; delta = (result[[1,1]] - result[[1,3]])/((u-d)*S); gamma = ((result[[1,1]] - result[[1,2]])/(u - s) - (result[[1,2]] - result[[1,3]])/(s - d))*2/(S^2 (u-d)); theta = (result[[1,2]]-(s-1)*S*delta- result[[-1,2]])*n/T; rho = -T*(value - S*delta); vega = T*S*S*gamma*\[Sigma]; {value, delta, gamma, theta, rho, vega}]\ \>", "Input", CellLabel->"In[29]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}, InitializationCell->True, AspectRatioFixed->True], Cell["\<\ CompCAEuroEvolveCall[S_, K_, n_, r_, q_, \[Sigma]_, T_] := Module[{a = afunc[n,r,q,T],b = bfunc[n, \[Sigma], T], disc = Discount[n,r,T],u, d, s, P, Q, R, result,payoffin, value, delta, \ gamma, theta, rho, vega}, u = up[a,b]; d = down[a,b]; s = stay[a,b]; P = disc*pfunc[b]; Q = disc*qfunc[b]; R = disc*rfunc[b]; payoffin =Join[ \t\t Table[Max[S s^node u^(n-node)-K,0], {node, 0,n, 1}], \t\t Table[Max[S d^node s^(n-node)-K,0], {node, 1,n, 1}]]; result = CompCAEuroEvolveList[payoffin, P, Q, R, n][[{-2,-1},{n,n+1,n+2}]]; value = result[[-1,2]]; delta = (result[[1,1]] - result[[1,3]])/((u-d)*S); gamma = ((result[[1,1]] - result[[1,2]])/(u - s) - (result[[1,2]] - result[[1,3]])/(s - d))*2/(S^2 (u-d)); theta = (result[[1,2]]-(s-1)*S*delta- result[[-1,2]])*n/T; rho = -T*(value - S*delta); vega = T*S*S*gamma*\[Sigma]; {value, delta, gamma, theta, rho, vega}]\ \>", "Input", CellLabel->"In[30]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}, InitializationCell->True, AspectRatioFixed->True], Cell["\<\ CompCAEuroEvolvePut[S_, K_, n_, r_, q_, \[Sigma]_, T_] := Module[{a = afunc[n,r,q,T],b = bfunc[n, \[Sigma], T], disc = Discount[n,r,T],u, d, s, P, Q, R, result,payoffin, value, delta, \ gamma, theta, rho, vega}, u = up[a,b]; d = down[a,b]; s = stay[a,b]; P = disc*pfunc[b]; Q = disc*qfunc[b]; R = disc*rfunc[b]; payoffin =Join[ \t\t Table[Max[K-S s^node u^(n-node),0], {node, 0,n, 1}], \t\t Table[Max[K-S d^node s^(n-node),0], {node, 1,n, 1}]]; result = CompCAEuroEvolveList[payoffin, P, Q, R, n][[{-2,-1},{n,n+1,n+2}]]; value = result[[-1,2]]; delta = (result[[1,1]] - result[[1,3]])/((u-d)*S); gamma = ((result[[1,1]] - result[[1,2]])/(u - s) - (result[[1,2]] - result[[1,3]])/(s - d))*2/(S^2 (u-d)); theta = (result[[1,2]]-(s-1)*S*delta- result[[-1,2]])*n/T; rho = -T*(value - S*delta); vega = T*S*S*gamma*\[Sigma]; {value, delta, gamma, theta, rho, vega}]\ \>", "Input", CellLabel->"In[31]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}, InitializationCell->True, AspectRatioFixed->True], Cell["\<\ EuroCAEvolveCall[S_, K_, n_, r_, q_, \[Sigma]_, T_] := Module[{a = afunc[n,r,q,T],b = bfunc[n, \[Sigma], T], disc = Discount[n,r,T],u, d, s, P, Q, R, result,payoffin, value, delta, \ gamma, theta, rho, vega}, u = up[a,b]; d = down[a,b]; s = stay[a,b]; P = disc*pfunc[b]; Q = disc*qfunc[b]; R = disc*rfunc[b]; payoffin =Join[ \t\t Table[Max[S s^node u^(n-node)-K,0], {node, 0,n, 1}], \t\t Table[Max[S d^node s^(n-node)-K,0], {node, 1,n, 1}]]; result = EuroCAEvolveList[payoffin, P, Q, R, n]; value = result[[-1,2]]; delta = (result[[1,1]] - result[[1,3]])/((u-d)*S); gamma = ((result[[1,1]] - result[[1,2]])/(u - s) - (result[[1,2]] - result[[1,3]])/(s - d))*2/(S^2 (u-d)); theta = (result[[1,2]]-(s-1)*S*delta- result[[-1,2]])*n/T; rho = -T*(value - S*delta); vega = T*S*S*gamma*\[Sigma]; {value, delta, gamma, theta, rho, vega}]\ \>", "Input", CellLabel->"In[32]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}, InitializationCell->True, AspectRatioFixed->True], Cell["\<\ EuroCAEvolvePut[S_, K_, n_, r_, q_, \[Sigma]_, T_] := Module[{a = afunc[n,r,q,T],b = bfunc[n, \[Sigma], T], disc = Discount[n,r,T],u, d, s, P, Q, R, result,payoffin, value, delta, \ gamma, theta, rho, vega}, u = up[a,b]; d = down[a,b]; s = stay[a,b]; P = disc*pfunc[b]; Q = disc*qfunc[b]; R = disc*rfunc[b]; payoffin =Join[ \t\t Table[Max[K-S s^node u^(n-node),0], {node, 0,n, 1}], \t\t Table[Max[K-S d^node s^(n-node),0], {node, 1,n, 1}]]; result = EuroCAEvolveList[payoffin, P, Q, R, n]; value = result[[-1,2]]; delta = (result[[1,1]] - result[[1,3]])/((u-d)*S); gamma = ((result[[1,1]] - result[[1,2]])/(u - s) - (result[[1,2]] - result[[1,3]])/(s - d))*2/(S^2 (u-d)); theta = (result[[1,2]]-(s-1)*S*delta- result[[-1,2]])*n/T; rho = -T*(value - S*delta); vega = T*S*S*gamma*\[Sigma]; {value, delta, gamma, theta, rho, vega}]\ \>", "Input", CellLabel->"In[33]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}, InitializationCell->True, AspectRatioFixed->True], Cell["\<\ In the following calculations we determine the prices of european \ options and their sensitivity functions, known as the greeks, for the \ following market values: The stock price is S = $100, the strike price is K = \ $102, the risk free interst rate r = 10%, the dividend yield q = 3%, \ volatility \[Sigma] = 30% and the time remaining tothe expiry date is T = 1 \ year. Here we choose n = 500 time steps.\ \>", "Text", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[CellGroupData[{ Cell[BoxData[ \({BlackScholesCall[100, \ 102, \ 0.3, \ 0.1, \ 0.03, \ 1], \ BlackScholesCallDelta[100, \ 102, \ 0.3, \ 0.1, \ 0.03, \ 1], BlackScholesCallGamma[100, \ 102, \ 0.3, \ 0.1, \ 0.03, \ 1], \n\t BlackScholesCallTheta[100, \ 102, \ 0.3, \ 0.1, \ 0.03, \ 1], \n\t\tBlackScholesCallRho[100, \ 102, \ 0.3, \ 0.1, \ 0.03, \ 1], \n\t\tBlackScholesCallVega[100, \ 102, \ 0.3, \ 0.1, \ 0.03, \ 1]}\)], "Input", CellLabel->"In[63]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \({13.81989016030748`, 0.6060445305008241`, 0.01227140698619314`, \(-8.382455841261937`\), 46.78456288977495`, 36.81422095857944`}\)], "Output", CellLabel->"Out[63]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(CompCAEuropeanOptionCall[100, 102, \ 500, \ 0.1, \ 0.03, \ 0.3, \ 1]\)], "Input", CellLabel->"In[64]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \({13.81880661883633`, 0.606189429469184`, 0.012276659579754353`, \(-8.388145454172324`\), 46.80013632808206`, 36.829978739263055`}\)], "Output", CellLabel->"Out[64]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(CompCAEuroEvolveCall[100, 102, \ 500, \ 0.1, \ 0.03, \ 0.3, \ 1]\)], "Input", CellLabel->"In[65]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \({13.818806618836332`, 0.6061894294691844`, 0.012276659579754658`, \(-8.388145454171436`\), 46.80013632808211`, 36.82997873926398`}\)], "Output", CellLabel->"Out[65]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(EuroCAEvolveCall[100, 102, \ 500, \ 0.1, \ 0.03, \ 0.3, \ 1]\)], "Input",\ CellLabel->"In[66]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \({13.818806618836328`, 0.606189429469184`, 0.012276659579754353`, \(-8.388145454171436`\), 46.80013632808207`, 36.829978739263055`}\)], "Output", CellLabel->"Out[66]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \({BlackScholesPut[100, \ 102, \ 0.3, \ 0.1, \ 0.03, \ 1], \ BlackScholesPutDelta[100, \ 102, \ 0.3, \ 0.1, \ 0.03, \ 1], BlackScholesPutGamma[100, \ 102, \ 0.3, \ 0.1, \ 0.03, \ 1], \n\t BlackScholesPutTheta[100, \ 102, \ 0.3, \ 0.1, \ 0.03, \ 1], \n\t\tBlackScholesPutRho[100, \ 102, \ 0.3, \ 0.1, \ 0.03, \ 1], \n\t\tBlackScholesPutVega[100, \ 102, \ 0.3, \ 0.1, \ 0.03, \ 1]}\)], "Input", CellLabel->"In[67]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \({9.068753445124528`, \(-0.36440100304768414`\), 0.01227140698619314`, \(-2.064450777940673`\), \ \(-45.508853749892914`\), 36.81422095857944`}\)], "Output", CellLabel->"Out[67]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(CompCAEuropeanOptionPut[100, 102, \ 500, \ 0.1, \ 0.03, \ 0.3, \ 1]\)], "Input", CellLabel->"In[68]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \({9.067669903652657`, \(-0.36431433255821166`\), 0.012276659579754995`, \(-2.069304736995292`\), \ \(-45.499103159473826`\), 36.82997873926498`}\)], "Output", CellLabel->"Out[68]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(CompCAEuroEvolvePut[100, 102, \ 500, \ 0.1, \ 0.03, \ 0.3, \ 1]\)], "Input", CellLabel->"In[69]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \({9.067669903652657`, \(-0.3643143325582128`\), 0.01227665957975536`, \(-2.0693047369961803`\), \ \(-45.49910315947393`\), 36.82997873926608`}\)], "Output", CellLabel->"Out[69]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(EuroCAEvolvePut[100, 102, \ 500, \ 0.1, \ 0.03, \ 0.3, \ 1]\)], "Input",\ CellLabel->"In[70]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \({9.067669903652657`, \(-0.3643143325582128`\), 0.01227665957975536`, \(-2.069304736995292`\), \(-45.49910315947393`\), 36.82997873926608`}\)], "Output", CellLabel->"Out[70]="] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], " ", ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["American Option Theory", "Section", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell["\<\ American options are intrinsically more difficult to analyse \ because of their property of early exercise, giving the holder of the option \ the right to exercise her option before the expiry date if she believes that \ it is unlikely that the underlying stock is either going to increase in the \ case of an american call or decrease in the case of an american put. The main \ emphasis of the solution of the Black-Scholes formula is that the stock has a \ lognormal distibution which menas that the expected future value at one time \ point - the maturity date can be reasonably described by a normal cumulative \ distribution function. But if the option can be exercised at any future time \ then there are an infinite number of such eventualities to calculate and the \ original argument breaks down. There are three alternative approaches \ however: a backward recursive binomial or trinomial model which can give \ approximate results for very large iterations, a quadratic approximation \ technique due to Baroni-Adesi and Whaley or lastly the determination of the \ moving free boundary, which is an associated problem concerning the structure \ of the time dependent curve which desribes those moments when exercise is \ optimal and the value of the option is best realised and placed in a bank \ account rather than left to depreciate with time. The recursive trinomial \ approach is described by the functions SSFastTriAmerCallFun[.] and \ SSFastTriAmerPutFun[.], but involves the use of procedural For[.] loops. It \ will be shown in the following section how these procedural constructs can be \ replaced by continuous cellular automata. \ \>", "Text", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell["\<\ We will first give the trinomial code for the american call and put \ options and then show that we can also replicate these values from an \ appropriately defined continuous cellular automata.\ \>", "Text", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \(\(OneTimeStepAmerFunTri = Compile[{{initial, \ _Real, \ 1}, {payoff, _Real, 1}, {p, \ _Real, \ 0}, \ {q, \ _Real, 0}, \ {r, \ _Real, \ 0}}, \[IndentingNewLine]MapThread[ Max, {payoff, p*Drop[initial, \(-2\)]\ + q*Drop[RotateLeft[initial, 2], \ \(-2\)] + r*Drop[RotateLeft[initial], \ \(-2\)]}]];\)\)], "Input", CellLabel->"In[34]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}, InitializationCell->True], Cell[BoxData[ \(\(ManyTimeStepAmerCallFunTri\ = \ \n Compile[{{initial, \ _Real, \ 1}, \ {p, _Real, \ 0}, \ {q, \ _Real, 0}, \ {r, \ _Real, 0}, \ {S, \ _Real, \ 0}, \ {u, \ _Real, \ 0}, \ {d, \ _Real, 0}, \ {s, \ _Real, 0}, {n, \ _Integer, \ 0}, \ {K, \ _Real, \ 0}}, \n Module[{new\ = \ initial, \ k = 1, \ R = 0, \ old = initial, \ payoff = Drop[initial, \(-2\)], finpay\ = \ {Max[S - K, 0]}}, \n For[k\ = \ 1, k <= n - 1, \ \(k++\), \ \n\((payoff\ = Join[Table[ Max[S\ s^R\ u^\((n - k - R)\) - K, 0], \ {R, \ 0, n - k, 1}], Table[Max[S\ d^R\ s^\((n - k - R)\) - K, 0], \ {R, 1, n - k, 1}]]; \[IndentingNewLine]old\ = \ new; \[IndentingNewLine]new\ = \ OneTimeStepAmerFunTri[old, payoff, \ p, q, r])\)]; \n old\ = \ new; \n new\ = \ \ OneTimeStepAmerFunTri[old, finpay, p, q, r]; \n Join[new, old]]];\)\)], "Input", CellLabel->"In[35]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}, InitializationCell->True], Cell[BoxData[ \(\(ManyTimeStepAmerPutFunTri\ = \ \n Compile[{{initial, \ _Real, \ 1}, \ {p, _Real, \ 0}, \ {q, \ _Real, 0}, \ {r, \ _Real, 0}, \ {S, \ _Real, \ 0}, \ {u, \ _Real, \ 0}, \ {d, \ _Real, 0}, \ {s, \ _Real, 0}, {n, \ _Integer, \ 0}, \ {K, \ _Real, \ 0}}, \n Module[{new\ = \ initial, \ k = 1, \ R = 0, \ old = initial, \ payoff = Drop[initial, \(-2\)], finpay\ = \ {Max[K - S, 0]}}, \n For[k\ = \ 1, k <= n - 1, \ \(k++\), \ \n\((payoff\ = Join[Table[ Max[K - S\ s^R\ u^\((n - k - R)\), 0], \ {R, \ 0, n - k, 1}], Table[Max[K - S\ d^R\ s^\((n - k - R)\), 0], \ {R, 1, n - k, 1}]]; \[IndentingNewLine]old\ = \ new; \[IndentingNewLine]new\ = \ OneTimeStepAmerFunTri[old, payoff, \ p, q, r])\)]; \n old\ = \ new; \n new\ = \ \ OneTimeStepAmerFunTri[old, finpay, p, q, r]; \n Join[new, old]]];\)\)], "Input", CellLabel->"In[36]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}, InitializationCell->True], Cell[BoxData[ \(SSFastTriAmerCallFun[S_, K_, n_, r_, q_, \[Sigma]_, T_] := Module[{a = afunc[n, r, q, T], b = bfunc[n, \[Sigma], T], disc = Discount[n, r, T], P, Q, R, value, delta, gamma, theta, rho, vega, result, payoffin}, \[IndentingNewLine]u = up[a, b]; \n\ \ \ \ d = down[a, b]; \n\ \ \ \ s = stay[a, b]; \[IndentingNewLine]P = disc\ pfunc[b]; \[IndentingNewLine]Q = disc\ qfunc[b]; \[IndentingNewLine]R = disc\ rfunc[b]; \[IndentingNewLine]payoffin = Join[Table[ Max[S\ s^node\ u^\((n - node)\) - K, 0], {node, 0, n, 1}], Table[Max[S\ d^node\ s^\((n - node)\) - K, 0], {node, 1, n, 1}]]; \[IndentingNewLine]result = ManyTimeStepAmerCallFunTri[payoffin, P, Q, R, S, u, d, s, n, K]; \[IndentingNewLine]value = result[\([1]\)]; \[IndentingNewLine]delta = \((result[\([2]\)] - result[\([4]\)])\)/\((\((u - d)\)*S)\); gamma = \((\((result[\([2]\)] - result[\([3]\)])\)/\((u - s)\) - \((result[\([3]\)] - result[\([4]\)])\)/\((s - d)\))\)*2/\((S^2\ \((u - d)\))\); \[IndentingNewLine]theta = \((result[\([3]\)] \ - \((s - 1)\)*S*delta - result[\([1]\)])\)* n/T; \[IndentingNewLine]rho = \(-T\)*\((value - S*delta)\); \[IndentingNewLine]vega = T*S*S*gamma*\[Sigma]; \[IndentingNewLine]{value, delta, gamma, theta, rho, vega}]\)], "Input", CellLabel->"In[37]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}, InitializationCell->True], Cell[BoxData[ \(SSFastTriAmerPutFun[S_, K_, n_, r_, q_, \[Sigma]_, T_] := Module[{a = afunc[n, r, q, T], b = bfunc[n, \[Sigma], T], disc = Discount[n, r, T], P, Q, R, value, delta, gamma, theta, rho, vega, result, payoffin}, \[IndentingNewLine]u = up[a, b]; \n\ \ \ \ d = down[a, b]; \n\ \ \ \ s = stay[a, b]; \[IndentingNewLine]P = disc\ pfunc[b]; \[IndentingNewLine]Q = disc\ qfunc[b]; \[IndentingNewLine]R = disc\ rfunc[b]; \[IndentingNewLine]payoffin = Join[Table[ Max[K - S\ s^node\ u^\((n - node)\), 0], {node, 0, n, 1}], Table[Max[K - S\ d^node\ s^\((n - node)\), 0], {node, 1, n, 1}]]; \[IndentingNewLine]result = ManyTimeStepAmerPutFunTri[payoffin, P, Q, R, S, u, d, s, n, K]; \[IndentingNewLine]value = result[\([1]\)]; \[IndentingNewLine]delta = \((result[\([2]\)] - result[\([4]\)])\)/\((\((u - d)\)*S)\); gamma = \((\((result[\([2]\)] - result[\([3]\)])\)/\((u - s)\) - \((result[\([3]\)] - result[\([4]\)])\)/\((s - d)\))\)*2/\((S^2\ \((u - d)\))\); \[IndentingNewLine]theta = \((result[\([3]\)] \ - \((s - 1)\)*S*delta - result[\([1]\)])\)* n/T; \[IndentingNewLine]rho = \(-T\)*\((value - S*delta)\); \[IndentingNewLine]vega = T*S*S*gamma*\[Sigma]; \[IndentingNewLine]{value, delta, gamma, theta, rho, vega}]\)], "Input", CellLabel->"In[38]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}, InitializationCell->True], Cell["\<\ The points to notice about the above code are that it is decidely \ procedural and that both option functions call either the compiled general \ procedure ManyTimeStepAmerCallFunTri[.] or ManyTimeStepAmerPutFunTri[.], both \ of which in turn repeatedly calls the single step function \ OneTimeStepAmerFunTri[.]. \ \>", "Text", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell["\<\ In the following calculations we determine the prices of american \ options and their sensitivity functions, known respectively as the delta, \ gamma, theta, rho and vega greeks, for the following market values: The stock \ price is S = $100, the strike price is K = $102, the risk free interst rate r \ = 10%, the dividend yield q = 0, volatility \[Sigma] = 30% and the time \ remaining tothe expiry date is T = 1 year. Here we choose n = 500 time steps.\ \ \>", "Text", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[CellGroupData[{ Cell[BoxData[ \(SSFastTriAmerCallFun[100, 102, 500, 0.1, 0, 0.3, 1]\)], "Input", CellLabel->"In[71]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \({15.72392811946277`, 0.6618780980141131`, 0.012188532494427841`, \(-10.534420700534675`\), 50.46388168194855`, 36.565597483283526`}\)], "Output", CellLabel->"Out[71]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(SSFastTriAmerPutFun[100, 102, 500, 0.1, 0, 0.3, 1]\)], "Input", CellLabel->"In[72]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \({9.307809168149863`, \(-0.4185961206126292`\), 0.017266561611763924`, \(-2.656507114288331`\), \ \(-51.16742122941278`\), 51.79968483529177`}\)], "Output", CellLabel->"Out[72]="] }, Open ]], Cell["\<\ If we now load in a package designed to calculate the above \ described theoretical values we can then see how well the trinomial tree \ schemes compare. The quadratic approximation is given by the function \ AmericanPutMBAW[.],while the moving boundary approach due to Carr,Jarrow and \ Myneni is desribed by the functions AmericanPutCJM[.],AmericanPutDeltaCJM[.] \ and AmericanPutGammaCJM[.].These last three describe the put option value,as \ well as its delta and gamma greek sensitivities.\ \>", "Text", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \(\(Needs["\"];\)\)], "Input", CellLabel->"In[39]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}, InitializationCell->True], Cell[CellGroupData[{ Cell[BoxData[ \(\(?AmericanPutCJM\)\)], "Input", CellLabel->"In[86]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \("AmericanPutCJM[price,strike,vol,r,t] returns an APPROXIMATE value for \ a vanilla zero-dividend American Put according to an approximation of the \ Carr-Jarrow-Myneni integral equation."\)], "Print", CellTags->"Info3307398330-2484739"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(?AmericanPutMBAW\)\)], "Input", CellLabel->"In[87]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \("AmericanPutMBAW[price,strike,vol,r,q,t] returns an APPROXIMATE value \ for a vanilla American Put according to an approximation of the quadratic \ analytical approximation."\)], "Print", CellTags->"Info3307398332-1287534"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \({AmericanPutMBAW[100, 102, \ 0.3, \ 0.1, \ 0, \ 1], \ AmericanPutCJM[100, 102, \ 0.3, \ 0.1, \ 1], AmericanPutDeltaCJM[100, 102, \ 0.3, \ 0.1, \ 1], \n\t AmericanPutGammaCJM[100, 102, \ 0.3, \ 0.1, \ 1]}\)], "Input", CellLabel->"In[73]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \({9.337921524904488`, 9.302966004262213`, \(-0.4184033734590755`\), 0.017241185981134406`}\)], "Output", CellLabel->"Out[73]="] }, Open ]], Cell["\<\ Clearly the trinomial model is giving accurate results up to the \ third decimal place for the Carr,Jarrow and Myneni model, but differs \ significantly from the Baroni-Adesi and Whaley quadratic formula, which is to \ be expected as it is only a second order approximation.\ \>", "Text", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[CellGroupData[{ Cell[BoxData[ \(SSFastTriAmerPutFun[100, 102, 500, 0.1, 0, 0.3, 1]\)], "Input", CellLabel->"In[76]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \({9.307809168149863`, \(-0.4185961206126292`\), 0.017266561611763924`, \(-2.656507114288331`\), \ \(-51.16742122941278`\), 51.79968483529177`}\)], "Output", CellLabel->"Out[76]="] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], " ", ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Continuous CAs for American Options", "Section", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "Unlike the european options we have to not only specify the initial payoff \ function but also the payoff structure which reflects the possible exercise \ values at different nodes in the trinomial lattice. These are not the final \ values of the lattice but are updated when compared with the expected values \ determined by a recursion technique. In this way the american options are ", StyleBox["conditional", FontSlant->"Italic"], " cellular automata, with an initial set of values reflecting the \ environment dictated by the overall payoff structure of path dependent \ options such as american derivatives. These payoffs will be later inserted in \ the code for the final continuous cellular american options." }], "Text", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \(\(With[{u = 1.1, d = 0.9, s = 1.002, S = 100, K = 102, n = 20}, callpayoffs\ = Table[Join[ Table[Max[S\ s^node\ u^\((n - k - node)\) - K, 0], \ {node, \ 0, n - k, 1}], Table[Max[S\ d^node\ s^\((n - k - node)\) - K, 0], \ {node, 1, n - k, 1}]], {k, 0, n}]];\)\)], "Input", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \(\(With[{u = 1.1, d = 0.9, s = 1.002, S = 100, K = 102, n = 20}, putpayoffs\ = Table[Join[ Table[Max[K - S\ s^node\ u^\((n - k - node)\), 0], \ {node, \ 0, n - k, 1}], Table[Max[K - S\ d^node\ s^\((n - k - node)\), 0], \ {node, 1, n - k, 1}]], {k, 0, n}]];\)\)], "Input", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell["\<\ As with the european options we hall need to use the \ padLeftRight[.] function to describe individual payoff values and \ paddedPayoffs[.] to denote the construction of payoff structures appropriate \ to american options. The main purpose of these supplementary functions \ padLeftRight and paddedPayoffs is to fill out the tree structure into a \ rectangular array suitable for repeated application of the EvolveStep \ procedure.\ \>", "Text", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[BoxData[{ \(\(padLeftRight[list_List, n_Integer] := PadLeft[list, n, {0}, \((n - Length[list])\)/2];\)\), "\[IndentingNewLine]", \(paddedPayoffs[payoff_List, payoffs_List] := \(padLeftRight[#, Length[payoff]] &\)\ /@ \ payoffs\)}], "Input", CellLabel->"In[40]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}, InitializationCell->True], Cell["\<\ As before there are a number of ways we can construct the \ appropriate cellular automata. Either we can specify the structure of the \ Step and Evolve List functions ourselves, or we can utilise the inbuilt \ CellularAutomaton[.] function. In the first case we will replicate the \ behaviour of the OneTimeStepAmerFunTri function and build a compiled \ procedure that shifts the previous array of cells depending upon whether we \ have moved into the up, down states or remained in the same state. Suppose \ that we are in the stae called initial, then at the next step we will be at \ the same price level with probability r we will be at the level down with \ probability q or finally we will be at the upper level with probability p. If \ we are at the same price level then the prices remain unchanged, whereas if \ we go down one level then the second value now occupies the leading position \ and is therefore shifted to the left, but if we go one level up then the \ first element is forced into the second position. In this way the expected \ future value of the next step of the process is given by: \ p*RotateRight[initial] +r*initial+q*RotateLeft[initial]. In order that first \ and last values at either end do not get incorrectly transposed, we remove \ them successively from each step by applying the procedure Take[.., {2,-2}]. \ However american options have the additional attribute of early exercise \ which translates into the payoff at any particular time being the maximum of \ either the expected future value or the actual exercised value. This \ necessitates mapping the Max function over the exercise payoff values and \ comparing them with the expected values. This is accomplished with the \ MapThread function and then repeatedly applied with the FoldList command. \ \ \>", "Text", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[BoxData[{ \(\(CompCAAmericanOptionEvolveStep\ = Compile[{{initial, \ _Real, \ 1}, {payoff, _Real, 1}, {p, \ _Real, \ 0}, \ {q, \ _Real, 0}, \ {r, \ _Real, \ 0}}, \[IndentingNewLine]MapThread[ Max, {payoff, Take[p*RotateRight[initial]\ + r*initial + q*RotateLeft[initial], {2, \(-2\)}]}]];\)\), "\n", \(\(CAAmericanOptionCompEvolveList[initial_List, payoffs_List, p_, q_, r_] := \[IndentingNewLine]FoldList[ CompCAAmericanOptionEvolveStep[#1, #2, p, q, r] &, initial, payoffs];\)\)}], "Input", CellLabel->"In[42]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}, InitializationCell->True], Cell["\<\ Of course both procedures can be collapsed into one line of code:\ \ \>", "Text", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \(\(CompCAAmericanOptionEvolveList = Compile[{{initial, \ _Real, \ 1}, {paddedPayoffs, _Real, 2}, {p, \ _Real, \ 0}, \ {q, \ _Real, 0}, \ {r, \ _Real, \ 0}}, FoldList[ MapThread[ Max, {p*RotateRight[#1] + r*#1 + q*RotateLeft[#1], #2}] &, initial, paddedPayoffs]];\)\)], "Input", CellLabel->"In[44]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}, InitializationCell->True], Cell["\<\ We can also avail ourselves of the inbuilt function ListCorrelate \ (or ListConvolve) which carries out discrete convolution of the kernel of \ probabilities {p,q,r} with the state of the system. This has the same effect \ as one step of the previous procedure but can be more optimised for large \ arrays. Again FoldList is applied for repeated application of the basic \ convolution as we move through time. \ \>", "Text", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[BoxData[{ \(\(CompCAAmerEvolveStep\ = Compile[{{initial, \ _Real, \ 1}, {payoff, _Real, 1}, {p, \ _Real, \ 0}, \ {q, \ _Real, 0}, \ {r, \ _Real, \ 0}}, \[IndentingNewLine]MapThread[ Max, {payoff, ListCorrelate[{p, r, q}, initial]}]];\)\), "\n", \(\(CAAmerCompEvolveList[initial_List, payoffs_List, p_, q_, r_] := \[IndentingNewLine]FoldList[ CompCAAmerEvolveStep[#1, #2, p, q, r] &, initial, payoffs];\)\)}], "Input", CellLabel->"In[45]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}, InitializationCell->True], Cell["\<\ As before both procedures can be collapsed into one line of code. \ But with the minor adjustment required by the fact that compiled functions \ only work on arrays of full rank and which therefore require the further \ application of the padLeftRight[.] function. Note also that the time \ dependent payoff structure is also padded out to match an array of full rank.\ \ \>", "Text", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \(\(CompCAAmerEvolveList = Compile[{{initial, \ _Real, \ 1}, {paddedPayoffs, _Real, 2}, {p, \ _Real, \ 0}, \ {q, \ _Real, 0}, \ {r, \ _Real, \ 0}}, FoldList[ MapThread[ Max, {padLeftRight[ListCorrelate[{p, r, q}, #1], Length[#1]], #2}] &, initial, paddedPayoffs], {{padLeftRight[_], _Real, 1}}];\)\)], "Input", CellLabel->"In[47]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}, InitializationCell->True], Cell["\<\ Finally we consider the use of the inbuilt function \ CellularAutomaton[.]. This undoubtedly constitutes the simplest use of the CA \ methodology and is also highly optimised. Again we are using the \ specification that requires a function in the first argument that convolves \ the probability vector {p,q,r} with the current state of the system. But \ unlike the situation encountered earlier for european options, we cannot \ avoid using the FoldList command as the lattice values have to be updated \ relative to the time dependent payoff structure.\ \>", "Text", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[BoxData[{ \(AmerCAEvolveStep[initial_List, payoff_List, p_, q_, r_] := MapThread[ Max, {padLeftRight[payoff, Length[initial]], \(CellularAutomaton[{# . {p, r, q} &, {}, 1}, initial, 1]\)[\([\(-1\)]\)]}]\), "\n", \(AmerCAEvolveList[initial_List, payoffs_List, p_, q_, r_] := FoldList[AmerCAEvolveStep[#1, #2, p, q, r] &, initial, payoffs]\)}], "Input", CellLabel->"In[48]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}, InitializationCell->True], Cell["\<\ Again both procedures can be collapsed into one line of code using \ pure functions.\ \>", "Text", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \(AmerCAEvolveList[initial_List, payoffs_List, p_, q_, r_] := FoldList[Function[{alist, blist}, MapThread[ Max, {padLeftRight[blist, Length[alist]], \(CellularAutomaton[{# . {p, r, q} &, {}, 1}, alist, 1]\)[\([\(-1\)]\)]}]\ ], initial, payoffs]\)], "Input", CellLabel->"In[50]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}, InitializationCell->True], Cell["\<\ In order to create the american option programs we merely have to \ take the existing trinomial function pattern and replace the functions \ ManyTimeStepAmerCallFunTri[.] and ManyTimeStepAmerPutFunTri[.] wherever we \ see them with the new CA functions CompCAAmericanOptionEvolveList, \ CompCAAmerEvolveList or AmerCAEvolveList in conjunction with the callpayoff \ and putpayoff matrix structure. The following functions not only determine \ the american option values but with very little further calculation, all of \ the sensitivity functions or greeks also.\ \>", "Text", CellMargins->{{60, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \(\(\(\ \)\(CompCAAmericanOptionEvolveCallFun[S_, K_, n_, r_, q_, \[Sigma]_, T_] := Module[{a = afunc[n, r, q, T], b = bfunc[n, \[Sigma], T], disc = Discount[n, r, T], P, Q, R, value, delta, gamma, theta, rho, vega, result, callpayoffs, node}, \[IndentingNewLine]u = up[a, b]; \n\ \ \ \ d = down[a, b]; \n\ \ \ \ s = stay[a, b]; \[IndentingNewLine]P = disc\ pfunc[b]; \[IndentingNewLine]Q = disc\ qfunc[b]; \[IndentingNewLine]R = disc\ rfunc[b]; \[IndentingNewLine]callpayoffs\ = Table[Join[ Table[Max[S\ s^node\ u^\((n - k - node)\) - K, 0], \ {node, \ 0, n - k, 1}], Table[Max[S\ d^node\ s^\((n - k - node)\) - K, 0], \ {node, 1, n - k, 1}]], {k, 0, n}]; \[IndentingNewLine]result\ = \ \ \(CompCAAmericanOptionEvolveList[callpayoffs[\([1]\)], paddedPayoffs[callpayoffs[\([1]\)], Rest[callpayoffs]], P, Q, R]\)[\([{\(-2\), \(-1\)}, {n, n + 1, n + 2}]\)]; \[IndentingNewLine]value = result[\([\(-1\), 2]\)]; \[IndentingNewLine]delta = \((result[\([1, 1]\)] - result[\([1, 3]\)])\)/\((\((u - d)\)*S)\); gamma = \((\((result[\([1, 1]\)] - result[\([1, 2]\)])\)/\((u - s)\) - \((result[\([1, 2]\)] - result[\([1, 3]\)])\)/\((s - d)\))\)*2/\((S^2\ \((u - d)\))\); \[IndentingNewLine]theta = \((result[\([1, 2]\)] - \((s - 1)\)*S*delta - result[\([\(-1\), 2]\)])\)* n/T; \[IndentingNewLine]rho = \(-T\)*\((value - S*delta)\); \[IndentingNewLine]vega = T*S*S*gamma*\[Sigma]; \[IndentingNewLine]{value, delta, gamma, theta, rho, vega}]\)\)\)], "Input", CellLabel->"In[51]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}, InitializationCell->True], Cell[BoxData[ \(CompCAAmericanOptionEvolvePutFun[S_, K_, n_, r_, q_, \[Sigma]_, T_] := Module[{a = afunc[n, r, q, T], b = bfunc[n, \[Sigma], T], disc = Discount[n, r, T], P, Q, R, value, delta, gamma, theta, rho, vega, result, putpayoffs, node}, \[IndentingNewLine]u = up[a, b]; \n\ \ \ \ d = down[a, b]; \n\ \ \ \ s = stay[a, b]; \[IndentingNewLine]P = disc\ pfunc[b]; \[IndentingNewLine]Q = disc\ qfunc[b]; \[IndentingNewLine]R = disc\ rfunc[b]; \[IndentingNewLine]putpayoffs\ = Table[Join[ Table[Max[K - S\ s^node\ u^\((n - k - node)\), 0], \ {node, \ 0, n - k, 1}], Table[Max[K - S\ d^node\ s^\((n - k - node)\), 0], \ {node, 1, n - k, 1}]], {k, 0, n}]; \[IndentingNewLine]result\ = \ \ \(CompCAAmericanOptionEvolveList[putpayoffs[\([1]\)], paddedPayoffs[putpayoffs[\([1]\)], Rest[putpayoffs]], P, Q, R]\)[\([{\(-2\), \(-1\)}, {n, n + 1, n + 2}]\)]; \[IndentingNewLine]value = result[\([\(-1\), 2]\)]; \[IndentingNewLine]delta = \((result[\([1, 1]\)] - result[\([1, 3]\)])\)/\((\((u - d)\)*S)\); gamma = \((\((result[\([1, 1]\)] - result[\([1, 2]\)])\)/\((u - s)\) - \((result[\([1, 2]\)] - result[\([1, 3]\)])\)/\((s - d)\))\)*2/\((S^2\ \((u - d)\))\); \[IndentingNewLine]theta = \((result[\([1, 2]\)] - \((s - 1)\)*S*delta - result[\([\(-1\), 2]\)])\)* n/T; \[IndentingNewLine]rho = \(-T\)*\((value - S*delta)\); \[IndentingNewLine]vega = T*S*S*gamma*\[Sigma]; \[IndentingNewLine]{value, delta, gamma, theta, rho, vega}]\)], "Input", CellLabel->"In[52]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}, InitializationCell->True], Cell[BoxData[ \(\(\(\ \)\(CompCAAmerEvolveCallFun[S_, K_, n_, r_, q_, \[Sigma]_, T_] := Module[{a = afunc[n, r, q, T], b = bfunc[n, \[Sigma], T], disc = Discount[n, r, T], P, Q, R, value, delta, gamma, theta, rho, vega, result, callpayoffs, node}, \[IndentingNewLine]u = up[a, b]; \n\ \ \ \ d = down[a, b]; \n\ \ \ \ s = stay[a, b]; \[IndentingNewLine]P = disc\ pfunc[b]; \[IndentingNewLine]Q = disc\ qfunc[b]; \[IndentingNewLine]R = disc\ rfunc[b]; \[IndentingNewLine]callpayoffs\ = Table[Join[ Table[Max[S\ s^node\ u^\((n - k - node)\) - K, 0], \ {node, \ 0, n - k, 1}], Table[Max[S\ d^node\ s^\((n - k - node)\) - K, 0], \ {node, 1, n - k, 1}]], {k, 0, n}]; \[IndentingNewLine]result\ = \ \(CompCAAmerEvolveList[ callpayoffs[\([1]\)], paddedPayoffs[callpayoffs[\([1]\)], Rest[callpayoffs]], P, Q, R]\)[\([{\(-2\), \(-1\)}, {n, n + 1, n + 2}]\)]; \[IndentingNewLine]value = result[\([\(-1\), 2]\)]; \[IndentingNewLine]delta = \((result[\([1, 1]\)] - result[\([1, 3]\)])\)/\((\((u - d)\)*S)\); gamma = \((\((result[\([1, 1]\)] - result[\([1, 2]\)])\)/\((u - s)\) - \((result[\([1, 2]\)] - result[\([1, 3]\)])\)/\((s - d)\))\)*2/\((S^2\ \((u - d)\))\); \[IndentingNewLine]theta = \((result[\([1, 2]\)] - \((s - 1)\)*S*delta - result[\([\(-1\), 2]\)])\)* n/T; \[IndentingNewLine]rho = \(-T\)*\((value - S*delta)\); \[IndentingNewLine]vega = T*S*S*gamma*\[Sigma]; \[IndentingNewLine]{value, delta, gamma, theta, rho, vega}]\)\)\)], "Input", CellLabel->"In[53]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}, InitializationCell->True], Cell[BoxData[ \(CompCAAmerEvolvePutFun[S_, K_, n_, r_, q_, \[Sigma]_, T_] := Module[{a = afunc[n, r, q, T], b = bfunc[n, \[Sigma], T], disc = Discount[n, r, T], P, Q, R, value, delta, gamma, theta, rho, vega, result, putpayoffs, node}, \[IndentingNewLine]u = up[a, b]; \n\ \ \ \ d = down[a, b]; \n\ \ \ \ s = stay[a, b]; \[IndentingNewLine]P = disc\ pfunc[b]; \[IndentingNewLine]Q = disc\ qfunc[b]; \[IndentingNewLine]R = disc\ rfunc[b]; \[IndentingNewLine]putpayoffs\ = Table[Join[ Table[Max[K - S\ s^node\ u^\((n - k - node)\), 0], \ {node, \ 0, n - k, 1}], Table[Max[K - S\ d^node\ s^\((n - k - node)\), 0], \ {node, 1, n - k, 1}]], {k, 0, n}]; \[IndentingNewLine]result\ = \ \(CompCAAmerEvolveList[ putpayoffs[\([1]\)], paddedPayoffs[putpayoffs[\([1]\)], Rest[putpayoffs]], P, Q, R]\)[\([{\(-2\), \(-1\)}, {n, n + 1, n + 2}]\)]; \[IndentingNewLine]value = result[\([\(-1\), 2]\)]; \[IndentingNewLine]delta = \((result[\([1, 1]\)] - result[\([1, 3]\)])\)/\((\((u - d)\)*S)\); gamma = \((\((result[\([1, 1]\)] - result[\([1, 2]\)])\)/\((u - s)\) - \((result[\([1, 2]\)] - result[\([1, 3]\)])\)/\((s - d)\))\)*2/\((S^2\ \((u - d)\))\); \[IndentingNewLine]theta = \((result[\([1, 2]\)] - \((s - 1)\)*S*delta - result[\([\(-1\), 2]\)])\)* n/T; \[IndentingNewLine]rho = \(-T\)*\((value - S*delta)\); \[IndentingNewLine]vega = T*S*S*gamma*\[Sigma]; \[IndentingNewLine]{value, delta, gamma, theta, rho, vega}]\)], "Input", CellLabel->"In[54]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}, InitializationCell->True], Cell[BoxData[ \(\(\(\ \)\(AmerCAEvolveCallFun[S_, K_, n_, r_, q_, \[Sigma]_, T_] := Module[{a = afunc[n, r, q, T], b = bfunc[n, \[Sigma], T], disc = Discount[n, r, T], P, Q, R, value, delta, gamma, theta, rho, vega, result, callpayoffs, node}, \[IndentingNewLine]u = up[a, b]; \n\ \ \ \ d = down[a, b]; \n\ \ \ \ s = stay[a, b]; \[IndentingNewLine]P = disc\ pfunc[b]; \[IndentingNewLine]Q = disc\ qfunc[b]; \[IndentingNewLine]R = disc\ rfunc[b]; \[IndentingNewLine]callpayoffs\ = Table[Join[ Table[Max[S\ s^node\ u^\((n - k - node)\) - K, 0], \ {node, \ 0, n - k, 1}], Table[Max[S\ d^node\ s^\((n - k - node)\) - K, 0], \ {node, 1, n - k, 1}]], {k, 0, n}]; \[IndentingNewLine]result\ = \ \(AmerCAEvolveList[ callpayoffs[\([1]\)], Rest[callpayoffs], P, Q, R]\)[\([{\(-2\), \(-1\)}, {n, n + 1, n + 2}]\)]; \[IndentingNewLine]value = result[\([\(-1\), 2]\)]; \[IndentingNewLine]delta = \((result[\([1, 1]\)] - result[\([1, 3]\)])\)/\((\((u - d)\)*S)\); gamma = \((\((result[\([1, 1]\)] - result[\([1, 2]\)])\)/\((u - s)\) - \((result[\([1, 2]\)] - result[\([1, 3]\)])\)/\((s - d)\))\)*2/\((S^2\ \((u - d)\))\); \[IndentingNewLine]theta = \((result[\([1, 2]\)] - \((s - 1)\)*S*delta - result[\([\(-1\), 2]\)])\)* n/T; \[IndentingNewLine]rho = \(-T\)*\((value - S*delta)\); \[IndentingNewLine]vega = T*S*S*gamma*\[Sigma]; \[IndentingNewLine]{value, delta, gamma, theta, rho, vega}]\)\)\)], "Input", CellLabel->"In[55]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}, InitializationCell->True], Cell[BoxData[ \(AmerCAEvolvePutFun[S_, K_, n_, r_, q_, \[Sigma]_, T_] := Module[{a = afunc[n, r, q, T], b = bfunc[n, \[Sigma], T], disc = Discount[n, r, T], P, Q, R, value, delta, gamma, theta, rho, vega, result, putpayoffs, node}, \[IndentingNewLine]u = up[a, b]; \n\ \ \ \ d = down[a, b]; \n\ \ \ \ s = stay[a, b]; \[IndentingNewLine]P = disc\ pfunc[b]; \[IndentingNewLine]Q = disc\ qfunc[b]; \[IndentingNewLine]R = disc\ rfunc[b]; \[IndentingNewLine]putpayoffs\ = Table[Join[ Table[Max[K - S\ s^node\ u^\((n - k - node)\), 0], \ {node, \ 0, n - k, 1}], Table[Max[K - S\ d^node\ s^\((n - k - node)\), 0], \ {node, 1, n - k, 1}]], {k, 0, n}]; \[IndentingNewLine]result\ = \ \(AmerCAEvolveList[ putpayoffs[\([1]\)], Rest[putpayoffs], P, Q, R]\)[\([{\(-2\), \(-1\)}, {n, n + 1, n + 2}]\)]; \[IndentingNewLine]value = result[\([\(-1\), 2]\)]; \[IndentingNewLine]delta = \((result[\([1, 1]\)] - result[\([1, 3]\)])\)/\((\((u - d)\)*S)\); gamma = \((\((result[\([1, 1]\)] - result[\([1, 2]\)])\)/\((u - s)\) - \((result[\([1, 2]\)] - result[\([1, 3]\)])\)/\((s - d)\))\)*2/\((S^2\ \((u - d)\))\); \[IndentingNewLine]theta = \((result[\([1, 2]\)] - \((s - 1)\)*S*delta - result[\([\(-1\), 2]\)])\)* n/T; \[IndentingNewLine]rho = \(-T\)*\((value - S*delta)\); \[IndentingNewLine]vega = T*S*S*gamma*\[Sigma]; \[IndentingNewLine]{value, delta, gamma, theta, rho, vega}]\)], "Input", CellLabel->"In[56]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}, InitializationCell->True], Cell[BoxData[ \(On[General::"\", General::"\"]\)], "Input", CellLabel->"In[57]:=", CellMargins->{{60, Inherited}, {Inherited, Inherited}}, InitializationCell->True], Cell["\<\ In the following calculations we determine the prices of american \ options and their sensitivity functions, known as the greeks, for the \ following market values: The stock price is S = $100, the strike price is K = \ $102, the risk free interst rate r = 10%, the dividend yield q = 3%, \ volatility \[Sigma] = 30% and the time remaining tothe expiry date is T = 1 \ year. 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-0.10000000000000001}, {0, 0}}, BoxBaselineShift -> -0.20000000000000001], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-0.074999999999999997, \ -0.085000000000000006}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica", "gridMathematica"->FormBox[ RowBox[ {"grid", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-0.17499999999999999, 0}, {0, 0}}]}], TextForm], "webMathematica"->FormBox[ RowBox[ {"web", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-0.17499999999999999, 0}, {0, 0}}]}], TextForm], Inherited}, LineSpacing->{1, 3}, CounterIncrements->"Text", FontFamily->"Verdana"], Cell[StyleData["Text", "Presentation"], CellMargins->{{24, 50}, {10, 10}}, LineSpacing->{1, 5}, FontSize->14], Cell[StyleData["Text", "SlideShow"], CellMargins->{{100, 50}, {10, 10}}, FontSize->14], Cell[StyleData["Text", "Printout"], CellMargins->{{2, 2}, {6, 6}}, TextJustification->0.5, Hyphenation->True, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["SmallText"], CellMargins->{{60, 10}, {6, 6}}, DefaultNewInlineCellStyle->"None", LineSpacing->{1, 3}, LanguageCategory->"NaturalLanguage", CounterIncrements->"SmallText", FontFamily->"Verdana", FontSize->9], Cell[StyleData["SmallText", "Presentation"], CellMargins->{{24, 50}, {8, 8}}, LineSpacing->{1, 5}, FontSize->10], Cell[StyleData["SmallText", "SlideShow"], CellMargins->{{100, 50}, {8, 8}}, LineSpacing->{1, 5}, FontSize->10], Cell[StyleData["SmallText", "Printout"], CellMargins->{{2, 2}, {5, 5}}, TextJustification->0.5, Hyphenation->True, FontSize->7] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Input/Output", "Section"], Cell["\<\ The cells in this section define styles used for input and output \ to the kernel. Be careful when modifying, renaming, or removing these \ styles, because the front end associates special meanings with these style \ names. Some attributes for these styles are actually set in FormatType Styles \ (in the last section of this stylesheet). \ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Input"], CellMargins->{{66, 10}, {5, 7}}, Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, DefaultFormatType->DefaultInputFormatType, "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, LanguageCategory->"Mathematica", FormatType->InputForm, ShowStringCharacters->True, NumberMarks->True, LinebreakAdjustments->{0.85, 2, 10, 0, 1}, CounterIncrements->"Input", FontWeight->"Bold"], Cell[StyleData["Input", "Presentation"], CellMargins->{{72, 50}, {8, 10}}, LineSpacing->{1, 0}, FontSize->16], Cell[StyleData["Input", "SlideShow"], CellMargins->{{100, 50}, {8, 10}}, LineSpacing->{1, 0}, FontSize->16], Cell[StyleData["Input", "Printout"], CellMargins->{{39, 0}, {4, 6}}, LinebreakAdjustments->{0.85, 2, 10, 1, 1}, FontSize->9] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["InputOnly"], CellMargins->{{66, 10}, {7, 7}}, Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, DefaultFormatType->DefaultInputFormatType, "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, LanguageCategory->"Mathematica", FormatType->InputForm, ShowStringCharacters->True, NumberMarks->True, LinebreakAdjustments->{0.85, 2, 10, 0, 1}, CounterIncrements->"Input", StyleMenuListing->None, FontWeight->"Bold"], Cell[StyleData["InputOnly", "Presentation"], CellMargins->{{72, Inherited}, {8, 10}}, LineSpacing->{1, 0}, FontSize->16], Cell[StyleData["InputOnly", "SlideShow"], CellMargins->{{100, Inherited}, {8, 10}}, LineSpacing->{1, 0}, FontSize->16], Cell[StyleData["InputOnly", "Printout"], CellMargins->{{39, 0}, {4, 6}}, LinebreakAdjustments->{0.85, 2, 10, 1, 1}, FontSize->9] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Output"], CellMargins->{{66, 10}, {7, 5}}, CellEditDuplicate->True, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, DefaultFormatType->DefaultOutputFormatType, "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, LanguageCategory->None, FormatType->InputForm, CounterIncrements->"Output"], Cell[StyleData["Output", "Presentation"], CellMargins->{{72, 50}, {10, 8}}, LineSpacing->{1, 0}, FontSize->16], Cell[StyleData["Output", "SlideShow"], CellMargins->{{100, 50}, {10, 8}}, LineSpacing->{1, 0}, FontSize->16], Cell[StyleData["Output", "Printout"], CellMargins->{{39, 0}, {6, 4}}, FontSize->9] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Message"], CellMargins->{{66, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, "TwoByteSyntaxCharacterAutoReplacement"->True, AutoStyleOptions->{"UnmatchedBracketStyle"->None}, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, LanguageCategory->None, FormatType->InputForm, CounterIncrements->"Message", StyleMenuListing->None, FontFamily->"Helvetica", FontSize->10, FontColor->RGBColor[0.6, 0.100008, 0.100008]], Cell[StyleData["Message", "Presentation"], CellMargins->{{72, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}, FontSize->14], Cell[StyleData["Message", "SlideShow"], CellMargins->{{100, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}, FontSize->14], Cell[StyleData["Message", "Printout"], CellMargins->{{39, Inherited}, {Inherited, Inherited}}, FontSize->7, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Print"], CellMargins->{{66, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, LanguageCategory->None, FormatType->InputForm, CounterIncrements->"Print", StyleMenuListing->None], Cell[StyleData["Print", "Presentation"], CellMargins->{{72, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}, FontSize->14], Cell[StyleData["Print", "SlideShow"], CellMargins->{{100, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}, FontSize->14], Cell[StyleData["Print", "Printout"], CellMargins->{{39, Inherited}, {Inherited, Inherited}}, FontSize->8] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Graphics"], CellMargins->{{4, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"GraphicsGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, LanguageCategory->None, FormatType->InputForm, CounterIncrements->"Graphics", ImageMargins->{{43, Inherited}, {Inherited, 0}}, StyleMenuListing->None, FontFamily->"Courier", FontSize->10], Cell[StyleData["Graphics", "Presentation"], ImageMargins->{{62, Inherited}, {Inherited, 0}}], Cell[StyleData["Graphics", "SlideShow"], ImageMargins->{{100, Inherited}, {Inherited, 0}}], Cell[StyleData["Graphics", "Printout"], ImageMargins->{{30, Inherited}, {Inherited, 0}}, Magnification->0.8] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["CellLabel"], LanguageCategory->None, StyleMenuListing->None, FontFamily->"Helvetica", FontSize->9, FontColor->RGBColor[0.392157, 0.396078, 0.717647]], Cell[StyleData["CellLabel", "Presentation"], FontSize->12], Cell[StyleData["CellLabel", "SlideShow"], FontSize->12], Cell[StyleData["CellLabel", "Printout"], FontFamily->"Courier", FontSize->8, FontSlant->"Italic", FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["FrameLabel"], LanguageCategory->None, StyleMenuListing->None, FontFamily->"Helvetica", FontSize->9], Cell[StyleData["FrameLabel", "Presentation"], FontSize->12], Cell[StyleData["FrameLabel", "SlideShow"], FontSize->12], Cell[StyleData["FrameLabel", "Printout"], FontFamily->"Courier", FontSize->8, FontSlant->"Italic", FontColor->GrayLevel[0]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Presentation Specific", "Section"], Cell[CellGroupData[{ Cell[StyleData["Author"], CellMargins->{{100, 27}, {2, 20}}, FontFamily->"Times", FontSize->24, FontSlant->"Italic"], Cell[StyleData["Author", "Presentation"], CellMargins->{{200, 27}, {2, 50}}, FontSize->28], Cell[StyleData["Author", "SlideShow"], CellMargins->{{200, 27}, {2, 50}}, FontSize->28], Cell[StyleData["Author", "Printout"], CellMargins->{{100, 27}, {2, 20}}, FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Affiliation"], CellMargins->{{100, 27}, {30, 12}}, FontFamily->"Times", FontSize->24, FontSlant->"Italic"], Cell[StyleData["Affiliation", "Presentation"], CellMargins->{{200, 27}, {2, 10}}, FontSize->28], Cell[StyleData["Affiliation", "SlideShow"], CellMargins->{{200, 27}, {2, 10}}, FontSize->28], Cell[StyleData["Affiliation", "Printout"], CellMargins->{{100, 27}, {2, 12}}, FontSize->14] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Header Graphic", "Section"], Cell[CellGroupData[{ Cell[StyleData["ConferenceGraphicCell"], ShowCellBracket->False, CellMargins->{{0, 0}, {0, 0}}, Evaluatable->False, PageBreakBelow->False, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, Background->GrayLevel[1], Magnification->1], Cell[StyleData["ConferenceGraphicCell", "Presentation"]], Cell[StyleData["ConferenceGraphicCell", "SlideShow"]], Cell[StyleData["ConferenceGraphicCell", "Printout"], FontSize->8] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Inline Formatting", "Section"], Cell["\<\ These styles are for modifying individual words or letters in a \ cell exclusive of the cell tag.\ \>", "Text"], Cell[StyleData["RM"], StyleMenuListing->None, FontWeight->"Plain", FontSlant->"Plain"], Cell[StyleData["BF"], StyleMenuListing->None, FontWeight->"Bold"], Cell[StyleData["IT"], StyleMenuListing->None, FontSlant->"Italic"], Cell[StyleData["TR"], StyleMenuListing->None, FontFamily->"Times", FontWeight->"Plain", FontSlant->"Plain"], Cell[StyleData["TI"], StyleMenuListing->None, FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], Cell[StyleData["TB"], StyleMenuListing->None, FontFamily->"Times", FontWeight->"Bold", FontSlant->"Plain"], Cell[StyleData["TBI"], StyleMenuListing->None, FontFamily->"Times", FontWeight->"Bold", FontSlant->"Italic"], Cell[StyleData["MR"], "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, StyleMenuListing->None, FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], Cell[StyleData["MO"], "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, StyleMenuListing->None, FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Italic"], Cell[StyleData["MB"], "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, StyleMenuListing->None, FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], Cell[StyleData["MBO"], "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, StyleMenuListing->None, FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Italic"], Cell[StyleData["SR"], StyleMenuListing->None, FontFamily->"Helvetica", FontWeight->"Plain", FontSlant->"Plain"], Cell[StyleData["SO"], StyleMenuListing->None, FontFamily->"Helvetica", FontWeight->"Plain", FontSlant->"Italic"], Cell[StyleData["SB"], StyleMenuListing->None, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Plain"], Cell[StyleData["SBO"], StyleMenuListing->None, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Italic"], Cell[CellGroupData[{ Cell[StyleData["SO10"], StyleMenuListing->None, FontFamily->"Helvetica", FontSize->10, FontWeight->"Plain", FontSlant->"Italic"], Cell[StyleData["SO10", "Printout"], StyleMenuListing->None, FontFamily->"Helvetica", FontSize->7, FontWeight->"Plain", FontSlant->"Italic"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Formulas and Programming", "Section"], Cell[CellGroupData[{ Cell[StyleData["InlineFormula"], CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Formula", ScriptLevel->1, SingleLetterItalics->True], Cell[StyleData["InlineFormula", "Presentation"], LineSpacing->{1, 5}, FontSize->16], Cell[StyleData["InlineFormula", "SlideShow"], LineSpacing->{1, 5}, FontSize->16], Cell[StyleData["InlineFormula", "Printout"], CellMargins->{{2, 0}, {6, 6}}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["DisplayFormula"], CellMargins->{{60, Inherited}, {Inherited, Inherited}}, CellHorizontalScrolling->True, DefaultFormatType->DefaultInputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Formula", ScriptLevel->0, SingleLetterItalics->True, UnderoverscriptBoxOptions->{LimitsPositioning->True}], Cell[StyleData["DisplayFormula", "Presentation"], LineSpacing->{1, 5}, FontSize->16], Cell[StyleData["DisplayFormula", "SlideShow"], CellMargins->{{100, 50}, {Inherited, Inherited}}, LineSpacing->{1, 5}, FontSize->16], Cell[StyleData["DisplayFormula", "Printout"], FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Program"], CellFrame->{{0, 0}, {0.5, 0.5}}, CellMargins->{{60, 4}, {0, 8}}, CellHorizontalScrolling->True, Hyphenation->False, LanguageCategory->"Formula", ScriptLevel->1, FontFamily->"Courier"], Cell[StyleData["Program", "Presentation"], CellMargins->{{24, 50}, {10, 10}}, LineSpacing->{1, 5}, FontSize->16], Cell[StyleData["Program", "SlideShow"], CellMargins->{{100, 50}, {10, 10}}, LineSpacing->{1, 5}, FontSize->16], Cell[StyleData["Program", "Printout"], CellMargins->{{2, 0}, {6, 6}}, FontSize->9] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Outline Styles", "Section"], Cell[CellGroupData[{ Cell[StyleData["Outline1"], CellMargins->{{60, 10}, {7, 7}}, CellGroupingRules->{"SectionGrouping", 50}, ParagraphIndent->-38, CounterIncrements->"Outline1", CounterAssignments->{{"Outline2", 0}, {"Outline3", 0}, {"Outline4", 0}}, FontSize->18, FontWeight->"Bold", CounterBoxOptions->{CounterFunction:>CapitalRomanNumeral}], Cell[StyleData["Outline1", "SlideShow"]], Cell[StyleData["Outline1", "Printout"], CounterBoxOptions->{CounterFunction:>CapitalRomanNumeral}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Outline2"], CellMargins->{{90, 10}, {7, 7}}, CellGroupingRules->{"SectionGrouping", 60}, ParagraphIndent->-27, CounterIncrements->"Outline2", CounterAssignments->{{"Outline3", 0}, {"Outline4", 0}}, FontSize->15, FontWeight->"Bold", CounterBoxOptions->{CounterFunction:>(Part[ CharacterRange[ "A", "Z"], #]&)}], Cell[StyleData["Outline2", "SlideShow"]], Cell[StyleData["Outline2", "Printout"], CounterBoxOptions->{CounterFunction:>(Part[ CharacterRange[ "A", "Z"], #]&)}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Outline3"], CellMargins->{{120, 10}, {7, 7}}, CellGroupingRules->{"SectionGrouping", 70}, ParagraphIndent->-21, CounterIncrements->"Outline3", CounterAssignments->{{"Outline4", 0}}, FontSize->12, CounterBoxOptions->{CounterFunction:>Identity}], Cell[StyleData["Outline3", "SlideShow"]], Cell[StyleData["Outline3", "Printout"], CounterBoxOptions->{CounterFunction:>Identity}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Outline4"], CellMargins->{{150, 10}, {7, 7}}, CellGroupingRules->{"SectionGrouping", 80}, ParagraphIndent->-18, CounterIncrements->"Outline4", FontSize->10, CounterBoxOptions->{CounterFunction:>(Part[ CharacterRange[ "a", "z"], #]&)}], Cell[StyleData["Outline4", "SlideShow"]], Cell[StyleData["Outline4", "Printout"]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Hyperlink Styles", "Section"], Cell["\<\ The cells below define styles useful for making hypertext \ ButtonBoxes. The \"Hyperlink\" style is for links within the same Notebook, \ or between Notebooks.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Hyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0.269993, 0.308507, 0.6], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonFrame->"None", ButtonNote->ButtonData}], Cell[StyleData["Hyperlink", "Presentation"], FontSize->16], Cell[StyleData["Hyperlink", "SlideShow"]], Cell[StyleData["Hyperlink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell["\<\ The following styles are for linking automatically to the on-line \ help system.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["MainBookLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0.269993, 0.308507, 0.6], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "MainBook", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["MainBookLink", "Presentation"], FontSize->16], Cell[StyleData["MainBookLink", "SlideShow"]], Cell[StyleData["MainBookLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["AddOnsLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0.269993, 0.308507, 0.6], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "AddOns", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["AddOnsLink", "Presentation"], FontSize->16], Cell[StyleData["AddOnsLink", "SlideShow"]], Cell[StyleData["AddOnsLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["RefGuideLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0.269993, 0.308507, 0.6], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "RefGuide", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["RefGuideLink", "Presentation"], FontSize->16], Cell[StyleData["RefGuideLink", "SlideShow"]], Cell[StyleData["RefGuideLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["RefGuideLinkText"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0.269993, 0.308507, 0.6], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "RefGuide", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["RefGuideLinkText", "Presentation"], FontSize->16], Cell[StyleData["RefGuideLinkText", "SlideShow"]], Cell[StyleData["RefGuideLinkText", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["GettingStartedLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0.269993, 0.308507, 0.6], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "GettingStarted", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["GettingStartedLink", "Presentation"], FontSize->16], Cell[StyleData["GettingStartedLink", "SlideShow"]], Cell[StyleData["GettingStartedLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["DemosLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0.269993, 0.308507, 0.6], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "Demos", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["DemosLink", "SlideShow"]], Cell[StyleData["DemosLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["TourLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0.269993, 0.308507, 0.6], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "Tour", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["TourLink", "SlideShow"]], Cell[StyleData["TourLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["OtherInformationLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0.269993, 0.308507, 0.6], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "OtherInformation", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["OtherInformationLink", "Presentation"], FontSize->16], Cell[StyleData["OtherInformationLink", "SlideShow"]], Cell[StyleData["OtherInformationLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["MasterIndexLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0.269993, 0.308507, 0.6], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "MasterIndex", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["MasterIndexLink", "SlideShow"]], Cell[StyleData["MasterIndexLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Headers and Footers", "Section"], Cell[StyleData["Header"], CellMargins->{{0, 0}, {4, 1}}, DefaultNewInlineCellStyle->"None", LanguageCategory->"NaturalLanguage", StyleMenuListing->None, FontSize->10, FontSlant->"Italic"], Cell[StyleData["Footer"], CellMargins->{{0, 0}, {0, 4}}, DefaultNewInlineCellStyle->"None", LanguageCategory->"NaturalLanguage", StyleMenuListing->None, FontSize->9, FontSlant->"Italic"], Cell[StyleData["PageNumber"], CellMargins->{{0, 0}, {4, 1}}, StyleMenuListing->None, FontFamily->"Times", FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell["Palette Styles", "Section"], Cell["\<\ The cells below define styles that define standard \ ButtonFunctions, for use in palette buttons.\ \>", "Text"], Cell[StyleData["Paste"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, Placeholder]}]&)}], Cell[StyleData["Evaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, 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ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[ ], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], All]}]&)}] }, Closed]], Cell[CellGroupData[{ Cell["Placeholder Styles", "Section"], Cell["\<\ The cells below define styles useful for making placeholder \ objects in palette templates.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Placeholder"], Placeholder->True, StyleMenuListing->None, FontSlant->"Italic", FontColor->RGBColor[0.890623, 0.864698, 0.384756], TagBoxOptions->{Editable->False, Selectable->False, StripWrapperBoxes->False}], Cell[StyleData["Placeholder", "Presentation"]], Cell[StyleData["Placeholder", "SlideShow"]], Cell[StyleData["Placeholder", "Printout"]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["PrimaryPlaceholder"], StyleMenuListing->None, DrawHighlighted->True, FontSlant->"Italic", Background->RGBColor[0.912505, 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If a cell's FormatType matches the name of one of the styles \ defined below, then that style is applied between the cell's style and its \ own options. 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For example, \ \"UnmatchedBracket\" style defines how unmatched bracket, curly bracket, and \ parenthesis characters are displayed (typically by coloring them to make them \ stand out).\ \>", "Text"], Cell[StyleData["UnmatchedBracket"], StyleMenuListing->None, FontColor->RGBColor[0.760006, 0.330007, 0.8]], Cell[StyleData["Completions"], StyleMenuListing->None, FontFamily->"Courier"] }, Closed]], Cell[CellGroupData[{ Cell["Styles from HelpBrowser", "Section"], Cell[CellGroupData[{ Cell[StyleData["MathCaption"], CellFrame->{{0, 0}, {0, 0.5}}, CellMargins->{{66, 12}, {2, 24}}, PageBreakBelow->False, CellFrameMargins->{{8, 8}, {8, 2}}, CellFrameColor->GrayLevel[0.700008], CellFrameLabelMargins->4, LineSpacing->{1, 1}, ParagraphSpacing->{0, 8}, StyleMenuListing->None, FontColor->GrayLevel[0.2]], Cell[StyleData["MathCaption", "Presentation"], FontSize->18], Cell[StyleData["MathCaption", "SlideShow"], CellMargins->{{100, 50}, {Inherited, Inherited}}], Cell[StyleData["MathCaption", 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0}, {6, 6}}, FontSize->9] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Usage"], ShowCellBracket->True, CellMargins->{{66, 4}, {8, 8}}, Evaluatable->True, CellGroupingRules->"InputGrouping", PageBreakWithin->False, GroupPageBreakWithin->False, CellLabelAutoDelete->False, CellLabelMargins->{{14, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultInputFormatType, ShowSpecialCharacters->Automatic, "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Mathematica", FormatType->StandardForm, ShowStringCharacters->True, NumberMarks->True, StyleMenuListing->None, FontWeight->"Bold"], Cell[StyleData["Usage", "Presentation"], FontSize->18], Cell[StyleData["Usage", "SlideShow"], CellMargins->{{100, 50}, {Inherited, Inherited}}], Cell[StyleData["Usage", "Printout"], ShowCellBracket->False, CellMargins->{{39, 0}, {6, 6}}, FontSize->9] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Notes"], ShowCellBracket->True, CellMargins->{{66, 4}, {8, 8}}, Evaluatable->True, CellGroupingRules->"InputGrouping", PageBreakWithin->False, GroupPageBreakWithin->False, CellLabelAutoDelete->False, CellLabelMargins->{{14, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultInputFormatType, ShowSpecialCharacters->Automatic, "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Mathematica", FormatType->StandardForm, ShowStringCharacters->True, NumberMarks->True, StyleMenuListing->None, FontWeight->"Bold"], Cell[StyleData["Notes", "Presentation"], FontSize->18], Cell[StyleData["Notes", "SlideShow"], CellMargins->{{100, 50}, {Inherited, Inherited}}], Cell[StyleData["Notes", "Printout"], ShowCellBracket->False, CellMargins->{{39, 0}, {6, 6}}, FontSize->9] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["InlineOutput"], ShowCellBracket->True, CellMargins->{{66, 4}, {8, 8}}, Evaluatable->True, CellGroupingRules->"InputGrouping", PageBreakWithin->False, GroupPageBreakWithin->False, CellLabelAutoDelete->False, CellLabelMargins->{{14, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultInputFormatType, ShowSpecialCharacters->Automatic, "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Mathematica", FormatType->StandardForm, ShowStringCharacters->True, NumberMarks->True, StyleMenuListing->None, FontWeight->"Bold"], Cell[StyleData["InlineOutput", "Presentation"], FontSize->18], Cell[StyleData["InlineOutput", "SlideShow"], CellMargins->{{100, 50}, {Inherited, Inherited}}], Cell[StyleData["InlineOutput", "Printout"], ShowCellBracket->False, CellMargins->{{39, 0}, {6, 6}}, FontSize->9] }, Closed]], Cell[CellGroupData[{ Cell["Emphasis Boxes and Pictures", "Subsection"], Cell[CellGroupData[{ Cell[StyleData["Box"], CellFrame->0.5, CellMargins->{{27, 12}, {0, 8}}, CellHorizontalScrolling->True, CellFrameColor->RGBColor[0.74902, 0.694118, 0.552941], StyleMenuListing->None, Background->RGBColor[0.964706, 0.929412, 0.839216], FrameBoxOptions->{BoxFrame->0.5, BoxMargins->True}, GridBoxOptions->{ColumnSpacings->1}], Cell[StyleData["Box", "Presentation"], FontSize->18], Cell[StyleData["Box", "SlideShow"], CellMargins->{{100, 50}, {Inherited, Inherited}}], Cell[StyleData["Box", "Printout"], CellMargins->{{2, 0}, {0, 8}}, FontSize->10, Background->GrayLevel[0.900008]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["DoubleBox"], CellFrame->0.5, CellMargins->{{27, 12}, {0, 8}}, CellHorizontalScrolling->True, CellFrameColor->RGBColor[0.74902, 0.694118, 0.552941], StyleMenuListing->None, Background->RGBColor[0.964706, 0.929412, 0.839216], FrameBoxOptions->{BoxFrame->0.5, BoxMargins->True}, GridBoxOptions->{ColumnSpacings->2, RowAlignments->Top}], Cell[StyleData["DoubleBox", "Presentation"], FontSize->18], Cell[StyleData["DoubleBox", "SlideShow"], CellMargins->{{100, 50}, {Inherited, 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CellMargins->{{100, 50}, {Inherited, Inherited}}], Cell[StyleData["2ColumnEvenBox", "Printout"], CellMargins->{{2, 0}, {0, 8}}, FontSize->10, Background->GrayLevel[0.900008]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["2ColumnSmallBox"], CellFrame->0.5, CellMargins->{{27, 12}, {0, 8}}, CellHorizontalScrolling->True, CellFrameColor->RGBColor[0.74902, 0.694118, 0.552941], LineIndent->0, StyleMenuListing->None, Background->RGBColor[0.964706, 0.929412, 0.839216], FrameBoxOptions->{BoxFrame->0.5, BoxMargins->True}, GridBoxOptions->{ColumnSpacings->1.5, ColumnWidths->0.35, ColumnAlignments->{Right, Left}}], Cell[StyleData["2ColumnSmallBox", "Presentation"], FontSize->18], Cell[StyleData["2ColumnSmallBox", "SlideShow"], CellMargins->{{100, 50}, {Inherited, Inherited}}], Cell[StyleData["2ColumnSmallBox", "Printout"], CellMargins->{{2, 0}, {0, 8}}, FontSize->10, Background->GrayLevel[0.900008]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["3ColumnBox"], CellFrame->0.5, 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ColumnAlignments->{Right, Center, Left}}], Cell[StyleData["3ColumnSmallBox", "Presentation"], FontSize->18], Cell[StyleData["3ColumnSmallBox", "SlideShow"], CellMargins->{{100, 50}, {Inherited, Inherited}}], Cell[StyleData["3ColumnSmallBox", "Printout"], CellMargins->{{2, 0}, {0, 8}}, FontSize->10, Background->GrayLevel[0.900008]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["4ColumnBox"], CellFrame->0.5, CellMargins->{{27, 12}, {0, 8}}, CellHorizontalScrolling->True, CellFrameColor->RGBColor[0.74902, 0.694118, 0.552941], SingleLetterItalics->False, LineIndent->0, StyleMenuListing->None, Background->RGBColor[0.964706, 0.929412, 0.839216], FrameBoxOptions->{BoxFrame->0.5, BoxMargins->True}, GridBoxOptions->{ColumnWidths->{0.13, 0.35, 0.13, 0.35}}], Cell[StyleData["4ColumnBox", "Presentation"], FontSize->18], Cell[StyleData["4ColumnBox", "SlideShow"], CellMargins->{{100, 50}, {Inherited, Inherited}}], Cell[StyleData["4ColumnBox", "Printout"], CellMargins->{{2, 0}, {0, 8}}, FontSize->10, Background->GrayLevel[0.900008]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["5ColumnBox"], CellFrame->0.5, CellMargins->{{27, 12}, {0, 8}}, CellHorizontalScrolling->True, CellFrameColor->RGBColor[0.74902, 0.694118, 0.552941], LineIndent->0, StyleMenuListing->None, Background->RGBColor[0.964706, 0.929412, 0.839216], FrameBoxOptions->{BoxFrame->0.5, BoxMargins->True}, GridBoxOptions->{ColumnWidths->0.202}], Cell[StyleData["5ColumnBox", "Presentation"], FontSize->18], Cell[StyleData["5ColumnBox", "SlideShow"], CellMargins->{{100, 50}, {Inherited, Inherited}}], Cell[StyleData["5ColumnBox", "Printout"], CellMargins->{{2, 0}, {0, 8}}, FontSize->9, Background->GrayLevel[0.900008]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["6ColumnBox"], CellFrame->0.5, CellMargins->{{27, 12}, {0, 8}}, CellHorizontalScrolling->True, CellFrameColor->RGBColor[0.74902, 0.694118, 0.552941], LineIndent->0, StyleMenuListing->None, Background->RGBColor[0.964706, 0.929412, 0.839216], FrameBoxOptions->{BoxFrame->0.5, BoxMargins->True}, GridBoxOptions->{ColumnWidths->{0.12, 0.22, 0.12, 0.12, 0.22, 0.12}}], Cell[StyleData["6ColumnBox", "Presentation"], FontSize->18], Cell[StyleData["6ColumnBox", "SlideShow"], CellMargins->{{100, 50}, {Inherited, Inherited}}], Cell[StyleData["6ColumnBox", "Printout"], CellMargins->{{2, 0}, {0, 8}}, FontSize->10, Background->GrayLevel[0.900008]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["FramedBox"], CellFrame->0.5, CellMargins->{{27, 12}, {0, 8}}, CellHorizontalScrolling->True, PageBreakWithin->False, CellFrameColor->RGBColor[0.74902, 0.694118, 0.552941], AutoIndent->False, AutoSpacing->False, LineIndent->0, StyleMenuListing->None, FontWeight->"Plain", Background->RGBColor[0.964706, 0.929412, 0.839216], GridBoxOptions->{RowSpacings->1.5, ColumnAlignments->{Left}}], Cell[StyleData["FramedBox", "Presentation"], FontSize->18], Cell[StyleData["FramedBox", "SlideShow"], CellMargins->{{100, 50}, {Inherited, Inherited}}], Cell[StyleData["FramedBox", "Printout"], CellMargins->{{2, 4}, {0, 8}}, FontSize->10, Background->GrayLevel[1]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["DefinitionBox"], CellFrame->0.5, CellMargins->{{27, 12}, {0, 8}}, CellHorizontalScrolling->True, PageBreakWithin->False, CellFrameColor->RGBColor[0.74902, 0.694118, 0.552941], AutoIndent->False, AutoSpacing->False, LineIndent->0, StyleMenuListing->None, FontWeight->"Plain", Background->RGBColor[0.964706, 0.929412, 0.839216], GridBoxOptions->{RowSpacings->1.5, ColumnSpacings->1, ColumnWidths->{0.4, 0.6}, ColumnAlignments->{Left}}], Cell[StyleData["DefinitionBox", "Presentation"], FontSize->18], Cell[StyleData["DefinitionBox", "SlideShow"], CellMargins->{{100, 50}, {Inherited, Inherited}}], Cell[StyleData["DefinitionBox", "Printout"], CellMargins->{{2, 4}, {0, 8}}, FontSize->10, Background->GrayLevel[1]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["DefinitionBox3Col"], CellFrame->0.5, CellMargins->{{27, 12}, {0, 8}}, 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Background->RGBColor[0.964706, 0.929412, 0.839216], GridBoxOptions->{RowSpacings->1.5, ColumnSpacings->1, ColumnWidths->{0.15, 0.35, 0.15, 0.35}, ColumnAlignments->{Left}}], Cell[StyleData["DefinitionBox4Col", "Presentation"], FontSize->18], Cell[StyleData["DefinitionBox4Col", "SlideShow"], CellMargins->{{100, 50}, {Inherited, Inherited}}], Cell[StyleData["DefinitionBox4Col", "Printout"], CellMargins->{{2, 4}, {0, 8}}, FontSize->10, Background->GrayLevel[1]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["DefinitionBox5Col"], CellFrame->0.5, CellMargins->{{27, 12}, {0, 8}}, CellHorizontalScrolling->True, PageBreakWithin->False, CellFrameColor->RGBColor[0.74902, 0.694118, 0.552941], AutoIndent->False, AutoSpacing->False, LineIndent->0, StyleMenuListing->None, FontWeight->"Plain", Background->RGBColor[0.964706, 0.929412, 0.839216], GridBoxOptions->{RowSpacings->1.5, ColumnSpacings->1, ColumnWidths->0.2, ColumnAlignments->{Left}}], Cell[StyleData["DefinitionBox5Col", "Presentation"], 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