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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.

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    $\begingroup$ Consider a portfolio which holds the call option, some proportion of the stock and some proportion of another asset whose value depends on the variance. This way, you make the market complete and can hedge the volatility risk. Then apply Itô’s Lemma to get the changes of this portfolio. Then find the trading strategy (number of the two assets you need to trade) to eliminate the stochastic terms. You’re then left with a deterministic portfolio change. $\endgroup$
    – Kevin
    Oct 7 '20 at 14:15
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    $\begingroup$ Which risks do you want to hedge? What is the end goal of this exercise? $\endgroup$ Oct 7 '20 at 15:14
  • $\begingroup$ @KeSchn this is exactly what I need. Do you know any sources that go through the math and the logic in debt? $\endgroup$
    – Modvinden
    Oct 8 '20 at 7:50
  • $\begingroup$ @Modvinden you should do your homework on your own. What have you tried thus far? Do you know Itô’s Lemma for two processes (and time)? Take the simple delta hedge from the Black-Scholes model and try to generalise each step. $\endgroup$
    – Kevin
    Oct 8 '20 at 8:09
  • $\begingroup$ @KeSchn What I have managed so far is the following. The price of a call option is $C=S_t e^{-qt} P_1 - Ke^{-rt} P_2$ where $P_1$ and $P_1$ in-the-money probability as definded in the original paper. Then we need $\Delta_C= e^{-qt}P_1$ of the underlying asset and $\Vega = S e^{-qt} \frac{\partial P_1}{\partial v_0} 2 \sqrt{v_0}-K e^{-rt} \frac{\partial P_2}{\partial v_0} 2 \sqrt{v_0}$ of the variance swap. Is this correct? $\endgroup$
    – Modvinden
    Oct 8 '20 at 11:31
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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.

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  • $\begingroup$ Thank you so much for the great response. So I can calculate $\Delta_1$ and $v_1$ by differentiating $C_1=S_t e^{−qt}P_1−Ke^{−rt}P_2$ wrt $S_t$ and $\sigma$. I understand this. However, I still need to find $\Delta_2$ and $v_2$ by differentiating $C_2$ wrt the underlying and its $sigma$. $\Delta_2$ and $v_2$ will then depend on the way I specify $C_2$ to be? I have read that $C_2$ is often a variance swap. How is $C_2$ often specified - I mean what expression/equation do people use for $C_2$? $\endgroup$
    – Modvinden
    Oct 9 '20 at 11:34
  • $\begingroup$ Within Heston (and actually that's the way the Heston PDE is derived) $C_2$ is another vanilla option. Of course you can take another instrument sensitive to volatility, but at this stage it will only make your life more complicated. For example, what is the sensitivity of the variance swap to the instantaneous volatility? That's not an easy question. My advice: stick to taking another option for $C_2$. Once you understand the nitty gritty then it's OK to talk to investment banks trying to sell you varswaps :-D $\endgroup$ Oct 9 '20 at 11:58
  • $\begingroup$ So $C_2$ is a vanilla option on a new underlying asset $S_{t,2}$? I want to implement this into my simulation, so then I need to specify dynamics of this option as well? And should this option also be generated with Heston? Thank you for the great answers!! $\endgroup$
    – Modvinden
    Oct 9 '20 at 12:10
  • $\begingroup$ Not another underlying, the same underlying, different strike and/or maturity. $\endgroup$ Oct 9 '20 at 12:11
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    $\begingroup$ There is no rule in theory. But you may want to choose a maturity and strike such that also the vanna of the total portfolio (target option + hedge instruments) is minimized. This way you reduce transaction costs. Vanna is the sensitivity of delta to vol or equivalently sensitivity of vega to underlying asset movements. Minimizing vanna will reduce the number of times you need to rebalance your hedge assuming you have some mismatch bandwidths. But for purely educational purposes you can choose any option. $\endgroup$ Oct 9 '20 at 12:19

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