Positive Stochastic Collocation
In the context of my thesis, I explored the use of stochastic collocation to capture the marginal densities of a positive asset. Indeed, most financial asset prices must be non-negative. But the classic stochastic collocation towards the normally distributed random variable, is not.
A simple tweak, proposed early on by Grzelak, is to assume absorption and use the put-call parity to price put options (which otherwise depend on the left tail). This sort of works most of the time, but a priori, there is no guarantee that we will end up with a positive put option price. As an extreme example, we may consider the case where the collocation price formula leads to \(V_{\textsf{call}}(K=0) < f \) where \(f \) is the forward price to maturity. The put-call parity relation applied at \(K=0 \) leads to \(V_{\textsf{put}}(K=0) = V_{\textsf{call}}(K=0)-f < 0 \). This means that for some strictly positive strike, the put option price will be negative, which is non-sensical. In reality, it thus implies that absorption must happen earlier, not at \(S=0 \), but at some strictly positive asset price. And then it is not so obvious to chose the right value in advance.
In the paper, I look at alternative ways of expressing the absorption, which do not have this issue. Intuitively however, one may wonder why we would go through the hassle of considering a distribution which may end up negative to model a positive price.
In finance, the assumption of lognormal distribution of price returns is very common. The most straighforward would thus be to collocate towards a lognormal variate (instead of a normal one), and use an increasing polynomial map from \([0, \infty) \) to \([0, \infty) \). There is no real numerical challenge to implement the idea. However it turns out not to work well, for reasons explained in the paper, one of those being a lack of invariance with regards to the lognormal distribution volatility.
This did not make it directly to the final thesis, because it was not core to it. Instead, I explore absorption with the Heston driver process (although, in hindsight, I should have just considered mapping 0 to 0 in the monotonic spline extrapolation). I recently added the paper on the simpler case of positive collocation with a normal or lognormal process to the arxiv.