Graduate Students Seminar

Speaker 1: Justin Towson

Title

The LIBOR Market Model (LMM) family of models and their use in interest rate risk management and hedging

Abstract

The LIBOR Market Model (LMM) and its variations are interest rate models that have been a standard in the finance and insurance industries since the economic crises during 2008-12. This class of models makes extensive use of Girsanov's Theorem and change-of-measure techniques to translate between current and future prices for modeling future cash flows using stochastic simulation. The set of modeled interest rates over time are then used in cash flow projection or actuarial models to value expected liabilities as well as to price derivatives and sensitivities to changes in inputs to the Black-Scholes formula (i.e. Greeks). Compared to the previous generation of interest rate models (e.g. Vasicek or Hull-White), in market models volatility smile is computationally easier to calibrate by allowing a term-structure of volatility targets over time and option strikes. Optimization techniques can then be used to calibrate the Local Volatility matrix to market data inputs. Prior to the COVID-19 pandemic, there has been a persistent negative interest rate environment in the Japanese and European economies for the past two decades and even in US real rates for the last few years. Since the LMM is a lognormal model of interest rates based on Black-Scholes volatilities, recent developments have incorporated a displacement term with the Displaced Diffusion LIBOR Market Model (DD-LMM) to shift negative negative rates to positive rates which can then be used as inputs. For more challenging market volatility term structures, there are two major classes of Stochastic Volatility LIBOR Market Models (SV-LMM) which fall into the Heston-family and SABR-family of SV models. The Heston model incorporates jump diffusion Poisson processes and a mean reverting term, though there has been some recent research on incorporating a mean reverting term in SABR volatilities as well. The LMM is used primarily to model risk-free interest rates such as government bonds or swaps, though can be incorporated into hybrid excess volatility models for modeling risky assets under risk neutrality. Cross-currency considerations also motivate the need to model interest rates in a domestic currency and price instruments in a foreign currency which also leads to the derivation of the class of Inflation Market Models (IMM). Beyond theoretical considerations, there are also many practical considerations during the calibration process which make up much of the body of current research.

Speaker 2: Lara Scott

Title

Chemoattractant Distribution: Analytical and Numerical Solutions to a PDE

Abstract

We use Drosophila egg chambers to study cell migration by creating a geometrically-realistic model of the egg chambers and then using that to determine the distribution of chemoattractants using a reaction-diffusion system. We have solved a one-dimensional reaction-diffusion partial differential equation analytically and plotted the analytical solution as well as an explicit finite difference and finite element solutions in Matlab. We then generalized the code to solve a two-dimensional reaction-diffusion partial differential equation so that we may eventually use Matlab to solve the partial differential equation using the actual geometry of the Drosophila egg chamber.