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 !
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$\begingroup$ Note that I've edited the title of your question because: "Black-Scholes with stochastic volatility" makes little sense. As soon as volatility is not constant, we cannot talk about "Black-Scholes". $\endgroup$– QuantupleMay 10, 2016 at 9:54
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2 Answers
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[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.
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1$\begingroup$ Glad it helped, maybe I should have mentioned that local volatility models allow you to perfectly fit all your observed vanilla option prices (while stochastic volatility models will have a hard time fitting them). But this comes at a cost: the behaviour of future implied volatility smiles are not realistic under the local volatility assumption. Also note that one nasty feature of stochastic volatility models is that they are not complete (hence the PDE will have an additional term accounting for the market price of volatility/variance risk), which is not the case in local volatility models. $\endgroup$ May 10, 2016 at 10:04
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1$\begingroup$ What do you mean by "one nasty feature of stochastic volatility models is that they are not complete" ? $\endgroup$– glorkMay 10, 2016 at 10:19
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1$\begingroup$ I'm talking about market completeness, since volatility is not a directly tradable asset but has its own source of stochasticity, this creates a non directly hedgable source of risk which needs to be accounted for somehow. More specifically, when deriving the BS PDE, the argument was that if you could continuously trade in the underlying in a certain proportion $\Delta_t=\partial V/\partial S(t)$ then your PF would be immune to fluctuations of the spot and hence evolve at a risk-free rate. Now that volatility kicks in, (...) $\endgroup$ May 10, 2016 at 10:35
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1$\begingroup$ (...) since there is not a single way to immunise your PF against volatility fluctuations because the market is not complete (you could use variance swaps, options etc.), this leads to an additional premium term. This is for instance rapidly discussed in the original Heston article when the author talks about moving from the physical measure to the risk-neutral measure. $\endgroup$ May 10, 2016 at 10:37
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1$\begingroup$ I'm talking about a self-financing hedging portfolio $\Pi_t = V_t - \Delta_t S_t$, if you apply Itô assuming $S_t$ follows a $GBM(.,\sigma^2)$ this gives you: $d\Pi_t = \frac{\partial V_t}{\partial t} dt + \frac{1}{2}\frac{\partial^2 V_t}{\partial S_t^2} \sigma^2 S_t^2 dt + (\frac{\partial V_t}{\partial S_t} - \Delta_t)dS_t$. Now, you see that the last term disappears if $\Delta_t=\frac{\partial V}{\partial S_t}$. $\endgroup$ May 11, 2016 at 14:45
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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).
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1$\begingroup$ Although I completely agree, I will play devil's advocate by saying that this might give the wrong idea to the OP. The equations you wrote are model-independent indeed. However, to be of any practical use, they require a good command of quadrature techniques + complex analysis nothing like the BS formula. $\endgroup$ May 10, 2016 at 13:16
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$\begingroup$ Tractability depends on what $\sigma_t$ follows. $\endgroup$– KiwiakosMay 10, 2016 at 15:27
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1$\begingroup$ Naturally. But except for VG (which, being a pure jump process, may not be what the OP was looking for given the SDE he provided), Lévy models with stochastic clock do not produce closed-form formulas that would be as easy to use as the BS pricing formula (numerical integration is lurking). $\endgroup$ May 10, 2016 at 16:10
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