Exercise: Stochastic Volatility and Local Volatility Models

Consider the following stochastic volatility model for a stock $S_t$ where the dynamics are governed by the following system of stochastic differential equations $($SDEs$)$:

$d{S}_t={}_t{S}_{t}dt+\sqrt[\phantom{2}]{v_t}{S}_{t}d{Z}_1(t)$

$d{v}_t=(-v_t)dt+\sqrt[\phantom{2}]{v_t}d{Z}_2(t)$

$($:$d{Z}_1(t)d{Z}_2(t)$:$)=dt$

where:

$S_t$ is the stock price at time $t,$

$v_t$ is the instantaneous variance at time $t,$

$d{Z}_1(t)$ and $d{Z}_2(t)$ are correlated Wiener processes with correlation $,$

${}_t$ is the drift rate of the stock,

$$ is the speed of mean reversion of the variance,

$$ is the long$-$term mean of the variance,

$$ is the volatility of variance $($also known as "volatility of volatility"$).$

Part $1$: Basic Analysis and the Black$-$Scholes Limit

Show that the model reduces to the Black$-$Scholes model when $=0.$

Derive the corresponding Black$-$Scholes pricing formula for a European call option by setting $=0$ and simplifying the SDEs. What does this imply about the volatility in the Black$-$Scholes model?

Part $2$: Derivation of the Option Pricing PDE

$2.$ Derive the PDE for the option price $V(S,v,t)$ under stochastic volatility.

Set up a risk$-$free portfolio using the option, the stock, and an additional derivative to hedge against changes in volatility. By requiring the portfolio to be risk$-$free, derive the partial differential equation $($PDE$)$ that the option price must satisfy under this stochastic volatility model.

$3.$ Interpret the market price of volatility risk $(S,v,t).$

In your derivation of the PDE, you will find a term involving $(S,v,t),$ known as the "market price of volatility risk." Discuss its role and significance in the option pricing framework.

Part $3$: Local Volatility Model

$4.$ Transition to a Local Volatility Model:

Assume that the stochastic volatility process is not explicitly modeled, but that the option prices reflect the implied volatility surface in the market. Using the Dupire equation for local volatility, express the local volatility ${}_{\text{l}\text{o}\text{c}\text{a}\text{l}}(S,t)$ in terms of the price of European call options $C(S_0,K,T),$ where $K$ is the strike price and $T$ is the expiration time. Derive the following relationship for local volatility:

${}_{\text{l}\text{o}\text{c}\text{a}\text{l}}^2(K,T)=\frac{\frac{\text{d}\text{e}\text{l}\text{C}}{\text{d}\text{e}\text{l}\text{T}}}{\frac{1}{2}{K}^2\frac{de{l}^{2}C}{del{K}^2}}$

Part $4$: Numerical Analysis

$5.$ Numerical Solution of the PDE:

Implement a finite difference method to solve the PDE you derived in Part $2$ for the price of a European call option under stochastic volatility. Assume reasonable parameter values for the model $($e$.$g$.,=2,=0\mathrm{.}04,=0\mathrm{.}5,=-0\mathrm{.}5,$ and initial volatility $v_0=0\mathrm{.}04).$

Use boundary conditions suitable for a European call option.

Compare the results obtained from the stochastic volatility model with those from the Black$-$Scholes model $($when $=0).$

Exploring the Impact of Volatility of Volatility:

Analyze the impact of different values of $($volatility of volatility$)$ on the option price. Plot the option prices as a function of strike for varying values of $,$ and comment on the effect of increased volatility of volatility on the shape of the implied volatility smile.

Part $5$: Extension to Exotic Options

$7.$ Pricing an Exotic Option:

Extend the analysis by pricing a barrier option $($e$.$g$.,$ a down$-$and$-$out call option$)$ using both the stochastic volatility model and the local volatility model. Compare the results from the two models and discuss how the different volatility assumptions impact the pricing of this exotic option.

Part $6$: Analytical vs Numerical Results

$8.$ Compare the Analytical Heston Model Solution:

Compare your numerical results for the European option obtained in Part $5$ with the analytical solution for the Heston model as provided. Use the characteristic function$-$based pricing formula for the Heston model and analyze any discrepancies between the numerical and analytical results.