May
3

## On the Number of State Variables in Options Pricing

Read an interesting paper last week "On the Number of State Variables in Options Pricing" by Gang Li, Chu Zhang. As the title suggests, this paper is trying to identify how many state variables are good enough to price an option, ideally the less variables the better.

The authors first review a few popular models for option pricing such as Black Scholes model, GARCH option pricing, and Stochastic volatility models, then they argue two possible sources of model misspecification, one is the omitted state variables, or factors, for instance, should we consider volatility smile? should we include jump into our pricing equation, etc. The other source of model misspecification is the functional form of the process for the state variables, including the specification of risk premiums associated with the state variables, this misspecification may be especially prone to error, or in another term, easily leads to model risk. Square root process or simple mean-reversion? or a combination of these two as some literature suggest.

In order to identify the necessary number of factors, the authors then use a nonparametric approach with state variables approximated by

By applying this methodology to S&P 500 index option, the authors find two factors are fairly enough, their results suggest that for S&P 500 options, adding jumps to the one-factor model with jump intensity and jump sizes is not enough, extending the volatility process to higher dimensions than two is of little use either. Therefore a promising direction to model options is to improve the specification of the two factor model.

Below is a residual analysis graph captured from the paper,

where M0 means without additional state variable, M1, M2 and M3 means one, two and three state variables, respectively, obviously for this example, two state variables perform very well, better than one state variable, and the extra gain of three variables is very small.

Should you are interested the nonlinear principal components analysis, I shared a Matlab toolbox at the post Nonlinear PCA toolbox, enjoy.

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The authors first review a few popular models for option pricing such as Black Scholes model, GARCH option pricing, and Stochastic volatility models, then they argue two possible sources of model misspecification, one is the omitted state variables, or factors, for instance, should we consider volatility smile? should we include jump into our pricing equation, etc. The other source of model misspecification is the functional form of the process for the state variables, including the specification of risk premiums associated with the state variables, this misspecification may be especially prone to error, or in another term, easily leads to model risk. Square root process or simple mean-reversion? or a combination of these two as some literature suggest.

In order to identify the necessary number of factors, the authors then use a nonparametric approach with state variables approximated by

**nonlinear principal components**extracted from the implied volatilities. Nonparametric approach helps to overcome the problem of function form misspecification, and**nonlinear principle component**helps to demonstrate the explanatory power of each factor, similar with a typical principle component analysis except the former is able to capture the nonlinear relationship among observation series, which is obviously the case for the implied volatilities.By applying this methodology to S&P 500 index option, the authors find two factors are fairly enough, their results suggest that for S&P 500 options, adding jumps to the one-factor model with jump intensity and jump sizes is not enough, extending the volatility process to higher dimensions than two is of little use either. Therefore a promising direction to model options is to improve the specification of the two factor model.

Below is a residual analysis graph captured from the paper,

where M0 means without additional state variable, M1, M2 and M3 means one, two and three state variables, respectively, obviously for this example, two state variables perform very well, better than one state variable, and the extra gain of three variables is very small.

Should you are interested the nonlinear principal components analysis, I shared a Matlab toolbox at the post Nonlinear PCA toolbox, enjoy.

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