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Martijn Pistorius

Publications and source records attributed to Martijn Pistorius.

At least 19 recordsLinked to original sources
arXiv2018

Dynamic Portfolio Optimization with Looping Contagion Risk

In this paper we consider a utility maximization problem with defaultable stocks and looping contagion risk. We assume that the default intensity of one company depends on the stock prices of itself and other companies, and the default of the company induces immediate drops in the stock prices of the surviving companies. We prove that the value function is the unique viscosity solution of the HJB equation. We also perform some numerical tests to compare and analyse the statistical distributions of the terminal wealth of log utility and power utility based on two strategies, one using the full information of intensity process and the other a proxy constant intensity process.

q-fin.MF↗
arXiv2017

On dynamic spectral risk measures, a limit theorem and optimal portfolio allocation

In this paper we propose the notion of continuous-time dynamic spectral risk-measure (DSR). Adopting a Poisson random measure setting, we define this class of dynamic coherent risk-measures in terms of certain backward stochastic differential equations. By establishing a functional limit theorem, we show that DSRs may be considered to be (strongly) time-consistent continuous-time extensions of iterated spectral risk-measures, which are obtained by iterating a given spectral risk-measure (such as Expected Shortfall) along a given time-grid. Specifically, we demonstrate that any DSR arises in the limit of a sequence of such iterated spectral risk-measures driven by lattice-random walks, under suitable scaling and vanishing time- and spatial-mesh sizes. To illustrate its use in financial optimisation problems, we analyse a dynamic portfolio optimisation problem under a DSR.

math.PR↗
arXiv2016

On Dynamic Deviation Measures and Continuous-Time Portfolio Optimisation

In this paper we propose the notion of dynamic deviation measure, as a dynamic time-consistent extension of the (static) notion of deviation measure. To achieve time-consistency we require that a dynamic deviation measures satisfies a generalised conditional variance formula. We show that, under a domination condition, dynamic deviation measures are characterised as the solutions to a certain class of backward SDEs. We establish for any dynamic deviation measure an integral representation, and derive a dual characterisation result in terms of additively $m$-stable dual sets. Using this notion of dynamic deviation measure we formulate a dynamic mean-deviation portfolio optimisation problem in a jump-diffusion setting and identify a subgame-perfect Nash equilibrium strategy that is linear as function of wealth by deriving and solving an associated extended HJB equation.

math.PR↗
arXiv2014

Randomisation and recursion methods for mixed-exponential Levy models, with financial applications

We develop a new Monte Carlo variance reduction method to estimate the expectation of two commonly encountered path-dependent functionals: first-passage times and occupation times of sets. The method is based on a recursive approximation of the first-passage time probability and expected occupation time of sets of a Levy bridge process that relies in part on a randomisation of the time parameter. We establish this recursion for general Levy processes and derive its explicit form for mixed-exponential jump-diffusions, a dense subclass (in the sense of weak approximation) of Levy processes, which includes Brownian motion with drift, Kou's double-exponential model and hyper-exponential jump-diffusion models. We present a highly accurate numerical realisation and derive error estimates. By way of illustration the method is applied to the valuation of range accruals and barrier options under exponential Levy models and Bates-type stochastic volatility models with exponential jumps. Compared with standard Monte Carlo methods, we find that the method is significantly more efficient.

math.PR↗
arXiv2014

The distribution of the supremum for spectrally asymmetric Lévy processes

In this article we derive formulas for the probability $P(\sup_{t\leq T} X(t)>u)$ $T>0$ and $P(\sup_{t<\infty} X(t)>u)$ where $X$ is a spectrally positive Lévy process with infinite variation. The formulas are generalizations of the well-known Takács formulas for stochastic processes with non-negative and interchangeable increments. Moreover, we find the joint distribution of $\inf_{t\leq T} Y(t)$ and $Y(T)$ where $Y$ is a spectrally negative Lévy process.

math.PR↗
arXiv2014

Convergence of BSΔEs driven by random walks to BSDEs: the case of (in)finite activity jumps with general driver

In this paper we present a weak approximation scheme for BSDEs driven by a Wiener process and an (in)finite activity Poisson random measure with drivers that are general Lipschitz functionals of the solution of the BSDE. The approximating backward stochastic difference equations (BSΔEs) are driven by random walks that weakly approximate the given Wiener process and Poisson random measure. We establish the weak convergence to the solution of the BSDE and the numerical stability of the sequence of solutions of the BSΔEs. By way of illustration we analyse explicitly a scheme with discrete step-size distributions.

math.PR↗
arXiv2014

Asymptotic independence of three statistics of maximal segmental scores

Let $ξ_1,ξ_2,\ldots$ be an iid sequence with negative mean. The $(m,n)$-segment is the subsequence $ξ_{m+1},\ldots,ξ_n$ and its \textit{score} is given by $\max\{\sum_{m+1}^nξ_i,0\}$. Let $R_n$ be the largest score of any segment ending at time $n$, $R^*_n$ the largest score of any segment in the sequence $ξ_{1},\ldots,ξ_n$, and $O_x$ the overshoot of the score over a level $x$ at the first epoch the score of such a size arises. We show that, under the Cramér assumption on $ξ_1$, asymptotic independence of the statistics $R_n$, $R_n^* -y$ and $O_{x+y}$ holds as $\min\{n,y,x\}\to\infty$. Furthermore, we establish a novel Spitzer-type identity characterising the limit law $O_\infty$ in terms of the laws of $(1,n)$-scores. As corollary we obtain: (1) a novel factorization of the exponential distribution as a convolution of $O_\infty$ and the stationary distribution of $R$; (2) if $y=γ^{-1}\log n$ (where $γ$ is the Cramér coefficient), our results, together with the classical theorem of Iglehart \cite{Iglehart}, yield the existence and explicit form of the joint weak limit of $(R_n, R_n^* -y,O_{x+y})$.

math.PR↗
arXiv2013

Buffer-overflows: joint limit laws of undershoots and overshoots of reflected processes

Let $τ(x)$ be the epoch of first entry into the interval $(x,\infty)$, $x>0$, of the reflected process $Y$ of a Lévy process $X$, and define the overshoot $Z(x) = Y(τ(x))-x$ and undershoot $z(x) = x - Y(τ(x)-)$ of $Y$ at the first-passage time over the level $x$. In this paper we establish, separately under the Cramér and positive drift assumptions, the existence of the weak limit of $(z(x), Z(x))$ as $x$ tends to infinity and provide explicit formulae for their joint CDFs in terms of the Lévy measure of $X$ and the renewal measure of the dual of $X$. We apply our results to analyse the behaviour of the classical M/G/1 queueing system at the buffer-overflow, both in a stable and unstable case.

math.PR↗
arXiv2013

Joint asymptotic distribution of certain path functionals of the reflected process

Let $τ(x)$ be the first time the reflected process $Y$ of a Levy processes $X$ crosses x>0. The main aim of the paper is to investigate the asymptotic dependence of the path functionals: $Y(t) = X(t) - \inf_{0\leq s\leq t}X(s)$, $M(t,x)=\sup_{0\leq s\leq t}Y(s)-x$ and $Z(x)=Y(τ(x))-x$. We prove that under Cramer's condition on X(1), the functionals $Y(t)$, $M(t,y)$ and $Z(x+y)$ are asymptotically independent as $\min\{t,y,x\}\to\infty$. We also characterise the law of the limiting overshoot $Z(\infty)$ of the reflected process. If, as $\min\{t,x\}\to\infty$, the quantity $t\te{-γx}$ has a positive limit ($γ$ denotes the Cramér coefficient), our results together with the theorem of Doney & Maller (2005) imply the existence and the explicit form of the joint weak limit $(Y(\infty),M(\infty),Z(\infty))$.

math.PR↗
arXiv2012

Fast computation of vanilla prices in time-changed models and implied volatilities using rational approximations

We present a new numerical method to price vanilla options quickly in time-changed Brownian motion models. The method is based on rational 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 transform method with respect to accuracy and speed appears to favour the newly developed method in the cases considered. We present error estimates for the option prices. Additionally, 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. 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 one can by the use of other common methods. Error estimates are presented for a wide range of parameters.

q-fin.CP↗
arXiv2011

Optimal dividend distribution under Markov-regime switching

We investigate the problem of optimal dividend distribution for a company in the presence of regime shifts. We consider a company whose cumulative net revenues evolve as a Brownian motion with positive drift that is modulated by a finite state Markov chain, and model the discount rate as a deterministic function of the current state of the chain. In this setting the objective of the company is to maximize the expected cumulative discounted dividend payments until the moment of bankruptcy, which is taken to be the first time that the cash reserves (the cumulative net revenues minus cumulative dividend payments) are zero. We show that, if the drift is positive in each state, it is optimal to adopt a barrier strategy at certain positive regime-dependent levels, and provide an explicit characterization of the value function as the fixed point of a contraction. In the case that the drift is small and negative in one state, the optimal strategy takes a different form, which we explicitly identify if there are two regimes. We also provide a numerical illustration of the sensitivities of the optimal barriers and the influence of regime-switching.

q-fin.GN↗
arXiv2010

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.

q-fin.PR↗
arXiv2010

Exotic derivatives under stochastic volatility models with jumps

In equity and foreign exchange markets the risk-neutral dynamics of the underlying asset are commonly represented by stochastic volatility models with jumps. In this paper we consider a dense subclass of such models and develop analytically tractable formulae for the prices of a range of first-generation exotic derivatives. We provide closed form formulae for the Fourier transforms of vanilla and forward starting option prices as well as a formula for the slope of the implied volatility smile for large strikes. A simple explicit approximation formula for the variance swap price is given. The prices of volatility swaps and other volatility derivatives are given as a one-dimensional integral of an explicit function. Analytically tractable formulae for the Laplace transform (in maturity) of the double-no-touch options and the Fourier-Laplace transform (in strike and maturity) of the double knock-out call and put options are obtained. The proof of the latter formulae is based on extended matrix Wiener-Hopf factorisation results. We also provide convergence results.

q-fin.PR↗
arXiv2009

Pricing and hedging barrier options in a hyper-exponential additive model

In this paper we develop an algorithm to calculate the prices and Greeks of barrier options in a hyper-exponential additive model with piecewise constant parameters. We obtain an explicit semi-analytical expression for the first-passage probability. The solution rests on a randomization and an explicit matrix Wiener-Hopf factorization. Employing this result we derive explicit expressions for the Laplace-Fourier transforms of the prices and Greeks of barrier options. As a numerical illustration, the prices and Greeks of down-and-in digital and down-and-in call options are calculated for a set of parameters obtained by a simultaneous calibration to Stoxx50E call options across strikes and four different maturities. By comparing the results with Monte-Carlo simulations, we show that the method is fast, accurate, and stable.

q-fin.PR↗
arXiv2009

On additive time-changes of Feller processes

In this note we generalise the Phillips theorem on the subordination of Feller processes by Levy subordinators to the class of additive subordinators (i.e. subordinators with independent but possibly nonstationary increments). In the case where the original Feller process is Levy we also express the time-dependent characteristics of the subordinated process in terms of the characteristics of the Levy process and the additive subordinator.

math.PR↗
arXiv2009

A transform approach to compute prices and greeks of barrier options driven by a class of Levy processes

In this paper we propose a transform method to compute the prices and greeks of barrier options driven by a class of Levy processes. We derive analytical expressions for the Laplace transforms in time of the prices and sensitivities of single barrier options in an exponential Levy model with hyper-exponential jumps. Inversion of these single Laplace transform yields rapid, accurate results. These results are employed to construct an approximation of the prices and sensitivities of barrier options in exponential generalised hyper-exponential (GHE) Levy models. The latter class includes many of the Levy models employed in quantitative finance such as the variance gamma (VG), KoBoL, generalised hyperbolic, and the normal inverse Gaussian (NIG) models. Convergence of the approximating prices and sensitivities is proved. To provide a numerical illustration, this transform approach is compared with Monte Carlo simulation in the cases that the driving process is a VG and a NIG Levy process. Parameters are calibrated to Stoxx50E call options.

q-fin.PR↗
arXiv2008

A method of moments approach to pricing double barrier contracts driven by a general class of jump diffusions

We present the method of moments approach to pricing barrier-type options when the underlying is modelled by a general class of jump diffusions. By general principles the option prices are linked to certain infinite dimensional linear programming problems. Subsequently approximating those systems by finite dimensional linear programming problems, upper and lower bounds for the prices of such options are found. As numerical illustration we apply the method to the valuation of several barrier-type options (double barrier knockout option, American corridor and double no touch) under a number of different models, including a case with deterministic interest rates, and compare with Monte Carlo simulation results. In all cases we find tight bounds with short execution times. Theoretical convergence results are also provided.

q-fin.CP↗