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Low-dimensional approximations of high-dimensional asset price models
We consider high-dimensional asset price models that are reduced in their dimension in order to reduce the complexity of the problem or the effect of the curse of dimensionality in the context of option pricing. We apply model order reduction (MOR) to obtain a reduced system. MOR has been previously studied for asymptotically stable controlled stochastic systems with zero initial conditions. However, stochastic differential equations modeling price processes are uncontrolled, have non-zero initial states and are often unstable. Therefore, we extend MOR schemes and combine ideas of techniques known for deterministic systems. This leads to a method providing a good pathwise approximation. After explaining the reduction procedure, the error of the approximation is analyzed and the performance of the algorithm is shown conducting several numerical experiments. Within the numerics section, the benefit of the algorithm in the context of option pricing is pointed out
Low-dimensional approximations of high-dimensional asset price models
We consider high-dimensional asset price models that are reduced in their dimension in order to reduce the complexity of the problem or the effect of the curse of dimensionality in the context of option pricing. We apply model order reduction (MOR) to obtain a reduced system. MOR has been previously studied for asymptotically stable controlled stochastic systems with zero initial conditions. However, stochastic differential equations modeling price processes are uncontrolled, have non-zero initial states and are often unstable. Therefore, we extend MOR schemes and combine ideas of techniques known for deterministic systems. This leads to a method providing a good pathwise approximation. After explaining the reduction procedure, the error of the approximation is analyzed and the performance of the algorithm is shown conducting several numerical experiments. Within the numerics section, the benefit of the algorithm in the context of option pricing is pointed out
High dimensional American options
Pricing single asset American options is a hard problem in mathematical finance. There are no closed form solutions available (apart from in the case of the perpetual option), so many approximations and numerical techniques have been developed. Pricing multiâasset (high dimensional) American options is still more difficult.
We extend the method proposed theoretically by Glasserman and Yu (2004) by employing regression basis functions that are martingales under geometric Brownian motion. This results in more accurate Monte Carlo simulations, and computationally cheap lower and upper bounds to the American option price. We have implemented these models in QuantLib, the openâsource derivatives pricing library. The code for many of the models discussed in this thesis can be downloaded from quantlib.org as part of a practical pricing and risk management library.
We propose a new type of multiâasset option, the âRadial Barrier Optionâ for which we find analytic solutions. This is a barrier style option that pays out when a barrier, which is a function of the assets and their correlations, is hit. This is a useful benchmark test case for Monte Carlo simulations and may be of use in approximating multiâasset American options. We use Laplace transforms in this analysis which can be applied to give analytic results for the hitting times of Bessel processes.
We investigate the asymptotic solution of the single asset BlackâScholesâMerton equation in the case of low volatility. This analysis explains the success of some American option approximations, and has the potential to be extended to basket options
Reduced Order Models for Pricing European and American Options under Stochastic Volatility and Jump-Diffusion Models
European options can be priced by solving parabolic partial(-integro)
differential equations under stochastic volatility and jump-diffusion models
like Heston, Merton, and Bates models. American option prices can be obtained
by solving linear complementary problems (LCPs) with the same operators. A
finite difference discretization leads to a so-called full order model (FOM).
Reduced order models (ROMs) are derived employing proper orthogonal
decomposition (POD). The early exercise constraint of American options is
enforced by a penalty on subset of grid points. The presented numerical
experiments demonstrate that pricing with ROMs can be orders of magnitude
faster within a given model parameter variation range
Optimal market making
Market makers provide liquidity to other market participants: they propose
prices at which they stand ready to buy and sell a wide variety of assets. They
face a complex optimization problem with both static and dynamic components.
They need indeed to propose bid and offer/ask prices in an optimal way for
making money out of the difference between these two prices (their bid-ask
spread). Since they seldom buy and sell simultaneously, and therefore hold long
and/or short inventories, they also need to mitigate the risk associated with
price changes, and subsequently skew their quotes dynamically. In this paper,
(i) we propose a general modeling framework which generalizes (and reconciles)
the various modeling approaches proposed in the literature since the
publication of the seminal paper "High-frequency trading in a limit order book"
by Avellaneda and Stoikov, (ii) we prove new general results on the existence
and the characterization of optimal market making strategies, (iii) we obtain
new closed-form approximations for the optimal quotes, (iv) we extend the
modeling framework to the case of multi-asset market making and we obtain
general closed-form approximations for the optimal quotes of a multi-asset
market maker, and (v) we show how the model can be used in practice in the
specific (and original) case of two credit indices
Pricing American Options by Exercise Rate Optimization
We present a novel method for the numerical pricing of American options based
on Monte Carlo simulation and the optimization of exercise strategies. Previous
solutions to this problem either explicitly or implicitly determine so-called
optimal exercise regions, which consist of points in time and space at which a
given option is exercised. In contrast, our method determines the exercise
rates of randomized exercise strategies. We show that the supremum of the
corresponding stochastic optimization problem provides the correct option
price. By integrating analytically over the random exercise decision, we obtain
an objective function that is differentiable with respect to perturbations of
the exercise rate even for finitely many sample paths. The global optimum of
this function can be approached gradually when starting from a constant
exercise rate.
Numerical experiments on vanilla put options in the multivariate
Black-Scholes model and a preliminary theoretical analysis underline the
efficiency of our method, both with respect to the number of
time-discretization steps and the required number of degrees of freedom in the
parametrization of the exercise rates. Finally, we demonstrate the flexibility
of our method through numerical experiments on max call options in the
classical Black-Scholes model, and vanilla put options in both the Heston model
and the non-Markovian rough Bergomi model
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
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