Bates Stochastic Volatility Jump model

The Bates and Scott option pricing models were designed to capture two features of the asset returns: 

- conditional volatility evolves over time in a stochastic but mean-reverting fashion

- the presence of occasional substantial outliers in the asset returns 

 

The two models combined the Heston model of SV with the Merton model of independent normally distributed jumps in the log asset price. 

 

The Bates model ignores IR risk, while the Scott model allows IR to be stochastic. 

Both models evaluate European option prices numerically, using the Fourier inversion approach of Heston. 

The Bates model also includes an approximation for pricing American options. 

 

The two models were historically important in showing that the tractable class of affine option pricing models includes jump processes as well as diffusion processes. 

 

In the Bates and Scott models, the risk-neutral processes for the underlying asset St and instantaneous variance Vt, are assumed to be of the form: 

Bates model

 

where b = cost of carry; Zt and Zvt are Wiener processes with correlation p; qt is an integer-valued Poisson counter with risk-neutral intensity Lambda* the counts the occurence of jumps; and k* is the random percentage jump size, with a Gaussian distribution Jump size distribution conditional upon the occurence of a jump. 

 

The Bates model assumes b is constant, while the Scott model assumes it is a linear combination of Vt and an additional state variable that follows an independent square-root process. 

 

Bates examines FX options, for which b is the domestic/foreign interest differential, while Scott examines nondividend paying stock options for which b is the cost of carry and equal to the risk-free IR. 

 

Characteristic function etc....

Related Models

Related affine models can be categorized along four lines: 

1. Alternate specifications of jump processes

2. Bates extension to stochastic-intensity jump processes 

3. Models in which the underlying volatility can also jump

4. Multifactor specifications.

 

 

 

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