1,077 research outputs found
Application of Operator Splitting Methods in Finance
Financial derivatives pricing aims to find the fair value of a financial
contract on an underlying asset. Here we consider option pricing in the partial
differential equations framework. The contemporary models lead to
one-dimensional or multidimensional parabolic problems of the
convection-diffusion type and generalizations thereof. An overview of various
operator splitting methods is presented for the efficient numerical solution of
these problems.
Splitting schemes of the Alternating Direction Implicit (ADI) type are
discussed for multidimensional problems, e.g. given by stochastic volatility
(SV) models. For jump models Implicit-Explicit (IMEX) methods are considered
which efficiently treat the nonlocal jump operator. For American options an
easy-to-implement operator splitting method is described for the resulting
linear complementarity problems.
Numerical experiments are presented to illustrate the actual stability and
convergence of the splitting schemes. Here European and American put options
are considered under four asset price models: the classical Black-Scholes
model, the Merton jump-diffusion model, the Heston SV model, and the Bates SV
model with jumps
Modelling FX smile : from stochastic volatility to skewness
Imperial Users onl
Stochastic Spot/Volatility Correlation in Stochastic Volatility Models and Barrier Option Pricing
Most models for barrier pricing are designed to let a market maker tune the
model-implied covariance between moves in the asset spot price and moves in the
implied volatility skew. This is often implemented with a local
volatility/stochastic volatility mixture model, where the mixture parameter
tunes that covariance. This paper defines an alternate model where the
spot/volatility correlation is a separate mean-reverting stochastic variable
which is itself correlated with spot. We also develop an efficient
approximation for barrier option and one touch pricing in the model based on
semi-static vega replication and compare it with Monte Carlo pricing. The
approximation works well in markets where the risk neutral drift is modest.Comment: 23 pages, 11 figure
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
Moment Methods for Exotic Volatility Derivatives
The latest generation of volatility derivatives goes beyond variance and
volatility swaps and probes our ability to price realized variance and sojourn
times along bridges for the underlying stock price process. In this paper, we
give an operator algebraic treatment of this problem based on Dyson expansions
and moment methods and discuss applications to exotic volatility derivatives.
The methods are quite flexible and allow for a specification of the underlying
process which is semi-parametric or even non-parametric, including
state-dependent local volatility, jumps, stochastic volatility and regime
switching. We find that volatility derivatives are particularly well suited to
be treated with moment methods, whereby one extrapolates the distribution of
the relevant path functionals on the basis of a few moments. We consider a
number of exotics such as variance knockouts, conditional corridor variance
swaps, gamma swaps and variance swaptions and give valuation formulas in
detail
The History of the Quantitative Methods in Finance Conference Series. 1992-2007
This report charts the history of the Quantitative Methods in Finance (QMF) conference from its beginning in 1993 to the 15th conference in 2007. It lists alphabetically the 1037 speakers who presented at all 15 conferences and the titles of their papers.
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