Deriving Heston Option Delta from Homogeneity

# Delta of an option under Heston model


# Delta of an option under Heston model












I am studying the Heston model. I have not had time to read the detailed derivation of the formula to compute option prices. The formula is given according to this thread: Heston Model Option Price Formula

I want to compute the delta of the option priced under Heston model: By looking at this expression, having read the thread on the link it seems the $S_t$ does not make part of $P_1$ and $P_2$ \begin{align} C(t\,,{{S}_{t}},{{v}_{t}},K,T)={{S}_{t}}{{P}_{1}}-K\,{{e}^{-r\tau }}{{P}_{2}} \end{align} So that I could take the direct derivative:

\begin{align} \frac{\partial C(t\,,{{S}_{t}},{{v}_{t}},K,T)}{\partial S_t}=\frac{\partial{{S}_{t}}{{P}_{1}}-K\,{{e}^{-r\tau }}{{P}_{2}}}{\partial S_t}=P_1 \end{align}

I do not know if it is this simple. I read that I should use some "Homogeineity property" but I do not know what that means.

Question:

Couls someone help me on the computation of the delta of the option price under Heston model?

## Answer by Kevin (score 14, accepted)

https://quant.stackexchange.com/a/61852

### Bad news: Your calculation is not quite correct

As you say, the initial price of a European call option is $$C(S_0;K,T)= S_0e^{-qT}\Pi_1-Ke^{-rT}\Pi_2. \tag{$\star$}$$ However, the exercise probabilities $\Pi_1$ and $\Pi_2$ depend on the stock price $S_0$ too! Thus, you need the product rule and the chain rule to differentiate the option price with respect to $S_0$. The same problem applies to the calculation of delta in the Black-Scholes model. This makes the calculation a bit lengthy, see here.

Note that formula $(\star$) applies to many models, not just the Black-Scholes model and the Heston model. The formula equally applies to the CEV model, the jump-diffusion models from Merton and Kou, pure jump processes (e.g. variance gamma model), etc. etc. It is a consequence of the change of numéraire technique.

### Good news: Option prices are homogeneous of order one

Suppose the stock price is modelled as $S_t=S_0e^{X_t}$, where $X_t$ is a stochastic process normalised to $X_0=0$ which does not depend on $S_0$. Thus, doubling today's stock price also doubles future stock prices. While not every model satisfies this property, a great deal do (e.g. all the ones I mentioned above). Recall that risk-neutral pricing suggests \begin{align*} C(S_0;K,T)=e^{-rT}\mathbb{E}^\mathbb{Q}_0\left[\max\{S_T-K,0\}\right]. \end{align*} Homogeneity of order one simply means that for any $\lambda>0$, \begin{align*} C(\lambda S_0;\lambda K,T)=e^{-rT}\mathbb{E}^\mathbb{Q}_0\left[\max\{\lambda S_T-\lambda K,0\}\right]=\lambda C(S_0;K,T). \end{align*} Differentiating both sides with respect to $\lambda$ (using the multivariate chain rule) gives $$ S_0\frac{\partial C}{\partial S_0}+K\frac{\partial C}{\partial K}=C. \tag{$\star\star$}$$ Comparing the coefficients in Equations ($\star$) and $(\star\star$), we get \begin{align*} \frac{\partial C}{\partial S_0} &= e^{-qT}\Pi_1>0,\\ \frac{\partial C}{\partial K} &= -e^{-rT}\Pi_2<0. \tag{$\star\star\star$} \end{align*}

### Some notes

Because $\Pi_1$ and $\Pi_2$ are probabilities and bounded between 0 and 1, we know that so is a call option's delta (ignoring dividend yields). You can use the put-call-parity to get a similar result for European put options. The calculation above is closely linked to Euler's Theorem on homogeneous functions. If you calculate $\Pi_1$ as an improper integral of the characteristic function of the log-stock price, $\varphi$, you can compute delta explicitly via $\frac{\partial \varphi(u)}{\partial S_0}=\frac{iu}{S_0}\varphi(u)$, which holds for homogeneous stock price models.

Equation ($\star\star\star)$ links the risk-neutral distribution function, $\Pi_2$, to an (observable) derivative of call option prices. Differentiating this equation once more with respect to the strike price $K$ yields the celebrated result from Breeden and Litzenberger (1978).