Risk Latte - Asset Price Dynamics: Which Stochastic Process explains the Volatility Skew best?

Asset Price Dynamics: Which Stochastic Process explains the Volatility Skew best?

Team Latte
December 16, 2010

In a recent interview, the interviewer – who’s a Managing Director and the Head of Trading for Equity Derivatives in a European bank – asked the following question to the interviewee: if you were using Monte Carlo simulation and valuing an exotic payoff on an equity index which displays strong volatility skew then which stochastic process would you use?

What is the correct answer? We all know that Black-Scholes model is inadequate to explain the volatility smile and the skew. In fact, geometric Brownian motion (GBM), on which the Black-Scholes model is predicated, is itself quite inadequate to explain the volatility skew that is observed in the options market.

There is a process called a Square Root stochastic process, which also goes by the name of Cox-Ross (CR) process that describes the observed volatility skew the best. In recent, years many practitioners have started using this process to value Asian (arithmetic average) options and some other exotic equity derivatives payoffs. CR is very similar to the Cox-Ingersoll-Ross (CIR) square root process used in rates modeling and that is indeed the basis for Heston’s stochastic volatility model.

A square root, Cox-Ross process is given by:

Where, is the Weiner process given by. This is an extremely useful process because of the additive property because of which it is a desirable process for valuation of Asian options. But that is not the most important property of a Cox-Ross process. The most important property of a Cox-Ross process is that it explains the volatility skew observed in the market. How?

Let’s write the above square root process differently and express the left hand side as a return process (as opposed to a change in price process).

It is obvious from the above return process that the instantaneous variance of the return is equal to . This is an inverse function of the equity (asset) price. Thus, lower the equity (asset) price the higher the variance and vice-versa. This fact is empirically found to be true.

Therefore, the Cox-Ross square root process is a better process than a geometric Brownian motion (GBM) when it comes to explaining the volatility skew.

In fact, along with the square root process, Cox and Ross also introduced the Constant Elasticity of Variance (CEV) stochastic model of asset prices which till this day is popular amongst many FX option traders. CEV process can be viewed as a more generalized form of the Cox-Ross process.

However, having said all this it must be mentioned that besides valuing Asian options, the Cox-Ross square root process has not been used extensively in valuation of other exotic equity option payoffs. Of course, the CIR and the CEV processes both are still quite popular and used by many quants to value rates and FX options respectively. But in the equity arena, somehow the application of the square root process to asset price has not caught on. Heston’s stochastic volatility model, which is now being used by equity quants, on the other hand uses the square root process but applied to the variance of the asset price.

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