# 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)