$\%$ ParametersS$0=12$; $\%$ Initial stock pricev$0=0\mathrm{.}01$; $\%$ Initial volatilityr $=0\mathrm{.}025$; $\%$ Risk$-$free interest rateT $=1$; $\%$ Time to expirationK $=10$; $\%$ Strike pricekappa $=1\mathrm{.}2$;theta $=0\mathrm{.}04$;sigma $=0\mathrm{.}3$;rho $=-0\mathrm{.}3$;$\%$ Number of time steps and pathsN $=1000$; $\%$ Number of time stepsM $=10000$; $\%$ Number of paths$\%$ Time vectordt $=$ T $/$ N;t $=0$:dt:T;$\%$ Generate correlated Wiener processesdW$1=$ randn$($M$,$ N$)*$ sqrt$($dt$)$;dW$2=$ rho $*$ dW$1+$ sqrt$(1-$ rho$^2)*$ randn$($M$,$ N$)*$ sqrt$($dt$)$;$\%$ Initialize matrices to store stock prices and volatilitiesS $=$ zeros$($M$,$ N$+1)$;v $=$ zeros$($M$,$ N$+1)$;$\%$ Set initial conditionsS$($:$,1)=$ S$0$;v$($:$,1)=$ v$0$;$\%$ Simulate paths using Euler$-$Maruyama methodfor i $=1$:N S$($:$,$i$+1)=$ S$($:$,$i$).*$ exp$(($r $-0\mathrm{.}5*$ v$($:$,$i$))*$ dt $+$ sqrt$($v$($:$,$i$)).*$ dW$1($:$,$i$))$; v$($:$,$i$+1)=$ v$($:$,$i$)+$ kappa $*($theta $-$ v$($:$,$i$))*$ dt $+$ sigma $*$ sqrt$($v$($:$,$i$)).*$ dW$2($:$,$i$)$; v$($:$,$i$+1)=$ max$($v$($:$,$i$+1),0)$; $\%$ Ensure volatility is non$-$negativeend$\%$ Plot a sample path of stock price and volatilityfigure;subplot$(2,1,1)$;plot$($t$,$ S$(1,$:$),\text{'}$b$\text{'})$;xlabel$(\text{'}$Time$\text{'})$;ylabel$(\text{'}$Stock Price'$)$;title$(\text{'}$Sample Path of Stock Price'$)$;subplot$(2,1,2)$;plot$($t$,$ v$(1,$:$),\text{'}$r$\text{'})$;xlabel$(\text{'}$Time$\text{'})$;ylabel$(\text{'}$Volatility$\text{'})$;title$(\text{'}$Sample Path of Volatility'$)$;$\%$ Calculate discounted payoff for each pathpayoff $=$ max$($S$($:$,$end$)-$ K$,0).*$ exp$(-$r $*$ T$)$;$\%$ Compute option priceoption$\_$price$\_$european $=$ mean$($payoff$)$;$\%$ Display option pricedisp$([\text{'}$European Call Option Price $($Standard Monte Carlo$)$: $\text{'},$ num$2$str$($option$\_$price$\_$european$)])$;$\%$ Define the digital call option payoff functiondigital$\_$payoff $=$ @$($S$,$ K$)$ double$($S $>$ K$)$;$\%$ Calculate discounted payoff for each pathdigital$\_$payoff$\_$values $=$ digital$\_$payoff$($S$($:$,$end$),$ K$).*$ exp$(-$r $*$ T$)$;$\%$ Compute option priceoption$\_$price$\_$digital $=$ mean$($digital$\_$payoff$\_$values$)$;$\%$ Display option pricedisp$([\text{'}$Digital Call Option Price: $\text{'},$ num$2$str$($option$\_$price$\_$digital$)])$;$\%$ Calculate discounted payoff for each path for European call optionPayoff$\_$European $=$ max$($S$($:$,$end$)-$ K$,0).*$ exp$(-$r $*$ T$)$;$\%$ Control variate: European call option price at expirationControl$\_$Variate$\_$European $=$ exp$(($r $-0\mathrm{.}5*$ theta$)*$ T$)*$ S$($:$,$end$)-$ K $*$ exp$(-$r $*$ T$)$;$\%$ Compute control variate coefficient for European call optioncovXY $=$ cov$($Payoff$\_$European, Control$\_$Variate$\_$European$)$;beta$\_$European $=-$covXY$(1,2)/$ var$($Control$\_$Variate$\_$European$)$;$\%$ Adjusted Payoffs using Control Variates for European call optionPayoff$\_$European$\_$Adjusted $=$ Payoff$\_$European $+$ beta$\_$European $*($Control$\_$Variate$\_$European $-$ mean$($Control$\_$Variate$\_$European$))$;$\%$ Compute option price with Control Variates for European call optionPrice$\_$European$\_$CV $=$ mean$($Payoff$\_$European$\_$Adjusted$)$;$\%$ Calculate discounted payoff for each path for digital call optionPayoff$\_$Digital $=($S$($:$,$end$)>$ K$).*$ exp$(-$r $*$ T$)$;$\%$ Control variate: Digital call option price at expirationControl$\_$Variate$\_$Digital $=($S$($:$,$end$)>$ K$).*$ exp$(-$r $*$ T$)$; $\%$ Same as the payoff for digital call$\%$ Compute control variate coefficient for digital call optioncovXY $=$ cov$($Payoff$\_$Digital, Control$\_$Variate$\_$Digital$)$;beta$\_$Digital $=-$covXY$(1,2)/$ var$($Control$\_$Variate$\_$Digital$)$;$\%$ Adjusted Payoffs using Control Variates for digital call optionPayoff$\_$Digital$\_$Adjusted $=$ Payoff$\_$Digital $+$ beta$\_$Digital $*($Control$\_$Variate$\_$Digital $-$ mean$($Control$\_$Variate$\_$Digital$))$;$\%$ Compute option price with Control Variates for digital call optionPrice$\_$Digital$\_$CV $=$ mean$($Payoff$\_$Digital$\_$Adjusted$)$;$\%$ Display resultsdisp$([\text{'}$European Call Option Price with Control Variates: $\text{'},$ num$2$str$($Price$\_$European$\_$CV$)])$;disp$([\text{'}$Digital Call Option Price with Control Variates: $\text{'},$ num$2$str$($Price$\_$Digital$\_$CV$)])$;$\%$ Compute MLMC estimatorsMC$\_$estimator$\_$European $=$ mean$($Payoff$\_$European$\_$Adjusted$)-$ mean$($Payoff$\_$European$)$;MC$\_$estimator$\_$Digital $=$ mean$($Payoff$\_$Digital$\_$Adjusted$)-$ mean$($Payoff$\_$Digital$)$;$\%$ Display resultsdisp$([\text{'}$European Call Option Price $($MLMC$)$: $\text{'},$ num$2$str$($MC$\_$estimator$\_$European$)])$;disp$([\text{'}$Digital Call Option Price $($MLMC$)$: $\text{'},$ num$2$str$($MC$\_$estimator$\_$Digital$)])$;$\%$ Compute MLMC estimators without antithetic treatmentMC$\_$estimator$\_$European$\_$without$\_$antithetic $=$ mean$($Payoff$\_$European$\_$Adjusted$)-$ mean$($Payoff$\_$European$)$;MC$\_$estimator$\_$Digital$\_$without$\_$antithetic $=$ mean$($Payoff$\_$Digital$\_$Adjusted$)-$ mean$($Payoff$\_$Digital$)$;$\%$ Display resultsdisp$([\text{'}$European Call Option Price $($MLMC without Antithetic Treatment$)$: $\text{'},$ num$2$str$($MC$\_$estimator$\_$European$\_$without$\_$antithetic$)])$;disp$([\text{'}$Digital Call Option Price $($MLMC without Antithetic Treatment$)$: $\text{'},$ num$2$str$($MC$\_$estimator$\_$Digital$\_$without$\_$antithetic$)])$;$\%$ Apply antithetic treatmentS$\_$fine$\_$antithetic $=$ S$0*$ ones$($N $+1,$ M$)$;v$\_$fine$\_$antithetic $=$ v$0*$ ones$($N $+1,$ M$)$;dW$1\_$fine$\_$antithetic $=-$dW$1$; $\%$ negate dW$1$ for antithetic variatesdW$2\_$fine$\_$antithetic $=-$dW$2$; $\%$ negate dW$2$ for antithetic variatesfor i $=1$:length$($sample$\_$sizes$)$ M $=$ sample$\_$sizes$($i$)$; $\%$ Re$-$simulate paths with new sample size dW$1=$ randn$($M$,$ N$)*$ sqrt$($dt$)$; dW$2=$ rho $*$ dW$1+$ sqrt$(1-$ rho$^2)*$ randn$($M$,$ N$)*$ sqrt$($dt$)$; S $=$ zeros$($M$,$ N$+1)$; v $=$ zeros$($M$,$ N$+1)$; S$($:$,1)=$ S$0$; v$($:$,1)=$ v$0$; for j $=1$:N S$($:$,$j$+1)=$ S$($:$,$j$).*$ exp$(($r $-0\mathrm{.}5*$ v$($:$,$j$))*$ dt $+$ sqrt$($v$($:$,$j$)).*$ dW$1($:$,$j$))$; v$($:$,$j$+1)=$ v$($:$,$j$)+$ kappa $*($theta $-$ v$($:$,$j$))*$ dt $+$ sigma $*$ sqrt$($v$($:$,$j$)).*$ dW$2($:$,$j$)$; v$($:$,$j$+1)=$ max$($v$($:$,$j$+1),0)$; end $\%$ European option payoff$\_$european $=$ max$($S$($:$,$end$)-$ K$,0).*$ exp$(-$r $*$ T$)$; option$\_$prices$\_$european$($i$)=$ mean$($payoff$\_$european$)$; $\%$ Digital option digital$\_$payoff$\_$values $=$ digital$\_$payoff$($S$($:$,$end$),$ K$).*$ exp$(-$r $*$ T$)$; option$\_$prices$\_$digital$($i$)=$ mean$($digital$\_$payoff$\_$values$)$;end$\%$ Plot convergence graphsfigure;subplot$(2,1,1)$;plot$($sample$\_$sizes, option$\_$prices$\_$european, $\text{'}$bo$-\text{'})$;xlabel$(\text{'}$Sample Size'$)$;ylabel$(\text{'}$Option Price'$)$;title$(\text{'}$Convergence of European Call Option Price'$)$;subplot$(2,1,2)$;plot$($sample$\_$sizes, option$\_$prices$\_$digital, $\text{'}$ro$-\text{'})$;xlabel$(\text{'}$Sample Size'

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