$\mathrm{restart}\:$

$\mathrm{with}\left(\mathrm{LinearAlgebra}\right)\:$$$\begin{gathered} \mathrm{with}\left(\mathrm{Finance}\right)\: \\ \mathrm{with}\left(\mathrm{Statistics}\right)\: \\ \mathrm{with}\left(\mathrm{plots}\right)\: \\ \mathrm{interface}\left(\mathrm{displayprecision}\=5\right)\: \end{gathered}$$

$\mathrm{Data}:=\mathrm{ImportMatrix}\left(\mathrm{FileTools}:-\mathrm{JoinPath}\left(\left[\mathrm{kernelopts}\left(\mathrm{datadir}\right)\,''finance''\,''SP-500-CALLS.txt''\right]\right)\right)$

$\mathrm{DF}≔\mathrm{DataFrame}\left(\mathrm{Data}\left[2..\,..\right]\,\mathrm{columns}\=\mathrm{Data}\left[1\,..\right]\right)$

$M\,N{≔}_{}\mathrm{upperbound}\left(\mathrm{DF}\right)$

$M,N≔308,10$

$\mathrm{S0}:=1377$

$\mathrm{S0}≔1377$

$r:=0.0532$

$r≔0.05320$

$d:=0.0182$

$d≔0.01820$

$\mathrm{ExerciseDates}≔\left[\mathrm{seq}\left(\mathrm{NthWeekday}\left(3\,\mathrm{Friday}\,\mathrm{DF}\left[i\,''Month''\right]\,\mathrm{DF}\left[i\,''Year''\right]\right)\,i\=1..M\right)\right]\:$

$\mathrm{Times}:=\mathrm{map}\left(t\to \mathrm{YearFraction}\left(''October\; 27,\; 2006''\,t\right)\,\mathrm{ExerciseDates}\right)\:$

${\mathrm{TimeExample}}_1≔\mathrm{YearFraction}\left(''October\; 27,\; 2006''\,\mathrm{NthWeekday}\left(3\,\mathrm{Friday}\, \mathrm{December}\,2006\right)\right)$

${\mathrm{TimeExample}}_1≔0.13425$

$\mathrm{Dec2006}≔\mathrm{DF}\left[\mathrm{DF}\left[''Month''\right] \=~ ''December'' \mathbf{and} \mathrm{DF}\left[''Year''\right] \=~ 2006\right]$

$\mathrm{L1}≔\mathrm{Matrix}\left(\left[\mathrm{seq}\left( \left[ \mathrm{Dec2006}\left[i\, ''Strike''\right]\, \mathrm{ImpliedVolatility}\left( \mathrm{Dec2006}\left[i\, ''Ask''\right]\, \mathrm{S0}\, \mathrm{Dec2006}\left[i\, ''Strike''\right]\, \mathrm{TimeExample}\left[1\right]\, r\, d \right) \right]\, i \= 1 .. \mathrm{NumRows}\left(\mathrm{Dec2006}\right)\right)\right]\,\mathrm{datatype}\=\mathrm{float}\left[8\right]\right)$

$\mathrm{dataplot}\left(\mathrm{L1}\left[..\,1\right]\,\mathrm{L1}\left[..\,2\right]\,\mathrm{gridlines}\,\mathrm{size}\=\left[600\,''golden''\right]\right)$

${\mathrm{TimeExample}}_2≔\mathrm{YearFraction}\left(''October\; 27,\; 2006''\,\mathrm{NthWeekday}\left(3\,\mathrm{Friday}\,\mathrm{December}\,2007\right)\right)$

${\mathrm{TimeExample}}_2≔1.15068$

$\mathrm{Dec2007}≔\mathrm{DF}\left[\mathrm{DF}\left[''Month''\right] \=~ ''December'' \mathbf{and} \mathrm{DF}\left[''Year''\right] \=~ 2007\right]$

$\mathrm{L2}≔\mathrm{Matrix}\left(\left[\mathrm{seq}\left( \left[ \mathrm{Dec2007}\left[i\, ''Strike''\right]\, \mathrm{ImpliedVolatility}\left( \mathrm{Dec2007}\left[i\, ''Ask''\right]\, \mathrm{S0}\, \mathrm{Dec2007}\left[i\, ''Strike''\right]\, \mathrm{TimeExample}\left[2\right]\, r\, d \right) \right]\, i \= 1 .. \mathrm{NumRows}\left(\mathrm{Dec2007}\right)\right)\right]\,\mathrm{datatype}\=\mathrm{float}\left[8\right]\right)$

$\mathrm{dataplot}\left(\mathrm{L2}\left[..\,1\right]\,\mathrm{L2}\left[..\,2\right]\,\mathrm{gridlines}\,\mathrm{size}\=\left[600\,''golden''\right]\right)$

$\mathrm{\Σ}:=\left(S\,T\,K\right)\→{\mathrm{\α}}_1\+{\mathrm{\α}}_2\ln \left(\frac{K}{S}\right)\+\frac{{\mathrm{\α}}_3}{\sqrt{T}}\+{\mathrm{\α}}_4{\ln \left(\frac{K}{S}\right)}^2\+\frac{{\mathrm{\α}}_5\ln \left(\frac{K}{S}\right)}{\sqrt{T}}\+\frac{{\mathrm{\α}}_6{\ln \left(\frac{K}{S}\right)}^2}{\sqrt{T}}$

$\mathrm{\Sigma }≔\left(S\,T\,K\right)\mapsto {\mathrm{\alpha }}_1+{\mathrm{\alpha }}_2\cdot \ln \left(\frac{K}{S}\right)+\frac{{\mathrm{\alpha }}_3}{\sqrt{T}}+{\mathrm{\alpha }}_4\cdot {\ln \left(\frac{K}{S}\right)}^2+\frac{{\mathrm{\alpha }}_5\cdot \ln \left(\frac{K}{S}\right)}{\sqrt{T}}+\frac{{\mathrm{\alpha }}_6\cdot {\ln \left(\frac{K}{S}\right)}^2}{\sqrt{T}}$

$C:=\mathrm{BlackScholesPrice}\left(\mathrm{S0}\,K\,T\,\mathrm{\Σ}\left(\mathrm{S0}\,T\,K\right)\,r\,d\right)$

$C≔1377.00000{\ⅇ}^{-0.01820T}\left(0.50000+0.50000\mathrm{erf}\left(\frac{0.70711\left(\ln \left(\frac{1377.00000}{K}\right)+0.03500T+0.50000{\left({\mathrm{\alpha }}_1+{\mathrm{\alpha }}_2\ln \left(0.00073K\right)+\frac{{\mathrm{\alpha }}_3}{\sqrt{T}}+{\mathrm{\alpha }}_4{\ln \left(0.00073K\right)}^2+\frac{{\mathrm{\alpha }}_5\ln \left(0.00073K\right)}{\sqrt{T}}+\frac{{\mathrm{\alpha }}_6{\ln \left(0.00073K\right)}^2}{\sqrt{T}}\right)}^{2}T\right)}{\left({\mathrm{\alpha }}_1+{\mathrm{\alpha }}_2\ln \left(0.00073K\right)+\frac{{\mathrm{\alpha }}_3}{\sqrt{T}}+{\mathrm{\alpha }}_4{\ln \left(0.00073K\right)}^2+\frac{{\mathrm{\alpha }}_5\ln \left(0.00073K\right)}{\sqrt{T}}+\frac{{\mathrm{\alpha }}_6{\ln \left(0.00073K\right)}^2}{\sqrt{T}}\right)\sqrt{T}}\right)\right)-1.00000{\ⅇ}^{-0.05320T}K\left(0.50000+0.50000\mathrm{erf}\left(\frac{0.70711\left(\ln \left(\frac{1377.00000}{K}\right)+0.03500T+0.50000{\left({\mathrm{\alpha }}_1+{\mathrm{\alpha }}_2\ln \left(0.00073K\right)+\frac{{\mathrm{\alpha }}_3}{\sqrt{T}}+{\mathrm{\alpha }}_4{\ln \left(0.00073K\right)}^2+\frac{{\mathrm{\alpha }}_5\ln \left(0.00073K\right)}{\sqrt{T}}+\frac{{\mathrm{\alpha }}_6{\ln \left(0.00073K\right)}^2}{\sqrt{T}}\right)}^{2}T\right)}{\left({\mathrm{\alpha }}_1+{\mathrm{\alpha }}_2\ln \left(0.00073K\right)+\frac{{\mathrm{\alpha }}_3}{\sqrt{T}}+{\mathrm{\alpha }}_4{\ln \left(0.00073K\right)}^2+\frac{{\mathrm{\alpha }}_5\ln \left(0.00073K\right)}{\sqrt{T}}+\frac{{\mathrm{\alpha }}_6{\ln \left(0.00073K\right)}^2}{\sqrt{T}}\right)\sqrt{T}}-0.70711\left({\mathrm{\alpha }}_1+{\mathrm{\alpha }}_2\ln \left(0.00073K\right)+\frac{{\mathrm{\alpha }}_3}{\sqrt{T}}+{\mathrm{\alpha }}_4{\ln \left(0.00073K\right)}^2+\frac{{\mathrm{\alpha }}_5\ln \left(0.00073K\right)}{\sqrt{T}}+\frac{{\mathrm{\alpha }}_6{\ln \left(0.00073K\right)}^2}{\sqrt{T}}\right)\sqrt{T}\right)\right)$

$B:=⟨⟨\mathrm{Times}⟩\|\mathrm{convert}\left(\mathrm{DF}\left[''Strike''\right]\,\mathrm{Matrix}\right)⟩$

$V≔\mathrm{convert}\left(\mathrm{DF}\left[''Ask''\right]\,\mathrm{Vector}\left[\mathrm{column}\right]\right)$

$\mathrm{\β1}≔\mathrm{NonlinearFit}\left(C\,B\, V\,\left[T\,K\right]\,\mathrm{parameternames}\=\left[{\mathrm{\α}}_1\,{\mathrm{\α}}_2\,{\mathrm{\α}}_3\,{\mathrm{\α}}_4\,{\mathrm{\α}}_5\,{\mathrm{\α}}_6\right]\,\mathrm{output}\=\mathrm{parametervector}\right)$

$\mathrm{\β1}≔\left[\right]$

$\mathrm{F1}:=\genfrac{}{}{0ex}{}{\mathrm{\Σ}\left(\mathrm{S0}\,T\,K\right)}{\phantom{\mathrm{\α}\=\mathrm{\β1}}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{\Σ}\left(\mathrm{S0}\,T\,K\right)}}{\mathrm{\α}\=\mathrm{\β1}}$

$\mathrm{F1}≔0.14393+0.12599\ln \left(\frac{K}{1377}\right)-\frac{0.01069}{\sqrt{T}}-0.03301{\ln \left(\frac{K}{1377}\right)}^2-\frac{0.30394\ln \left(\frac{K}{1377}\right)}{\sqrt{T}}+\frac{0.40480{\ln \left(\frac{K}{1377}\right)}^2}{\sqrt{T}}$

$\mathrm{plot3d}\left(\mathrm{F1}\,T\=0..2.5\,K\=600..1500\,\mathrm{view}\=0..2.6\right)$

$\mathrm{animate}\left(\mathrm{plot}\,\left[\mathrm{F1}\,K\=600..1500\,\mathrm{thickness}\=3\right]\,T\=0.1..3\,\mathrm{axes}\=\mathrm{boxed}\, \mathrm{gridlines}\,\mathrm{size}\=\left[600\,''golden''\right]\right)$

$U:=\mathrm{Vector}\left(1..M\right)$

$$\begin{gathered} \mathbf{for} i \mathbf{to} M \mathbf{do} \\ {U}_i:=\mathrm{ImpliedVolatility}\left(V_i\,\mathrm{S0}\,B_{i\,2}\,B_{i\,1}\,r\,d\right) \\ \mathbf{end} \mathbf{do}\: \end{gathered}$$

$\mathrm{\β2}≔\mathrm{NonlinearFit}\left(\mathrm{\Σ}\left(\mathrm{S0}\,T\,K\right)\,B\,U\,\left[T\,K\right]\,\mathrm{parameternames}\=\left[{\mathrm{\α}}_1\,{\mathrm{\α}}_2\,{\mathrm{\α}}_3\,{\mathrm{\α}}_4\,{\mathrm{\α}}_5\,{\mathrm{\α}}_6\right]\,\mathrm{output}\=\mathrm{parametervector}\right)$

$\mathrm{\β2}≔\left[\right]$

$\mathrm{F2}:=\genfrac{}{}{0ex}{}{\mathrm{\Σ}\left(\mathrm{S0}\,T\,K\right)}{\phantom{\mathrm{\α}\=\mathrm{\β2}}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{\Σ}\left(\mathrm{S0}\,T\,K\right)}}{\mathrm{\α}\=\mathrm{\β2}}$

$\mathrm{F2}≔0.13011+0.17037\ln \left(\frac{K}{1377}\right)+\frac{0.00115}{\sqrt{T}}-0.26597{\ln \left(\frac{K}{1377}\right)}^2-\frac{0.36133\ln \left(\frac{K}{1377}\right)}{\sqrt{T}}+\frac{0.28070{\ln \left(\frac{K}{1377}\right)}^2}{\sqrt{T}}$

$\mathrm{plot3d}\left(\mathrm{F2}\,T\=0..2.5\,K\=600..1500\,\mathrm{view}\=0..2\right)$

$\mathrm{animate}\left(\mathrm{plot}\,\left[\mathrm{F2}\,K\=600..1500\,\mathrm{thickness}\=3\right]\,T\=0.1..3\, \mathrm{gridlines}\, \mathrm{axes} \= \mathrm{boxed}\,\mathrm{size}\=\left[600\,''golden''\right]\right)$

$P≔\mathrm{dataplot}\left(\mathrm{L1}\left[..\,1\right]\,\mathrm{L1}\left[..\,2\right]\right)\:$

$\mathrm{P1}:=\mathrm{plot}\left(\genfrac{}{}{0ex}{}{\mathrm{F1}}{\phantom{T\={\mathrm{TimeExample}}_1}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{F1}}}{T\={\mathrm{TimeExample}}_1}\,K\=800..1500\,\mathrm{thickness}\=3\,\mathrm{color}\=\mathrm{red}\right)\:$

$\mathrm{P2}:=\mathrm{plot}\left(\genfrac{}{}{0ex}{}{\mathrm{F2}}{\phantom{T\={\mathrm{TimeExample}}_1}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{F2}}}{T\={\mathrm{TimeExample}}_1}\,K\=800..1500\,\mathrm{thickness}\=3\,\mathrm{color}\=\mathrm{green}\right)\:$

$\mathrm{display}\left(P\,\mathrm{P1}\,\mathrm{P2}\,\mathrm{axes}\=\mathrm{boxed}\,\mathrm{gridlines}\,\mathrm{size}\=\left[600\,''golden''\right]\right)$

$P≔\mathrm{dataplot}\left(\mathrm{L2}\left[..\,1\right]\,\mathrm{L2}\left[..\,2\right]\right)\:$

$\mathrm{P1}:=\mathrm{plot}\left(\genfrac{}{}{0ex}{}{\mathrm{F1}}{\phantom{T\={\mathrm{TimeExample}}_2}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{F1}}}{T\={\mathrm{TimeExample}}_2}\,K\=1000..1500\,\mathrm{thickness}\=3\,\mathrm{color}\=\mathrm{red}\right)\:$

$\mathrm{P2}:=\mathrm{plot}\left(\genfrac{}{}{0ex}{}{\mathrm{F2}}{\phantom{T\={\mathrm{TimeExample}}_2}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{F2}}}{T\={\mathrm{TimeExample}}_2}\,K\=1000..1500\,\mathrm{thickness}\=3\,\mathrm{color}\=\mathrm{green}\right)\:$

$\mathrm{display}\left(P\,\mathrm{P1}\,\mathrm{P2}\,\mathrm{gridlines}\,\mathrm{axes}\=\mathrm{boxed}\,\mathrm{size}\=\left[600\,''golden''\right]\right)$

$\mathrm{restart}$

$\mathrm{with}\left(\mathrm{Finance}\right)\:$

$\mathrm{\Σ}:={\mathrm{\α}}_1\+{\mathrm{\α}}_2\ln \left(\frac{K}{\mathrm{S0}}\right)\+\frac{{\mathrm{\α}}_3\cdot 1}{\sqrt{T}}\+{\mathrm{\α}}_4{\ln \left(\frac{K}{\mathrm{S0}}\right)}^2\+\frac{{\mathrm{\α}}_5\ln \left(\frac{K}{\mathrm{S0}}\right)}{\sqrt{T}}\+\frac{{\mathrm{\α}}_6{\ln \left(\frac{K}{\mathrm{S0}}\right)}^2}{\sqrt{T}}$

$\mathrm{\Sigma }≔{\mathrm{\alpha }}_1+{\mathrm{\alpha }}_2\ln \left(\frac{K}{\mathrm{S0}}\right)+\frac{{\mathrm{\alpha }}_3}{\sqrt{T}}+{\mathrm{\alpha }}_4{\ln \left(\frac{K}{\mathrm{S0}}\right)}^2+\frac{{\mathrm{\alpha }}_5\ln \left(\frac{K}{\mathrm{S0}}\right)}{\sqrt{T}}+\frac{{\mathrm{\alpha }}_6{\ln \left(\frac{K}{\mathrm{S0}}\right)}^2}{\sqrt{T}}$

${\mathrm{\α}}_1≔0.1581\:$${\mathrm{\α}}_2:=-0.2777\:$${\mathrm{\α}}_3:=-0.0050\:$${\mathrm{\α}}_4:=-0.5011\:$${\mathrm{\α}}_5≔0.1103\:$${\mathrm{\α}}_6:=0.5909\:$

$\mathrm{S0}:=100$

$\mathrm{S0}≔100$

$\mathrm{\Σ}$

$0.1581-0.2777\ln \left(\frac{K}{100}\right)-\frac{0.0050}{\sqrt{T}}-0.5011{\ln \left(\frac{K}{100}\right)}^2+\frac{0.1103\ln \left(\frac{K}{100}\right)}{\sqrt{T}}+\frac{0.5909{\ln \left(\frac{K}{100}\right)}^2}{\sqrt{T}}$

$\mathrm{plot3d}\left(\mathrm{\Σ}\,K\=50..150\,T\=0.1..2\right)$

$C:=\mathrm{BlackScholesPrice}\left(\mathrm{S0}\,K\,T\,\mathrm{\Σ}\,0.05\,0.01\,\'\mathrm{call}\'\right)\:$

$P:=\mathrm{BlackScholesPrice}\left(\mathrm{S0}\,K\,T\,\mathrm{\Σ}\,0.05\,0.01\,\'\mathrm{put}\'\right)\:$

$V:=\mathrm{LocalVolatilitySurface}\left(0.05\,0.01\,C\,K\,T\right)\:$

$\mathrm{S1}:=⟨\mathrm{seq}\left(i\,i\=50..200\right)⟩$

$\mathrm{T1}:=⟨\mathrm{seq}\left(0.02i\,i\=1..50\right)⟩$

$V:=\mathrm{LocalVolatility}\left(C\,\mathrm{S1}\,\mathrm{T1}\,0.05\,0.01\,T\,K\right)$

$\mathrm{LV}:=\mathrm{LocalVolatility}\left(C\,S\,T\,0.05\,0.01\,T\,K\right)\:$

$\mathrm{P1} ≔ \mathrm{plot3d}\left(\mathrm{LV}\,S\=50..150\,T\=0.05..1\, \mathrm{color} \= \mathrm{blue}\right)\:$

$\mathrm{P2}:=\mathrm{plot3d}\left(\mathrm{\Σ}\,K\=50..150\,T\=0.05..1\, \mathrm{color} \= \mathrm{red}\right)\:$

${\mathrm{plots}}_{\mathrm{display}}\left(\mathrm{P1}\,\mathrm{P2}\,\mathrm{view}\=0..1.5\right)$

$\mathrm{LVS}:=\mathrm{LocalVolatilitySurface}\left(\mathrm{T1}\,\mathrm{S1}\,V\right)\:$

$\mathrm{IVS}:=\mathrm{ImpliedVolatilitySurface}\left(\mathrm{\Σ}\,T\,K\right)\:$

$X:=\mathrm{BlackScholesProcess}\left(100\,\mathrm{LVS}\,0.05\,0.01\right)$

$X≔\mathrm{\_X0}$

$\mathrm{PathPlot}\left(X\left(t\right)\,t\=0..1\,\mathrm{timesteps}\=20\,\mathrm{replications}\=20\,\mathrm{thickness}\=3\,\mathrm{color}\=\mathrm{red}..\mathrm{yellow}\,\mathrm{axes}\= \mathrm{boxed}\, \mathrm{gridlines}\,\mathrm{size}\=\left[600\,''golden''\right]\right)$

$\mathrm{P1}:=\mathrm{subs}\left(K\=\mathrm{S0}x\,\frac{1P}{\mathrm{S0}}\right)\:$

$\mathrm{C1}:=\mathrm{subs}\left(K\=\mathrm{S0}x\,\frac{1C}{\mathrm{S0}}\right)\:$

$T:=\'T\'\:$

$\mathrm{TT}:=\mathrm{ImpliedTrinomialTree}\left(100\,0.05\,0.01\,\mathrm{IVS}\,1\,100\right)\:$

$\mathrm{GetLocalVolatility}\left(\mathrm{TT}\,99\,99\,0.05\right)$

$0.1531851114$

$\mathrm{IVS}\left(1.0\,100\right)$

$0.1531000000$

$\mathrm{ExpectedValue}\left(\max \left(X\left(1\right)-100\,0\right)\,\mathrm{timesteps}\=100\,\mathrm{replications}\={10}^4\right)$

$\left[\mathrm{value}=8.562255135\,\mathrm{standarderror}=0.09642955646\right]$

$E:=\mathrm{EuropeanOption}\left(t\→\max \left(t-100\,0\right)\,1.0\right)\:$

$\mathrm{LatticePrice}\left(E\,\mathrm{TT}\,0.0\right)$

$8.477754630$

$\mathrm{ExpectedValue}\left(\max \left({\∫}_0^{1}X\left(u\right)\ⅆu-100\,0\right)\,\mathrm{timesteps}\=100\,\mathrm{replications}\={10}^3\right)$

$\left[\mathrm{value}=4.311947370\,\mathrm{standarderror}=0.1787554842\right]$