Last week's Learning Curve discussed several alternative volatility smile models, all with the common feature that the volatility was a known deterministic function. However there is good reason to believe that the volatility smile can be explained by random fluctuations of the volatility. Time series analysis on data from various markets suggests that volatility should be viewed as a random process exhibiting mean reversion. Moreover, volatility seems to be correlated with asset returns.
STOCHASTIC VOLATILITY MODELS
Several models have been proposed for a stochastic volatility process (*t,t * [0,T]).
These processes may be separated into two groups, namely lognormal processes and processes exhibiting mean reversion.
A mean reverting model allowing for a correlation between volatility and asset has been proposed by Heston. In this model the call price can be obtained easily by evaluating a one dimensional integral. The volatility process is specified by a short and a long volatility *0 and *(infinity), the reversion speed * > 0 and a volatility (of volatility) parameter * >= 0,
Assuming *t < *(infinity), this equation implies that, at a later time t+dt, the volatility will on average have moved up, since *2(infinity) - *2t > 0 and the expectation of * * dW is zero. Similarly, the volatility will move down if *t is above the long volatility. The reversion speed determines the strength of the force pulling the volatility *t to the limit *(infinity) as timet increases. Figure 1 shows this effect for various values of *.
Above we showed how short and long volatility determine the term structure of implied volatility. Now the relationship between the volatility smile and the correlation * between asset and volatility is explained. Assume first that there is a positive correlation. Then positive asset returns will on average be coupled with an increase of volatility which in turn leads to a fatter right tail of the asset distribution: The implied volatility curve increases with strike. This is a typical pattern for commodities markets where volatility decreases with increasing asset prices. Equity markets, however, exhibit negative skewness which may be reformulated in terms of a negative correlation between assets and volatility. Exchange rate markets generally do not have a pronounced correlation. This behaviour is evidenced in figure 2.
The volatility parameter * fattens both tails symmetrically and thus effects the kurtosis of the implied volatility as illustrated in figure 3.
COMPARISON OF MODELS
In a market where volatility smiles are becoming more important and new products are emerging which attempt to capture non-Black Scholes effects it is crucial to have volatility smile pricing models. We do not believe that any existing model can capture all the properties of a real market situation, and it is advisable for a trading desk to have several alternative in place.
Stochastic volatility models share the disadvantage that, since volatility is not a tradable asset, there is no hedging strategy eliminating all market risk related with volatility. In calibrating these models, one has to choose a free risk premium parameter. Deterministic volatility models are complete and are therefore independent of these risk preferences.
However non stochastic volatility models do not capture the dynamics of volatility.
There are products for which this dynamics is crucial. Two examples are the currently popular volatility swaps and options on cliquet (ie. sum of forward-started European options) structures. For these products a stochastic volatility model appears to be more suitable.
Deterministic volatility models can in principle be fitted to any given market situation thus providing a consistent pricing and hedging scheme. In contrast, stochastic volatility models, allowing only a limited number of parameters, can only approximately match market prices. The latest developments in the market are on combining features of both these models.
This week's Learning Curve was written by Oliver Brockhaus and Douglas Long, quantitative analysts in the equity derivatives group at Deutsche Bankin London.
