JUMP DIFFUSION AND STOCHASTIC VOLATILITY
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Derivatives

JUMP DIFFUSION AND STOCHASTIC VOLATILITY

The Black-Scholes model makes the assumption that volatility of the underlying is constant.

The Black-Scholes model makes the assumption that volatility of the underlying is constant. However, by observing asset movements in the market place it is evident that this is not the case. The following diagram illustrates typical market implied volatilites for foreign exchange, commodities and equities versus the flat Black-Scholes assumption. The shape of the implied volatility is commonly known as a smile.

 

 

 

 

 

 

 

 

 

 

To model a non-constant volatility, one must leave the lognormal Black-Scholes framework. A much more general approach treats instantaneous (local) volatility as deterministic, but we know from common experience that volatility is not deterministic. To go beyond this, we must treat volatility itself as stochastic: the Heston model takes this approach.

In the Heston model we have two stochastic processes, one for the asset process, the other for variance, as follows:

 

 

 

 

 

 

 

 

and there is a correlation * between the two driving Brownian motions W. * is the market price of volatility risk.

The Heston model is mean-reverting and allows for a correlation between the volatility and asset level. A typical path for both processes can be seen in the following diagram (volatility and asset strongly anti-correlated):

 

 

 

 

 

 

 

 

 

 

The parameters required for determining the Heston process, that is to say, short vol, long vol, reversion speed, vol of vol and correlation, can be calibrated to market data.  

As an alternative to stochastic volatility in the attempt to replicate the smile, the asset price can be modeled as a jump-diffusion, as specified by Merton:

 

 

where * is a deterministic volatility parameter, and M is defined as

 

 

 

 

and where N is a Poisson (step) process whose intensity µ determines the frequency and instants of jump events. The *i are independent standard Gaussian variables; * and * define, respectively, the mean size and the standard deviation of the jump amplitude.

The jump-diffusion model behaves like the Black-Scholes model, but the paths are at some random times instantly shifted by a random amount, each shift being independent from the past. A typical path looks like (the jumps are on average negative):

 

 

 

 

 

 

 

 

 

This model features fast computation of European options prices and hedge ratios and provides a good fit to the smile for short maturities. The parameters needed to determine the model, that is to say, volatility, jump intensity, jump size mean and variance, can be obtained via calibration to market data. A drawback of this model is that the smile rapidly becomes flat as maturity increases.

To remedy this situation, the next step is to couple the jump-diffusion and the Heston models, also allowing for jumps in the volatility. The dynamics of the asset price is as above and volatility is then specified as follows:

 

 

where N is the Poisson process introduced earlier and j>0 is the (fixed) size of the jumps in the volatility. The jumps are taken to be positive to ensure the volatility never becomes negative. Calibration to market implied volatilities again specifies the model parameters.

The Heston-Jump-diffusion (HJD) model combines the advantages of stochastic volatility and jump-diffusion models. In fact, the individual advantages of one tend to correct the weaknesses of the other. The individual effect of jumps and stochastic volatility is shown in the diagrams below.

 


 

 

 

 

 

 

 

 

The plot above is obtained by subtracting the implied volatilities obtained in the Heston model from those obtained in the complete HJD model. It is clear that the effect of the jumps in the HJD model is a strong skew for short maturities. Term structure is absent, as the smile flattens very quickly as maturity increases. This was expected, since the jump-diffusion model produces this kind of implied volatility surface.

The plot below results from the same operation as the former, except the implied volatilities obtained from a pure jump diffusion model are subtracted from those coming from HJD. The effect of Heston dynamics is thus to give a global orientation to the implied volatility surface, as well as some term structure.

The effects of jumps and stochastic volatility in the HJD model are non-overlapping and complement each other, correcting the weaknesses of each model. This model has the important property that it features both a strong skew for short maturities and a persistent smile effect in time. This is essential in order to price consistently those products that depend on both the volatility surface today and in the future. Missing one of these could lead to mispricing these options and taking unexpected risk. The effect of volatility jumps is mainly to change the overall level of the volatility surface, an effect that is accumulated with time. Empirical research on implied volatility for different indices shows that the HJD model outperforms both the Heston model and the jump-diffusion model in the replication of the implied volatility surface.

More general results can be obtained: one can introduce jumps in the asset price as before and jumps in the volatility process of two varieties: one is completely independent of the asset price process, the other happens with some probability when there is a jump in the underlying asset. All the jumps can have random size, but one has to ensure that the volatility always remains positive. Such a model allows market participants to fit the entire implied volatility surface without a need to separately fit between short-term and long-term options.

 

This week's Learning Curve was written by Marcus Overhaus, managing director and global head of quantitative research at Deutsche Bankin London.

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