The following Module is prepared from the research paper titled “Series Expansion of the SABR Joint Density” by Dr. Qi Wu. The paper was published in the Mathematical Finance journal in November 2010.

The paper provides a series-expansion form of equation which is used price and hedge contracts that are sensitive to forward smile risk (e.g., barrier options, Swaptions).

Series Expansion of the SABR Joint Density

Current Price
Current Vol [0-1]
rho [-1,1]
Time to Maturity (year)
Vol_of_Vol [0-1]
Discount Rate

Pricing of forward starting instruments require the joint transition density function of that asset/instrument.In the aforementioned paper, the author constructed an expansion of the joint density through a hierarchy of parabolic equations after applying total volatility-of-volatility scaling and a near-Gaussian coordinate transformation. Then an existence result was established to characterize the truncation error and explicit joint density formulas for the first three orders were provided. The results for the joint transition density serve as a basis for managing forward smile risk. Through numerical calculations, our implemented module provides the joint density and marginal density for European call options.

This Module Implemented by Md. Atiqul Islam Mollah & Md. Mizanur Rahman nur