Path Integral and Asset Pricing

Abstract

We give a pragmatic/pedagogical discussion of using Euclidean path integral in asset pricing. We then illustrate the path integral approach on short-rate models. By understanding the change of path integral measure in the Vasicek/Hull-White model, we can apply the same techniques to "lesstractable" models such as the Black-Karasinski model. We give explicit formulas for computing the bond pricing function in such models in the analog of quantum mechanical "semiclassical" approximation. We also outline how to apply perturbative quantum mechanical techniques beyond the "semiclassical" approximation, which are facilitated by Feynman diagrams.

Volatility smile as relativistic effect

“…Functional integration was originally applied to asset (option) pricing in [Bouchaud and Sornette, 1994], and subsequently in [Baaquie, 1997], [Otto, 1998], [Kleinert, 2002], etc. See [Kakushadze, 2015b] for a more detailed list.…”

Coping With Negative Short Rates

“…12 One way to deal with this is to consider short-rate models of the form r t = r 0 f (X t )/f (0), where f (y) is a positive function, e.g., f (y) = exp(y), which is the Black-Karasinski model. The path integral treatment of such models was discussed in (Kakushadze, 2014). …”

“…The fit in Model-2, which has fewer (to wit, 3) parameters than Model-1, 32 is actually better than in 26 Equivalently, we can simply set t = 0, so χ(t) = r 0 . 27 Typically, there are not that many maturities available, so reconstructing χ(s) = ∂ s η(t, s) + χ(t) and ν(s) =χ(s) would involve piecewise polynomial splines.…”

Feynman path integrals and asymptotic expansions for transition probability densities of some Lévy driven financial markets

“…In [4] a general formula to price European path-dependent options on multidimensional assets is obtained and implemented by means of various flexible and efficient algorithms. In a recent paper [12] explicit formulas are given for computing the bond pricing function in Black-Karasinski model in the analog of quantum mechanical "semiclassical" approximation.…”