# 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?

• @Kevin It is perfect! Thanks! – Pedro Gomes Mar 20 at 22:48

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).