TERM STRUCTURE MODEL IMPLEMENTATION CALIBRATION AND VALIDATION 2016

TERM STRUCTURE MODEL IMPLEMENTATION, CALIBRATION, AND VALIDATION 2016 MFM WINTER MODELING WORKSHOP, MASTER OF FINANCIAL MATHEMATICS, UNIVERSITY OF MINNESOTA MENTOR: MARCO PERZICHILLI, PARTICIPANTS: XINYUAN JIANG, CHONG WANG, VARUN SOOD, XIYAO CHENG, UJAE KANG, KAKA WANG

DERIVING SIMPLY COMPOUNDED SPOT INTEREST RATE: • The simply-compounded spot interest rate prevailing at time t for the maturity T is denoted by L(t, T) and is the constant rate at which an investment has to be made to produce an amount of one unit of currency at maturity, starting from P(t, T) units of currency at time t, when accruing occurs proportionally to the investment time. in formulas a: a. Brigo D. , Mercurio F. Interest Rate Models - Theory and Practice (Springer, 2006)(ISBN 3540221492)(1014 s)

Short rate and simulating plausible distribution if it. • A short-rate model, in the context of interest rate derivatives, is a mathematical model that describes the future evolution of interest rates by describing the future evolution of the short rate, usually written rt. • In order to simulate plausible distribution of short-rates, we used G 2++ model.

Short rate and simulating plausible distribution if it. (Continued) • Where phi(t) is calculated as

Simulation of the short rate… • The terminal short rate at the end of 5 years for the 10 year maturity rates can be simulated…

At year 5, the distribution of the rt • It fits very closely to the normal distribution with 1, 000 iterations (Red is normally fitted curve, Blue is simulated output.

Calibrating the parameters of G 2++ • We use the latest market caplet information to calibrated the parameters of G 2++.

Extracting implied volatility from G 2++ model then calibrate G 2++ model to tie out to the market volatility implied in the at the money caplet • Below formula is used to derive caplet price based on the G 2++ model.

Practical Application: using Excel Solver, one can draw parameters of G 2++ model. Below is the results of the simulation. On the left hand side is based on the latest market caplet’s implied volatility, on the right is an academic stress test scenaro with much higher implied volatility. (10 year short rate at the end of 5 th year)

Support Materials • BRIGO D. , MERCURIO F. INTEREST RATE MODELS - THEORY AND PRACTICE (SPRINGER, 2006)(ISBN 3540221492)(1014 S)_FB • Marco Perzichilli’s VBA codes related to key financial math equations and simulation algorithm.