# How does out-of-sample option pricing work in practice?

When estimating in-sample option prices, one usually estimates the structural parameters $\theta_t$ using all information up to time $t$, and then prices the option at time $t$ using the obtained parameters and other inputs like the spot price $S_t$ and strike price $X$ etc. The price is then given by: $\hat{c}_t = f(S_t, X, ..., \hat{\theta}_t)$ where $f(\cdot)$ is a function corresponding to a particular option pricing model, such as the Black Scholes formula.

In an out-of-sample framework, the approach described in many papers (see e.g., Baksi et al. 1997) is to first estimates the structural parameters using all information up to time $t$, and then to use these along with the other input variables at time $t+1$. So: $\hat{c}_{t+1} = f(S_{t+1}, X, ..., \hat{\theta}_t)$.

My question is:

1. Is this correct? And are the forecasts genuine out-of-sample forecasts?
2. Why do you use the input variables at time $t+1$? Aren't those unknown? If they were known you could optimize the parameter estimates by using all the information up to time $t+1$; hence the call price $c_{t+1}$ would also be known.
• What do you mean exactly with in and out of sample? For most all vanilla options the market accepted models contain stochastic processes that are martingales and hence to value an option at time t you only need the inputs to the option pricing model at time t. If you look to forecast an option price at t+1 then you also need to predict the inputs to such model for t+1. – Matthias Wolf Jul 19 '13 at 14:29
• I havent seen any paper that discusses how to forecast the inputs, thats why I am wondering. – JohnAndrews Jul 19 '13 at 15:31
• And that is what the question is about. I try to figure out the exact meaning of in and out of sample like I read in almost every paper that evaluates an option pricing model. – JohnAndrews Jul 31 '13 at 15:23

I checked out a paper which deals with out-of-sample option pricing (http://repec.kse.org.ua/pdf/KSE_dp38.pdf, especially following pp. 40-) and I believe it is a sound approach to test whether the addition of structural parameters ads value in pricing capability to more parsimonious models.

Their approach is to

• derive additional parameters (I use the term parameter as in parameterized model, additional in the hopes of obtaining a better fit between the model outputs and actual prices), hoping to derive a model that results in a better fit.
• However, the danger is to overfit such models and they use "out-of-sample" tests in order to verify whether the improvement in fit is merely a function of overfitting or whether the additional parameters display true added forecasting value.
• The structural parameters are fixed to t but all other option pricing model inputs are varied over time when calculating option prices at t+1, t+2,...,t+n, hence they speak of out-of-sample testing of structural parameters.
• The authors' conclusion of this particular paper is that indeed the approach, that alternative models take by inclusion of structural fitted parameters, is a viable way to model option prices aside the general stochastic volatility models. They state that out-of-sample tests have shown that such parameterized models are able to correctly capture the volatility term structure and smile.

In summary, I would answer question1 with a "reserved" yes if you mean with out-of-sample the testing of structure parameters rather than all model inputs (as described above other time varying parameters, such as stock price,..., are not out-of-sample). The second question I would answer in that the out-of-sample structural parameters at t are used to estimate option prices at (t+1,...t+n), and therefore the other parameters are kept varying by t to zero in on the structural parameter fitness, not on the other parameters.

So, my hunch is that the papers you came across focus on the structural parameters, only, when they mean "out-of-sample testing", while you maybe thought that out-of-sample testing includes all model inputs and thus were perplexed why stock prices at t+1...t+n are used to estime option prices over the same t+1...t+n.

• Good comment. In that sense it would make sense that the out-of-sample framework is used to `control' for overfitting and hence to determine whether for instance the additional complexity due to extra parameters is indeed the result of the proposed methodology. – JohnAndrews Aug 4 '13 at 19:06
• Indeed. They just isolated the parameters whose predictive power they want to test. – Matthias Wolf Aug 5 '13 at 2:54