Large-time option pricing using the Donsker-Varadhan LDP-correlated stochastic volatility with stochastic interest rates and jumps. (English) Zbl 1357.91047

Forde, Martin; Kumar, Rohini

Ann. Appl. Probab. 26, No. 6, 3699-3726 (2016).

Summary: We establish a large-time large deviation principle (LDP) for a general mean-reverting stochastic volatility model with nonzero correlation and sublinear growth for the volatility coefficient, using the Donsker-Varadhan LDP [M. D. Donsker and S. R. S. Varadhan, Commun. Pure Appl. Math. 36, 183–212 (1983; Zbl 0512.60068)] for the occupation measure of a Feller process under mild ergodicity conditions. We verify that these conditions are satisfied when the process driving the volatility is an Ornstein-Uhlenbeck (OU) process with a perturbed (sublinear) drift. We then translate these results into large-time asymptotics for call options and implied volatility and we verify our results numerically using Monte Carlo simulation. Finally, we extend our analysis to include a CIR short rate process and an independent driving Lévy process.

MSC:

91G20	Derivative securities (option pricing, hedging, etc.)
60F10	Large deviations
60H30	Applications of stochastic analysis (to PDEs, etc.)
60J60	Diffusion processes
60J25	Continuous-time Markov processes on general state spaces
60G51	Processes with independent increments; Lévy processes
91G60	Numerical methods (including Monte Carlo methods)
65C05	Monte Carlo methods

Keywords:

stochastic volatility model; large deviations; implied volatility asymptotics; ergodic processes; occupation measures; Ornstein-Uhlenbeck process; Lévy process

Citations:

Zbl 0512.60068