On accurate and efficient valuation of financial contracts under models with jumps

The aim of this thesis is to develop efficient valuation methods for nancial contracts under models with jumps and stochastic volatility, and to present their rigorous mathematical underpinning. For efficient risk management, large books of exotic options need to be priced and hedged under models that are exible enough to describe the observed option prices at speeds close to real time. To do so, hundreds of vanilla options, which are quoted in terms of implied volatility, need to be calibrated to market prices quickly and accurately on a regular basis. With this in mind we develop efficient methods for the evaluation of (i) vanilla options, (ii) implied volatility and (iii) common path-dependent options. Firstly, we derive a new numerical method for the classical problem of pricing vanilla options quickly in time-changed Brownian motion models. The method is based on ra- tional function approximations of the Black-Scholes formula. Detailed numerical results are given for a number of widely used models. In particular, we use the variance-gamma model, the CGMY model and the Heston model without correlation to illustrate our results. Comparison to the standard fast Fourier option pricing method with respect to speed appears to favour our newly developed method in the cases considered. Secondly, we use this method to derive a procedure to compute, for a given set of arbitrage-free European call option prices, the corresponding Black-Scholes implied volatility surface. In order to achieve this, rational function approximations of the inverse of the Black-Scholes formula are used. We are thus able to work out implied volatilities more efficiently than is possible using other common methods. Error estimates are presented for a wide range of parameters. Thirdly, we develop a new Monte Carlo variance reduction method to estimate the expectations of path-dependent functionals, such as first-passage times and occupation times, under a class of stochastic volatility models with jumps. The method is based on a recursive approximation of the rst-passage time probabilities and expected oc- cupation times of Levy bridge processes that relies in part on a randomisation of the time- parameter. We derive the explicit form of the recursive approximation in the case of bridge processes corresponding to the class of Levy processes with mixed-exponential jumps, and present a highly accurate numerical realisation. This class includes the linear Brownian motion, Kou's double-exponential jump-di usion model and the hyper-exponential jump- difusion model, and it is dense in the class of all Levy processes. We determine the rate of convergence of the randomisation method and con rm it numerically. Subsequently, we combine the randomisation method with a continuous Euler-Maruyama scheme to es- timate path-functionals under stochastic volatility models with jumps. Compared with standard Monte Carlo methods, we nd that the method is signi cantly more efficient. To illustrate the efficiency of the method, it is applied to the valuation of range accruals and barrier options.Open Acces

Swing Options Valuation: a BSDE with Constrained Jumps Approach

We introduce a new probabilistic method for solving a class of impulse control problems based on their representations as Backward Stochastic Differential Equations (BSDEs for short) with constrained jumps. As an example, our method is used for pricing Swing options. We deal with the jump constraint by a penalization procedure and apply a discrete-time backward scheme to the resulting penalized BSDE with jumps. We study the convergence of this numerical method, with respect to the main approximation parameters: the jump intensity $\lambda$, the penalization parameter $p > 0$ and the time step. In particular, we obtain a convergence rate of the error due to penalization of order $(\lambda p)^{\alpha - \frac{1}{2}}, \forall \alpha \in \left(0, \frac{1}{2}\right)$. Combining this approach with Monte Carlo techniques, we then work out the valuation problem of (normalized) Swing options in the Black and Scholes framework. We present numerical tests and compare our results with a classical iteration method.Comment: 6 figure

Joint Distribution of Passage Times of an Ornstein-Uhlenbeck Diffusion and Real-Time Computational Methods for Financial Sensitivities

This thesis analyses two broad problems: the computation of financial sensitivities, which is a computationally expensive exercise, and the evaluation of barriercrossing probabilities which cannot be approximated to reach a certain precision in certain circumstances. In the former case, we consider the computation of the parameter sensitivities of large portfolios and also valuation adjustments. The traditional approach to compute sensitivities is by the finite-difference approximation method, which requires an iterated implementation of the original valuation function. This leads to substantial computational costs, no matter whether the valuation was implemented via numerical partial differential equation methods or Monte Carlo simulations. However, we show that the adjoint algorithmic differentiation algorithm can be utilised to calculate these price sensitivities reliably and orders of magnitude faster compared to standard finite-difference approaches. In the latter case, we consider barrier-crossing problems of Ornstein-Uhlenbeck diffusions. Especially in the case where the barrier is difficult to reach, the problem turns into a rare event occurrence approximation problem. We prove that it cannot be estimated accurately and robustly with direct Monte Carlo methods because of the irremovable bias and Monte Carlo error. Instead, we adopt a partial differential equation method alongside the eigenfunction expansion, from which we are able to calculate the distribution and the survival functions for the maxima of a homogeneous Ornstein-Uhlenbeck process in a single interval. By the conditional independence property of Markov processes, the results can be further extended to inhomogeneous cases and multiple period barrier-crossing problems, both of which can be efficiently implemented by quadrature and Monte Carlo integration methods

On two-sided controls of a linear diffusion

siirretty Doriast

Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in thePHLX Deutschemark Options

An efficient method is developed for pricing American options on combination stochastic volatility/jump-diffusion processes when jump risk and volatility risk are systematic and nondiversifiable, thereby nesting two major option pricing models. The parameters implicit in PHLX-traded Deutschemark options of the stochastic volatility/jump- diffusion model and various submodels are estimated over 1984-91, and are tested for consistency with the $/DM futures process and the implicit volatility sample path. The parameters implicit in options are found to be inconsistent with the time series properties of implicit volatilities, but qualitatively consistent with log- differenced futures prices. No economically significant implicit expectations of exchange rate jumps were found in full-sample estimation, which is consistent with the reduced leptokurtosis of$/DM weekly exchange rate changes over 1984-91 relative to earlier periods.

Stochastic Control Representations for Penalized Backward Stochastic Differential Equations

This paper shows that penalized backward stochastic differential equation (BSDE), which is often used to approximate and solve the corresponding reflected BSDE, admits both optimal stopping representation and optimal control representation. The new feature of the optimal stopping representation is that the player is allowed to stop at exogenous Poisson arrival times. The convergence rate of the penalized BSDE then follows from the optimal stopping representation. The paper then applies to two classes of equations, namely multidimensional reflected BSDE and reflected BSDE with a constraint on the hedging part, and gives stochastic control representations for their corresponding penalized equations.Comment: 24 pages in SIAM Journal on Control and Optimization, 201