Integrated Delta and Leverage Behavior in Heston Options

# Integrated Delta does not seem to be smooth (ATM, Heston)


# Integrated Delta does not seem to be smooth (ATM, Heston)












I am interested in an integrated call option that removes the dependence on time, $$I(S)=\int_0^\infty C(S,t)\text{d}t.$$ Because the value of a call option is a smooth function, I expect this integral to be smooth too. By Fubini, the delta of $I$ is just the integrated delta of a normal call option and should be smooth too.

Below I plot the integrated call option value $I(S)$, its delta $\Delta(S)$, the option leverage $\Lambda(S)=\Delta(S)\frac{S}{I(S)}$ and the ratio $\frac{S}{I(S)}$

I consider the Heston model with two initial variances, $v_0=0.01^2$ and $v_0=0.4^2$. The strike price is $K=1$.

Question: While the plots for $I(S)$ and $\frac{S}{I(S)}$ (Panels A and D) look nice, the option leverage in Panel C looks weird (= hump-shaped) for the low variance case at $S_0\approx K$. Why is this? It doesn't look like this in the Black-Scholes case (not shown here) nor in the Heston model with high variance (broken line in Panel C). Panel D suggests that this hump comes from including $\Delta$ in the formula for $\Lambda$ because $\frac{S}{I(S)}$ on its own looks fine. Indeed, Panel B hints to me that Delta does not perfectly ''smooth-paste'' at $S_0\approx K$ (hard to see by the eye).

I understand that for low values of $v_0$, the call is similar to a forward and this may cause problems but (1) for the Black Scholes model, we never get a humped shape even if $\sigma^2$ is extremely close to zero and (2) the mean-reversion of the Heston model would suggest that variance will increase again, so there is much `optionality' left.

Thus, I guess (hope) the humped shape more a computational issue than a model feature. I evaluate the integral $I(S)$ using Matlab's `intgeral2` function (which employs adaptive quadrature) and Lewis' (2001) formula (integrating along the contour $\{z\in\mathbb{C}:\text{Im}(z)=1\}$ and using the Black Scholes case as control variate, re-scale the variance process). I get similar plots if I change the methodology (no control variate, use different contour, compute the probabilities $\Pi_1$ and $\Pi_2$ separately). I don't think that any other numerical technique (2D MC simulation or 3D finite differences) outperforms the pricing by Fourier inversion.

Any ideas of why there is the humped-shaped and how to eliminate it? Finally, it simply doesn't make sense economically if we interpret $\Lambda$ as % change in the option value given a % move in $S$. This ratio shouldn't be humped shaped.

For completeness, I use the model parameters

- $r=0.01$ (interest rate)

- $q=0.03$ (dividend yield; note: $q>0$ is necessary for the integral to remain finite)

- $\kappa=2$ (speed of mean reversion)

- $\theta=0.03$ (long term mean)

- $\xi=0.2$ (vol of var)

- $\rho=-0.7$ (correlation coefficient)