44 research outputs found
SPECTRAL METHODS FOR VOLATILITY DERIVATIVES
In the first quarter of 2006 Chicago Board Options Exchange (CBOE) introduced, as one of the listed products, options on its implied volatility index (VIX). This opened the challenge of developing a pricing framework that can simultaneously handle European options, forward-starts, options on the realized variance and options on the VIX. In this paper we propose a new approach to this problem using spectral methods. We define a stochastic volatility model with jumps and local volatility, which is almost stationary, and calibrate it to the European options on the S&P 500 for a broad range of strikes and maturities. We then extend the model, by lifting the corresponding Markov generator, to keep track of relevant path information, namely the realized variance. The lifted generator is too large a matrix to be diagonalized numerically. We overcome this diculty by developing a new semi-analytic algorithm for block-diagonalization. This method enables us to evaluate numerically the joint distribution between the underlying stock price and the realized variance which in turn gives us a way of pricing consistently the European options, general accrued variance payos as well as forward-starts and VIX options.Volatility derivatives; operator methods
Asymptotic formulae for implied volatility in the Heston model
In this paper we prove an approximate formula expressed in terms of
elementary functions for the implied volatility in the Heston model. The
formula consists of the constant and first order terms in the large maturity
expansion of the implied volatility function. The proof is based on saddlepoint
methods and classical properties of holomorphic functions.Comment: Presentation in Section 2 has been improved. Theorem 3.1 has been
slightly generalised. Figures 2 and 3 now include the at-the-money point
Continuously monitored barrier options under Markov processes
In this paper we present an algorithm for pricing barrier options in
one-dimensional Markov models. The approach rests on the construction of an
approximating continuous-time Markov chain that closely follows the dynamics of
the given Markov model. We illustrate the method by implementing it for a range
of models, including a local Levy process and a local volatility
jump-diffusion. We also provide a convergence proof and error estimates for
this algorithm.Comment: 35 pages, 5 figures, to appear in Mathematical Financ