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arXiv2011

Conditional Density Models for Asset Pricing

We model the dynamics of asset prices and associated derivatives by consideration of the dynamics of the conditional probability density process for the value of an asset at some specified time in the future. In the case where the price process is driven by Brownian motion, an associated "master equation" for the dynamics of the conditional probability density is derived and expressed in integral form. By a "model" for the conditional density process we mean a solution to the master equation along with the specification of (a) the initial density, and (b) the volatility structure of the density. The volatility structure is assumed at any time and for each value of the argument of the density to be a functional of the history of the density up to that time. In practice one specifies the functional modulo sufficient parametric freedom to allow for the input of additional option data apart from that implicit in the initial density. The scheme is sufficiently flexible to allow for the input of various types of data depending on the nature of the options market and the class of valuation problem being undertaken. Various examples are studied in detail, with exact solutions provided in some cases.

q-fin.PR↗

arXiv2012

Storage option an Analytic approach

The mathematical problem of the static storage optimisation is formulated and solved by means of a variational analysis. The solution obtained in implicit form is shedding light on the most important features of the optimal exercise strategy. We show how the solution depends on different constraint types including carry cost and cycling constraint. We investigate the relation between intrinsic and stochastic solutions. In particular we give another proof that the stochastic problem has a "bang-bang" optimal exercise strategy. We also show why the optimal stochastic exercise decision is always close to the intrinsic one. In the second half we develop a perturbation analysis to solve the stochastic optimisation problem. The obtained approximate solution allows us to estimate the time value of the storage option. In particular we find an answer to rather academic question of asymptotic time value for the mean reversion parameter approaching zero or infinity. We also investigate the differences between swing and storage problems. The analytical results are compared with numerical valuations and found to be in a good agreement.

q-fin.CP↗

arXiv2010

Ruin probability in the presence of risky investments

We consider an insurance company in the case when the premium rate is a bounded non-negative random function $c_\zs{t}$ and the capital of the insurance company is invested in a risky asset whose price follows a geometric Brownian motion with mean return $a$ and volatility $σ>0$. If $β:=2a/σ^2-1>0$ we find exact the asymptotic upper and lower bounds for the ruin probability $Ψ(u)$ as the initial endowment $u$ tends to infinity, i.e. we show that $C_*u^{-β}\leΨ(u)\le C^*u^{-β}$ for sufficiently large $u$. Moreover if $c_\zs{t}=c^*e^{γt}$ with $γ\le 0$ we find the exact asymptotics of the ruin probability, namely $Ψ(u)\sim u^{-β}$. If $β\le 0$, we show that $Ψ(u)=1$ for any $u\ge 0$.

q-fin.RM↗

arXiv2011

Applications of the quadratic covariation differentiation theory: variants of the Clark-Ocone and Stroock's formulas

In a 2006 article (\cite{A1}), Allouba gave his quadratic covariation differentiation theory for Itô's integral calculus. He defined the derivative of a semimartingale with respect to a Brownian motion as the time derivative of their quadratic covariation and a generalization thereof. He then obtained a systematic differentiation theory containing a fundamental theorem of stochastic calculus relating this derivative to Itô's integral, a differential stochastic chain rule, a differential stochastic mean value theorem, and other differentiation rules. Here, we use this differentiation theory to obtain variants of the Clark-Ocone and Stroock formulas, with and without change of measure. We prove our variants of the Clark-Ocone formula under $L^{2}$-type conditions; with no Malliavin calculus, without the use of weak distributional or Radon-Nikodym type derivatives, and without the significant machinery of the Hida-Malliavin calculus. Unlike Malliavin or Hida-Malliavin calculi, the form of our variant of the Clark-Ocone formula under change of measure is as simple as it is under no change of measure, and without requiring any further differentiability conditions on the Girsanov transform integrand beyond Novikov's condition. This is due to the invariance under change of measure of the first author's derivative in \cite{A1}. The formulations and proofs are natural applications of the differentiation theory in \cite{A1} and standard Itô integral calculus. Iterating our Clark-Ocone formula, we obtain variants of Stroock's formula. We illustrate the applicability of these formulas by easily, and without Hida-Malliavin methods, obtaining the representation of the Brownian indicator $F=\mathbb{I}_{[K,\infty)}(W_{T})$, which is not standard Malliavin differentiable, and by applying them to digital options in finance. We then identify the chaos expansion of the Brownian indicator.

math.PR↗

arXiv2010

A Semigroup Point Of View On Splitting Schemes For Stochastic (Partial) Differential Equations

We construct normed spaces of real-valued functions with controlled growth on possibly infinite-dimensional state spaces such that semigroups of positive, bounded operators $(P_t)_{t\ge 0}$ thereon with $\lim_{t\to 0+}P_t f(x)=f(x)$ are in fact strongly continuous. This result applies to prove optimal rates of convergence of splitting schemes for stochastic (partial) differential equations with linearly growing characteristics and for sets of functions with controlled growth. Applications are general Da Prato-Zabczyk type equations and the HJM equations from interest rate theory.

math.PR↗

arXiv2010

A simple discretization scheme for nonnegative diffusion processes, with applications to option pricing

A discretization scheme for nonnegative diffusion processes is proposed and the convergence of the corresponding sequence of approximate processes is proved using the martingale problem framework. Motivations for this scheme come typically from finance, especially for path-dependent option pricing. The scheme is simple: one only needs to find a nonnegative distribution whose mean and variance satisfy a simple condition to apply it. Then, for virtually any (path-dependent) payoff, Monte Carlo option prices obtained from this scheme will converge to the theoretical price. Examples of models and diffusion processes for which the scheme applies are provided.

q-fin.CP↗

arXiv2012

Multidimensional dynamic risk measure via conditional g-expectation

This paper deals with multidimensional dynamic risk measures induced by conditional $g$-expectations. A notion of multidimensional $g$-expectation is proposed to provide a multidimensional version of nonlinear expectations. By a technical result on explicit expressions for the comparison theorem, uniqueness theorem and viability on a rectangle of solutions to multidimensional backward stochastic differential equations, some necessary and sufficient conditions are given for the constancy, monotonicity, positivity, homogeneity and translatability properties of multidimensional conditional $g$-expectations and multidimensional dynamic risk measures; we prove that a multidimensional dynamic $g$-risk measure is nonincreasingly convex if and only if the generator $g$ satisfies a quasi-monotone increasingly convex condition. A general dual representation is given for the multidimensional dynamic convex $g$-risk measure in which the penalty term is expressed more precisely. It is shown that model uncertainty leads to the convexity of risk measures. As to applications, we show how this multidimensional approach can be applied to measure the insolvency risk of a firm with interacted subsidiaries; optimal risk sharing for $\protectγ$-tolerant $g$-risk measures is investigated. Insurance $g$-risk measure and other ways to induce $g$-risk measures are also studied at the end of the paper.

q-fin.RM↗

arXiv2010

Cumulant Expansion and Monthly Sum Derivative

Cumulant expansion is used to derive accurate closed-form approximation for Monthly Sum Options in case of constant volatility model. Payoff of Monthly Sum Option is based on sum of $N$ caped (and probably floored) returns. It is noticed, that $1/\sqrt{N}$ can be used as a small parameter in Edgeworth expansion. First two leading terms of this expansion are calculated here. It is shown that the suggest closed-form approximation is in a good agreement with numerical results for typical mode parameters.

q-fin.PR↗

arXiv2010

Calibration of One- and Two-Factor Models For Valuation of Energy Multi-Asset Derivative Contracts

We study historical calibration of one- and two-factor models that are known to describe relatively well the dynamics of energy underlyings such as spot and index natural gas or oil prices at different physical locations or regional power prices. We take into account uneven frequency of data due to weekends, holidays, and possible missing data. We study the case when several one- and two-factor models are used in the joint model with correlated model factors and present examples of joint calibration for daily natural gas prices at several locations in the US and for regional hourly power prices.

q-fin.PR↗

arXiv2010

Solving Optimal Dividend Problems via Phase-type Fitting Approximation of Scale Functions

The optimal dividend problem by De Finetti (1957) has been recently generalized to the spectrally negative Lévy model where the implementation of optimal strategies draws upon the computation of scale functions and their derivatives. This paper proposes a phase-type fitting approximation of the optimal strategy. We consider spectrally negative Lévy processes with phase-type jumps as well as meromorphic Lévy processes (Kuznetsov et al., 2010a), and use their scale functions to approximate the scale function for a general spectrally negative Lévy process. We obtain analytically the convergence results and illustrate numerically the effectiveness of the approximation methods using examples with the spectrally negative Lévy process with i.i.d. Weibull-distributed jumps, the β-family and CGMY process.

q-fin.CP↗

arXiv2010

Wavelet-Based Prediction for Governance, Diversification and Value Creation Variables

We study the possibility of completing data bases of a sample of governance, diversification and value creation variables by providing a well adapted method to reconstruct the missing parts in order to obtain a complete sample to be applied for testing the ownership-structure/diversification relationship. It consists of a dynamic procedure based on wavelets. A comparison with Neural Networks, the most used method, is provided to prove the efficiency of the here-developed one. The empirical tests are conducted on a set of French firms.

q-fin.ST↗

arXiv2011

A Numerical Study of Radial Basis Function Based Methods for Options Pricing under the One Dimension Jump-diffusion Model

The aim of this chapter is to show how option prices in jump-diffusion models can be computed using meshless methods based on Radial Basis Function (RBF) interpolation. The RBF technique is demonstrated by solving the partial integro-differential equation (PIDE) in one-dimension for the American put and the European vanilla call/put options on dividend-paying stocks in the Merton and Kou jump-diffusion models. The radial basis function we select is the Cubic Spline. We also propose a simple numerical algorithm for finding a finite computational range of an improper integral term in the PIDE so that the accuracy of approximation of the integral can be improved. Moreover, the solution functions of the PIDE are approximated explicitly by RBFs which have exact forms so we can easily compute the global integral by any kind of numerical quadrature. Finally, we will not only show numerically that our scheme is second order accurate in both spatial and time variables in a European case but also second order accurate in spatial variables and first order accurate in time variables in an American case.

q-fin.CP↗

arXiv2010

Heat Kernel Interest Rate Models with Time-Inhomogeneous Markov Processes

We consider a heat kernel approach for the development of stochastic pricing kernels. The kernels are constructed by positive propagators, which are driven by time-inhomogeneous Markov processes. We multiply such a propagator with a positive, time-dependent and decreasing weight function, and integrate the product over time. The result is a so-called weighted heat kernel that by construction is a supermartingale with respect to the filtration generated by the time-inhomogeneous Markov processes. As an application, we show how this framework naturally fits the information-based asset pricing framework where time-inhomogeneous Markov processes are utilized to model partial information about random economic factors. We present examples of pricing kernel models which lead to analytical formulae for bond prices along with explicit expressions for the associated interest rate and market price of risk. Furthermore, we also address the pricing of fixed-income derivatives within this framework.

q-fin.CP↗

arXiv2010

Generalised Wishart Processes

We introduce a stochastic process with Wishart marginals: the generalised Wishart process (GWP). It is a collection of positive semi-definite random matrices indexed by any arbitrary dependent variable. We use it to model dynamic (e.g. time varying) covariance matrices. Unlike existing models, it can capture a diverse class of covariance structures, it can easily handle missing data, the dependent variable can readily include covariates other than time, and it scales well with dimension; there is no need for free parameters, and optional parameters are easy to interpret. We describe how to construct the GWP, introduce general procedures for inference and predictions, and show that it outperforms its main competitor, multivariate GARCH, even on financial data that especially suits GARCH. We also show how to predict the mean of a multivariate process while accounting for dynamic correlations.

stat.ME↗

arXiv2011

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 $λ$, the penalization parameter $p > 0$ and the time step. In particular, we obtain a convergence rate of the error due to penalization of order $(λp)^{α- \frac{1}{2}}, \forall α\in \left(0, \frac{1}{2}\right)$. 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.

q-fin.CP↗

arXiv2011

Duality in Robust Utility Maximization with Unbounded Claim via a Robust Extension of Rockafellar's Theorem

We study the convex duality method for robust utility maximization in the presence of a random endowment. When the underlying price process is a locally bounded semimartingale, we show that the fundamental duality relation holds true for a wide class of utility functions on the whole real line and unbounded random endowment. To obtain this duality, we prove a robust version of Rockafellar's theorem on convex integral functionals and apply Fenchel's general duality theorem.

q-fin.CP↗

arXiv2011

Sensitivity analysis of the early exercise boundary for American style of Asian options

In this paper we analyze American style of floating strike Asian call options belonging to the class of financial derivatives whose payoff diagram depends not only on the underlying asset price but also on the path average of underlying asset prices over some predetermined time interval. The mathematical model for the option price leads to a free boundary problem for a parabolic partial differential equation. Applying fixed domain transformation and transformation of variables we develop an efficient numerical algorithm based on a solution to a non-local parabolic partial differential equation for the transformed variable representing the synthesized portfolio. For various types of averaging methods we investigate the dependence of the early exercise boundary on model parameters.

q-fin.CP↗

arXiv2011

GPGPUs in computational finance: Massive parallel computing for American style options

The pricing of American style and multiple exercise options is a very challenging problem in mathematical finance. One usually employs a Least-Square Monte Carlo approach (Longstaff-Schwartz method) for the evaluation of conditional expectations which arise in the Backward Dynamic Programming principle for such optimal stopping or stochastic control problems in a Markovian framework. Unfortunately, these Least-Square Monte Carlo approaches are rather slow and allow, due to the dependency structure in the Backward Dynamic Programming principle, no parallel implementation; whether on the Monte Carlo levelnor on the time layer level of this problem. We therefore present in this paper a quantization method for the computation of the conditional expectations, that allows a straightforward parallelization on the Monte Carlo level. Moreover, we are able to develop for AR(1)-processes a further parallelization in the time domain, which makes use of faster memory structures and therefore maximizes parallel execution. Finally, we present numerical results for a CUDA implementation of this methods. It will turn out that such an implementation leads to an impressive speed-up compared to a serial CPU implementation.

q-fin.CP↗