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arXiv2009

Indifference price with general semimartingales

For utility functions $u$ finite valued on $\mathbb{R}$, we prove a duality formula for utility maximization with random endowment in general semimartingale incomplete markets. The main novelty of the paper is that possibly non locally bounded semimartingale price processes are allowed. Following Biagini and Frittelli \cite{BiaFri06}, the analysis is based on the duality between the Orlicz spaces $(L^{\widehat{u}}, (L^{\widehat{u}})^*)$ naturally associated to the utility function. This formulation enables several key properties of the indifference price $π(B)$ of a claim $B$ satisfying conditions weaker than those assumed in literature. In particular, the indifference price functional $π$ turns out to be, apart from a sign, a convex risk measure on the Orlicz space $L^{\widehat{u}}$.

q-fin.PR↗

arXiv2009

Shaping tail dependencies by nesting box copulas

We introduce a family of copulas which are locally piecewise uniform in the interior of the unit cube of any given dimension. Within that family, the simultaneous control of tail dependencies of all projections to faces of the cube is possible and we give an efficient sampling algorithm. The combination of these two properties may be appealing to risk modellers.

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arXiv2010

Strong Taylor approximation of stochastic differential equations and application to the Lévy LIBOR model

In this article we develop a method for the strong approximation of stochastic differential equations (SDEs) driven by Lévy processes or general semimartingales. The main ingredients of our method is the perturbation of the SDE and the Taylor expansion of the resulting parameterized curve. We apply this method to develop strong approximation schemes for LIBOR market models. In particular, we derive fast and precise algorithms for the valuation of derivatives in LIBOR models which are more tractable than the simulation of the full SDE. A numerical example for the Lévy LIBOR model illustrates our method.

math.PR↗

arXiv2009

An application to credit risk of a hybrid Monte Carlo-Optimal quantization method

In this paper we use a hybrid Monte Carlo-Optimal quantization method to approximate the conditional survival probabilities of a firm, given a structural model for its credit defaul, under partial information. We consider the case when the firm's value is a non-observable stochastic process $(V_t)_{t \geq 0}$ and inverstors in the market have access to a process $(S_t)_{t \geq 0}$, whose value at each time t is related to $(V_s, s \leq t)$. We are interested in the computation of the conditional survival probabilities of the firm given the "investor information". As a application, we analyse the shape of the credit spread curve for zero coupon bonds in two examples.

q-fin.CP↗

arXiv2009

A stochastic reachability approach to portfolio construction in finance industry

In finance industry portfolio construction deals with how to divide the investors' wealth across an asset-classes' menu in order to maximize the investors' gain. Main approaches in use at the present are based on variations of the classical Markowitz model. However, recent evolutions of the world market showed limitations of this method and motivated many researchers and practitioners to study alternative methodologies to portfolio construction. In this paper we propose one approach to optimal portfolio construction based on recent results on stochastic reachability, which overcome some of the limits of current approaches. Given a sequence of target sets that the investors would like their portfolio to stay within, the optimal portfolio allocation is synthesized in order to maximize the joint probability for the portfolio value to fulfill the target sets requirements. A case study in the US market is given which shows benefits from the proposed methodology in portfolio construction. A comparison with traditional approaches is included.

q-fin.PM↗

arXiv2009

Pricing Bermudan options using nonparametric regression: optimal rates of convergence for lower estimates

The problem of pricing Bermudan options using Monte Carlo and a nonparametric regression is considered. We derive optimal non-asymptotic bounds for a lower biased estimate based on the suboptimal stopping rule constructed using some estimates of continuation values. These estimates may be of different nature, they may be local or global, with the only requirement being that the deviations of these estimates from the true continuation values can be uniformly bounded in probability. As an illustration, we discuss a class of local polynomial estimates which, under some regularity conditions, yield continuation values estimates possessing this property.

q-fin.PR↗

arXiv2009

On the uniqueness of classical solutions of Cauchy problems

Given that the terminal condition is of at most linear growth, it is well known that a Cauchy problem admits a unique classical solution when the coefficient multiplying the second derivative (i.e., the volatility) is also a function of at most linear growth. In this note, we give a condition on the volatility that is necessary and sufficient for a Cauchy problem to admit a unique solution.

math.AP↗

arXiv2011

High order discretization schemes for stochastic volatility models

In usual stochastic volatility models, the process driving the volatility of the asset price evolves according to an autonomous one-dimensional stochastic differential equation. We assume that the coefficients of this equation are smooth. Using Itô's formula, we get rid, in the asset price dynamics, of the stochastic integral with respect to the Brownian motion driving this SDE. Taking advantage of this structure, we propose - a scheme, based on the Milstein discretization of this SDE, with order one of weak trajectorial convergence for the asset price, - a scheme, based on the Ninomiya-Victoir discretization of this SDE, with order two of weak convergence for the asset price. We also propose a specific scheme with improved convergence properties when the volatility of the asset price is driven by an Orstein-Uhlenbeck process. We confirm the theoretical rates of convergence by numerical experiments and show that our schemes are well adapted to the multilevel Monte Carlo method introduced by Giles [2008a, 2008b].

math.PR↗

arXiv2009

Gauge Invariance, Geometry and Arbitrage

In this work, we identify the most general measure of arbitrage for any market model governed by Itô processes. We show that our arbitrage measure is invariant under changes of numéraire and equivalent probability. Moreover, such measure has a geometrical interpretation as a gauge connection. The connection has zero curvature if and only if there is no arbitrage. We prove an extension of the Martingale pricing theorem in the case of arbitrage. In our case, the present value of any traded asset is given by the expectation of future cash-flows discounted by a line integral of the gauge connection. We develop simple strategies to measure arbitrage using both simulated and real market data. We find that, within our limited data sample, the market is efficient at time horizons of one day or longer. However, we provide strong evidence for non-zero arbitrage in high frequency intraday data. Such events seem to have a decay time of the order of one minute.

q-fin.PR↗

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↗

arXiv2009

Double Kernel estimation of sensitivities

This paper adresses the general issue of estimating the sensitivity of the expectation of a random variable with respect to a parameter characterizing its evolution. In finance for example, the sensitivities of the price of a contingent claim are called the Greeks. A new way of estimating the Greeks has been recently introduced by Elie, Fermanian and Touzi through a randomization of the parameter of interest combined with non parametric estimation techniques. This paper studies another type of those estimators whose interest is to be closely related to the score function, which is well known to be the optimal Greek weight. This estimator relies on the use of two distinct kernel functions and the main interest of this paper is to provide its asymptotic properties. Under a little more stringent condition, its rate of convergence equals the one of those introduced by Elie, Fermanian and Touzi and outperforms the finite differences estimator. In addition to the technical interest of the proofs, this result is very encouraging in the dynamic of creating new type of estimators for sensitivities.

q-fin.CP↗

arXiv2009

Optimal double stopping time

We consider the optimal double stopping time problem defined for each stopping time $S$ by $v(S)=\esssup\{E[ψ(τ_1, τ_2) | \F_S], τ_1, τ_2 \geq S \}$. Following the optimal one stopping time problem, we study the existence of optimal stopping times and give a method to compute them. The key point is the construction of a {\em new reward} $ϕ$ such that the value function $v(S)$ satisfies $v(S)=\esssup\{E[ϕ(τ) | \F_S], τ\geq S \}$. Finally, we give an example of an american option with double exercise time.

math.PR↗

arXiv2009

On the rates of convergence of simulation based optimization algorithms for optimal stopping problems

In this paper we study simulation based optimization algorithms for solving discrete time optimal stopping problems. This type of algorithms became popular among practioneers working in the area of quantitative finance. Using large deviation theory for the increments of empirical processes, we derive optimal convergence rates and show that they can not be improved in general. The rates derived provide a guide to the choice of the number of simulated paths needed in optimization step, which is crucial for the good performance of any simulation based optimization algorithm. Finally, we present a numerical example of solving optimal stopping problem arising in option pricing that illustrates our theoretical findings.

math.OC↗

arXiv2009

Stock Market Trading Via Stochastic Network Optimization

We consider the problem of dynamic buying and selling of shares from a collection of $N$ stocks with random price fluctuations. To limit investment risk, we place an upper bound on the total number of shares kept at any time. Assuming that prices evolve according to an ergodic process with a mild decaying memory property, and assuming constraints on the total number of shares that can be bought and sold at any time, we develop a trading policy that comes arbitrarily close to achieving the profit of an ideal policy that has perfect knowledge of future events. Proximity to the optimal profit comes with a corresponding tradeoff in the maximum required stock level and in the timescales associated with convergence. We then consider arbitrary (possibly non-ergodic) price processes, and show that the same algorithm comes close to the profit of a frame based policy that can look a fixed number of slots into the future. Our analysis uses techniques of Lyapunov Optimization that we originally developed for stochastic network optimization problems.

q-fin.PM↗

arXiv2009

Defaultable bonds with an infinite number of Levy factors

A market with defaultable bonds where the bond dynamics is in a Heath-Jarrow-Morton setting and the forward rates are driven by an infinite number of Levy factors is considered. The setting includes rating migrations driven by a Markov chain. All basic types of recovery are investigated. We formulate necessary and sufficient conditions (generalized HJM conditions) under which the market is arbitrage free. Connections with consistency conditions are discussed.

q-fin.CP↗

arXiv2013

Linear stochastic volatility models

In this paper we investigate general linear stochastic volatility models with correlated Brownian noises. In such models the asset price satisfies a linear SDE with coefficient of linearity being the volatility process. This class contains among others Black-Scholes model, a log-normal stochastic volatility model and Heston stochastic volatility model. For a linear stochastic volatility model we derive representations for the probability density function of the arbitrage price of a financial asset and the prices of European call and put options. A closed-form formulae for the density function and the prices of European call and put options are given for log-normal stochastic volatility model. We also obtain present some new results for Heston and extended Heston stochastic volatility models.

q-fin.PR↗

arXiv2009

Optimal Stopping for Dynamic Convex Risk Measures

We use martingale and stochastic analysis techniques to study a continuous-time optimal stopping problem, in which the decision maker uses a dynamic convex risk measure to evaluate future rewards. We also find a saddle point for an equivalent zero-sum game of control and stopping, between an agent (the "stopper") who chooses the termination time of the game, and an agent (the "controller", or "nature") who selects the probability measure.

math.PR↗

arXiv2021

Geometric Arbitrage Theory and Market Dynamics Reloaded

We have embedded the classical theory of stochastic finance into a differential geometric framework called Geometric Arbitrage Theory and show that it is possible to: --Write arbitrage as curvature of a principal fibre bundle. --Parameterize arbitrage strategies by its holonomy. --Give the Fundamental Theorem of Asset Pricing a differential homotopic characterization. --Characterize Geometric Arbitrage Theory by five principles and show they they are consistent with the classical theory of stochastic finance. --Derive for a closed market the equilibrium solution for market portfolio and dynamics in the cases where: -->Arbitrage is allowed but minimized. -->Arbitrage is not allowed. --Prove that the no-free-lunch-with-vanishing-risk condition implies the zero curvature condition.

q-fin.CP↗