Black Scholes implied vol of SVJ model

Question

Under the SVJ model https://en.wikipedia.org/wiki/Stochastic_volatility_jump, what is the formula of the Black Scholes (log-normal) implied vol for an option with strike $K$ and time to maturity $T$ (assume the spot price is $1$)?

Answer 1

In general, any European payoff (eg. European option) can be priced by integrating the payoff over the density of the log-returns of the underlying. However, the SVJ model does not omit any closed-form density and therefore we cannot derive any analytical pricing formula. But, it does have a closed-form characteristic function, which can be used to recover the density via the Fourier inverse transform. This is the premise of Fourier pricing.

For the pricing of European options, we have a selection of methods for deriving the call price, but for brevity we will work with Lewis approach where the Fourier transforms are expressed in terms of the characteristic function of the log price. Therefore under the assumption of constant interest rate $r$ and dividend yield $q$, the call price of a European option under the SVJ model can be found via: \begin{equation} C_{SVJ}(S,K,t,T) = Se^{-q(T-t)} - \frac{\sqrt{SK}e^{-\frac{(r+q)(T-t)}{2}}}{\pi} \int_0^\infty \frac{Re\left(e^{iuk} \Phi(u-\frac{i}{2},t,T) \right)}{u^2+\frac{1}{4}}\: du \end{equation} for $S$ being the spot price at time $t$, K is the strike with $k=\log(S/K)+(r-q)(T-t)$ and $T$ is the maturity. Furthermore, $\Phi(\cdot)$ is the characteristic function of the standardized log-price for the SVJ model, $X_T=\log(S_T/S_t)-(r+q)(T-t)$. Now, to find the implied volatility of the SVJ model you simply equate the call prices of the SVJ model with the call prices for Black-Scholes model: \begin{equation} C_{SVJ}(S,K,t,T) = C_{BS}(S,\sigma,K,t,T) \end{equation} and solve for $\sigma$. There is no analytical solution to this problem and one has to use numerical solvers to find the implied volatility surface of the SVJ model. If you calculate the BS call prices following the Lewis approach, then you can subtract the Black-Scholes call prices on both sides, which nets you with:

$${ \int_0^\infty \frac{Re\left(e^{iuk} \left[\Phi_{SVJ}(u-\frac{i}{2},t,T)-\Phi_{BS}(u-\frac{i}{2},t,T,\sigma)\right] \right)}{u^2+\frac{1}{4}}\: du = 0,}$$

which is probably the closest thing you can get, for an equation of the implied volatility. The above equation gives us a simple but implicit relationship between the implied volatility surface and the characteristic function of the underlying stock process. This is also described in Gatheral, The volatility surface (around p. 60), where he assumes that $r=q=0$. Again, the only unknown value in the above integral is $\sigma$, which can be found using numerical methods as pointed out in the comment below.

Answer 2

I found the following paper which answers my question somehow. https://www.scaillet.ch/pdfs/asymptotics.pdf