Pricing European Calls under Stochastic Volatility

# Stochastic volatility


# Stochastic volatility












Suppose we have : $\frac{dS_{t}}{S_{t}}= \sigma dW_{t}$ with $\sigma_{t}$ a stochastic volatility process. How to compute $\mathbb{E}^{Q}[(S_{T}-K)+]$ ? Is there a BS alike formula : "$S_{0}N(d+)-Ke^{-rT}N(d-)$" ? Tx !

## Answer by Quantuple (score 7, accepted)

https://quant.stackexchange.com/a/25925

[Short answer]

No closed-form formula in general. You need to resort to numerical methods. Monte Carlo is preferred by most practitioners but you could also use Finite Difference schemes (and sometimes even Fourier inversion techniques depending on the model used and the instruments to be priced).

[Long answer]

One usually distinguishes between 2 classes of (pure diffusion) models characterised by the SDE (*)

$$ d S_t = \sigma_t S_t dW_t $$

- local volatility models are such that $\sigma_t := \sigma(t,S_t)$, see seminal work of Dupire in that area. Except for the degenerate case $\sigma_t = \sigma(t)$, no closed form formulas exist and one has to resort to numerical methods such as Finite Differences schemes to solve the pricing PDE (which can be shown to be a straightforward generalisation of the Black-Scholes PDE) or Monte Carlo to simulate paths of the process $(S_t)_{t\geq 0}$ by discretising the SDE mentioned above (there are numerous possible discretisation schemes).

- stochastic volatility models where $\sigma_t$ (or sometimes the instantaneous variance $v_t = \sigma_t^2$) possesses its own source of stochasticity - i.e. its own driving Brownian motion, correlated or not to that which drives the spot price - you therefore find yourself with a system of SDEs, one for the stock price, one for the volatility/variance, see seminal work of Heston, Schobel-Zhu, Stein & Stein and many others in that area. Although Finite Difference and Monte Carlo methods can be used for stochastic volatility models as well, these models were first made popular because they allowed to derive semi closed-form solutions expressed as Fourier transforms for simple instruments (typically European vanilla options and forward starts). These Fourier inversions can be made blazing fast: much faster than any Monte Carlo or Finite Difference scheme. Still, this will never be as fast as a plain evaluation of the BS pricing formula.

(*) There exist other types of diffusion models that are not described by the SDE mentioned above, notably the class of time-changed Lévy models mentioned in @Kiwiakos answer.

## Answer by Kiwiakos (score 2)

https://quant.stackexchange.com/a/25926

The answer is yes. In fact, there always exist a 'Black Scholes like' formula. Easy to show too. If the risk neutral distribution of the price has cumulative density $P$ and probability density $p$, then

$$ E(S-K)^+=E((S-K)\ 1_{S>K})=E(S\ 1_{S>K})-K\ E(1_{S>K}) $$

The second expectation is just $P(K)$, ie the probability that the option ends up in the money.

The first expectation is a bit trickier, but it can be written as $\int_K^\infty s p(s) ds$. The trick is to multiply and divide with $S_0$ and acknowledge that since under the risk neutral measure the price is a martingale, then $S_0=\int_0^\infty s p(s) ds $.

Then the first expression is written as $$ E(S\ 1_{S>K})=S_0 \frac{\int_K^\infty s p(s) ds}{\int_0^\infty s p(s) ds} = S_0 P^*(K) $$ You can confirm that the fraction (as a function of the strike ie $P^*$ above) is indeed a cumulative density. It is positive, increasing, and integrates to one. It is actually the Delta of the option.

And we have written the price in a 'Black-Scholes like' way as $$ C = S_0 P^*(K) - K\ P(K) $$ Spot times Delta minus Strike times Exercise Probability.

This is the expression Heston gives in his paper, which was the first with a semi-closed form for stoch vol. Then it was generalised in Bakshi and Madan. Also this expression is widely used for Levy models, which are subordinated Brownian motions (ie random volatility). VG, NIG, etc.

In practice of course the distribution might not be readily available, and we might need to compute the quantities numerically. Or take other shortcuts that do not exploit this representation (eg Carr Madan).