Hedging in the Heston Model

Question

I have simulated an underlying stock price, $S_t$ and a stochastic variance process, $v_t$ with the following stochastic differential equations from the Heston Universe: $$ dS_t = \mu S_tdt + \sqrt{v_t}dW_{1,t} $$ $$ dv_t = \kappa(\theta-v_t)dt + \sigma\sqrt{v_t}dW_{2,t} $$ Further, I have prices call options in the above Heston model with: $$ C=S_t e^{−qt}P_1−Ke^{−rt}P_2 $$ where $P_1$ and $P_2$ are in-the-money probability as definded in the original paper [1].

To do a perfect hedge of the call option in the simulated Heston world, I need a proportion of the stock and some proportion of another asset whose value depends on the variance. I want to be able to implement it in my simulations.

My question is now: how do I calculate these proportions?

I have calculated $\Delta_C = e^{−qt}P_1$ and $Vega=Se^{−qt} \frac{∂P_1}{∂v_0}2\sqrt{v_0}−Ke^{−rt}\frac{∂P_2}{∂v_0}2\sqrt{v_0}$, and I'm thinking that these are the quantaties to to buy of the underlying asset and of a another asset whose value depends on the variance to hedge the option.

However, I'm not sure that this is correct. Is this all that is needed to hedge an option in the Heston Universe?

[1] Heston, Steven L. (1993). "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options". The Review of Financial Studies. 6 (2): 327–343. doi: 10.1093/rfs/6.2.327. JSTOR 2962057.

Answer 1

Let's denote the option you need to hedge by $C_1$, which I am assuming you have sold (if you bought it then just turn the signs around). Under Heston you will need to hedge both its delta and its vega.

You can use the underlying $S$ to hedge the delta, but not to hedge vega. The most straightforward way to hedge the vega of $C_1$ is to buy another option in the market, call it $C_2$.

Let $\nu_1$ be the vega of $C_1$ and $\nu_2$ the vega of $C_2$ then the number $n$ of options $C_2$ you need to buy to hedge vega is

\begin{equation} n = \nu_1 / \nu_2 \end{equation}

However, you know that $C_2$ also has delta, call it $\Delta_2$, and $C_1$ has delta, call it $\Delta_1$. To neutralize these delta's you need to buy a certain amount $m$ of $S$, which is easily solved to be

\begin{equation} m = - \Delta_1 + n\Delta_2 \end{equation}

That's all there is to it, except of course that you need to calculate these greeks under the Heston model. Fortunately for Heston there is a closed form solution for the price and you can then differentiate the price wrt to $S$ and $\sigma$ to find delta and vega.