\(\ell_1\)-constrained implied transition densities (Q6049315)

This article presents a practical method of constructing an implied price surface and, more importantly, risk-neutral transition density functions from market option price data, which is the crucial building block of the local volatility surface. Section 2 starts with Fengler's quadratic programming, introduces two additional conditions to Fengler's no-arbitrage conditions. More precisely, the authors construct algorithm using the \(l_1\) norm instead of Fengler's \(l_2\) norm and add two extra constraints (eq. (12), (14)), and one optional constraint (eq. (15)) to Fengler's no-arbitrage constraints. Lastly, the authors recursively construct fitted splines from short to long maturities while avoiding the calendar arbitrage. The extracted risk-neutral transition density functions accurately recover the market option prices. Section 4 reports numerical experiment results.