Smart expansion and fast calibration for jump diffusion

Using Malliavin calculus techniques, we derive an analytical formula for the price of European options, for any model including local volatility and Poisson jump process. We show that the accuracy of the formula depends on the smoothness of the payoff function. Our approach relies on an asymptotic expansion related to small diffusion and small jump frequency/size. Our formula has excellent accuracy (the error on implied Black-Scholes volatilities for call option is smaller than 2 bp for various strikes and maturities). Additionally, model calibration becomes very rapid.asymptotic expansion; Malliavin calculus; volatility skew and smile; small diffusion process; small jump frequency/size

Pricing Inflation and Interest Rates Derivatives with Macroeconomic Foundations

I develop a model to price inflation and interest rates derivatives using continuous-time dynamics linked to monetary macroeconomic models: in this approach the reaction function of the central bank, the bond market liquidity, and expectations play an important role. The model explains the effects of non-standard monetary policies (like quantitative easing or its tapering) on derivatives pricing. A first adaptation of the discrete-time macroeconomic DSGE model is proposed, and some changes are made to use it for pricing: this is respectful of the original model, but it soon becomes clear that moving to continuous time brings significant benefits. The continuous-time model is built with no-arbitrage assumptions and economic hypotheses that are inspired by the DSGE model. Interestingly, in the proposed model the short rates dynamics follow a time-varying Hull-White model, which simplifies the calibration. This result is significant from a theoretical perspective as it links the new theory proposed to a well-established model. Further, I obtain closed forms for zero-coupon and year-on-year inflation payoffs. The calibration process is fully separable, which means that it is carried out in many simple steps that do not require intensive computation. The advantages of this approach become apparent when doing risk analysis on inflation derivatives: because the model explicitly takes into account economic variables, a trader can assess the impact of a change in central bank policy on a complex book of fixed income instruments, which is not straightforward when using standard models. The analytical tractability of the model makes it a candidate to tackle more complex problems, like inflation skew and counterparty/funding valuation adjustments (known by practitioners as XVA): both problems are interesting from a theoretical and an applied point of view, and, given their computational complexity, benefit from a tractable model. In both cases the results are promising.Open Acces