European Contingent Claims Research Articles

Discrete-time approximation for backward stochastic differential equations driven by G-Brownian motion

In this paper, we study the discrete-time approximation schemes for a class of backward stochastic differential equations driven by [Formula: see text]-Brownian motion ([Formula: see text]-BSDEs) which corresponds to the hedging pricing of European contingent claims. By introducing an auxiliary extended [Formula: see text]-expectation space, we propose a class of [Formula: see text]-schemes to discrete [Formula: see text]-BSDEs in this space. With the help of nonlinear stochastic analysis techniques and numerical analysis tools, we prove that our schemes admit half-order convergence for approximating [Formula: see text]-BSDE in the general case. In some special cases, our schemes can achieve a first-order convergence rate. Finally, we give an implementable numerical scheme for [Formula: see text]-BSDEs based on Peng’s central limit theorem and illustrate our convergence results with numerical examples.

Risk measures based on weak optimal transport

In this paper, we study convex risk measures with weak optimal transport penalties. In a first step, we show that these risk measures allow for an explicit representation via a nonlinear transform of the loss function. In a second step, we discuss computational aspects related to the nonlinear transform as well as approximations of the risk measures using, for example, neural networks. Our setup comprises a variety of examples, such as classical optimal transport penalties, parametric families of models, divergence risk measures, uncertainty on path spaces, moment constraints, and martingale constraints. In a last step, we show how to use the theoretical results for the numerical computation of worst-case losses in an insurance context and no-arbitrage prices of European contingent claims after quoted maturities in a model-free setting.

Wrong Way Risk corrections to CVA in CIR reduced-form models

In this paper we provide an efficient methodology to compute the credit value adjustment of a European contingent claim subject to some default event concerning the issuer solvability, when the underlying and the default event are correlated. In particular, in a Black and Scholes market/CIR intensity-default model, we consider a second order expansion around the origin of a vulnerable call option with respect to a correlation parameter rho, which may be used to describe the wrong way risk of the contract, measuring the dependence between the underlying asset price and the option’s issuer default intensity. Numerical implementations of this approach are compared with the benchmark Monte Carlo simulations.

Duality theory for exponential utility-based hedging in the Almgren–Chriss model

AbstractIn this paper we obtain a duality result for the exponential utility maximization problem where trading is subject to quadratic transaction costs and the investor is required to liquidate her position at the maturity date. As an application of the duality, we treat utility-based hedging in the Bachelier model. For European contingent claims with a quadratic payoff, we compute the optimal trading strategy explicitly.

Hedging options in a hidden Markov‐switching local‐volatility model via stochastic flows and a Monte‐Carlo method

AbstractThe hedging of European contingent claims in a continuous‐time hidden Markov‐regime‐switching diffusion model is discussed using stochastic flows of diffeomorphisms and Monte‐Carlo simulations. Specifically, the price dynamics of an underlying risky asset are governed by a continuous‐time hidden Markov‐modulated local‐volatility model. Filtering theory is used to estimate the unobservable drift given observable price information and to define a filtered market with complete observations. The delta–hedge ratio of a European option is derived using a martingale representation and stochastic flows of diffeomorphisms. The numerical computation of the delta–hedge ratio is estimated via Monte‐Carlo simulations. Numerical results for illustrating the proposed method and the (relative) importance of the impacts of the information risk and the local‐volatility parametrizations on the delta–hedge ratio are provided for the case of European call options.

Approximate value adjustments for European claims

We consider the problem of computing the Value Adjustment of European contingent claims when the possibility of default of either party is considered, possibly including also funding and collateralization requirements.As shown in Brigo et al. (Brigo, Liu, Pallavicini, and Sloth (2016b), Brigo, Francischello, and Pallavicini (2016a)), this leads to a more articulate variety of Value Adjustments (XVA) that introduce some nonlinear features. When exploiting a reduced-form approach for the default times, the adjusted price can be characterized as the solution to a possibly nonlinear Backward Stochastic Differential Equation (BSDE). The expectation representing the solution of the BSDE is usually quite hard to compute even in a Markovian setting, and one might resort either to the discretization of the Partial Differential Equation characterizing it or to Monte Carlo Simulations. Both choices are computationally quite expensive, In this paper, we suggest an alternative method based on an appropriate change of numéraire and a Taylor polynomial expansion when intensities are represented by affine processes correlated with the asset price. The numerical discussion at the end of this work shows that, at least in the case of the Cox-Ingersoll-Ross (CIR) intensity model, even the simple first-order approximation has a remarkable computational efficiency.

Static Replication of European Multi-Asset Options with Homogeneous Payoff

ABSTRACT The replication of any European contingent claim by a static continuous portfolio of calls and puts, formally proven by [Carr, Peter, and Dilip Madan. 1998. “Towards a Theory of Volatility Trading.” In Volatility: New Estimation Techniques for Pricing Derivatives, Vol. 29, edited by Robert A. Jarrow, 417–427. Risk books.] extends to multi-asset claims with absolutely homogeneous payoff. Using sophisticated tools from integral geometry, we show how such claims may be replicated with a continuum of vanilla basket calls and derive closed-form solutions to replicate two-asset best-of and worst-of options. We also derive a novel mathematical formula to invert the Radon transform which we apply to obtain a tractable expression of the joint implied distribution. Consequently, a large class of multi-asset options admit a model-free price enforced by arbitrage, just as single-asset European claims do.

Static Replication of European Multi-Asset Options with Homogeneous Payoff

The replication of any European contingent claim by a static continuous portfolio of calls and puts, formally proven by Carr and Madan (1998), extends to multi-asset claims with absolutely homogeneous payoff. Using sophisticated tools from integral geometry, we show how such claims may be replicated with a continuum of vanilla basket calls and derive closed-form solutions to replicate two-asset best-of and worst-of options. We also derive a novel mathematical formula to invert the Radon transform which we apply to obtain a tractable expression of the joint implied distribution. Consequently, a large class of multi-asset options admit a model-free price enforced by arbitrage, just as single-asset European claims do.

LOCALLY RISK-MINIMIZING HEDGING FOR EUROPEAN CONTINGENT CLAIMS WRITTEN ON NON-TRADABLE ASSETS WITH COMMON JUMP RISK

This article investigates the optimal hedging problem of the European contingent claims written on non-tradable assets. We assume that the risky assets satisfy jump diffusion models with a common jump process which reflects the correlated jump risk. The non-tradable asset and jump risk lead to an incomplete financial market. Hence, the cross-hedging method will be used to reduce the potential risk of the contingent claims seller. First, we obtain an explicit closed-form solution for the locally risk-minimizing hedging strategies of the European contingent claims by using the Föllmer–Schweizer decomposition. Then, we consider the hedging for a European call option as a special case. The value of the European call option under the minimal martingale measure is derived by the Fourier transform method. Next, some semi-closed solution formulae of the locally risk-minimizing hedging strategies for the European call option are obtained. Finally, some numerical examples are provided to illustrate the sensitivities of the optimal hedging strategies. By comparing the optimal hedging strategies when the underlying asset is a non-tradable asset or a tradable asset, we find that the liquidity risk has a significant impact on the optimal hedging strategies.

Penalty Methods for Bilateral XVA Pricing in European and American Contingent Claims by a PDE Model

Taking into account default risk in the valuation of financial derivatives has become increasingly important, especially after the 2007-2008 financial crisis. Under some assumptions, the valuation of financial derivatives, including a value adjustment to account for default risk (the so-called XVA), gives rise to a nonlinear PDE. We propose numerical methods for handling the nonlinearity in this PDE, the most efficient of which are discrete penalty iteration methods. We first formulate a penalty iteration method for the case of European contingent claims, and study its convergence. We then extend the method to the case of American contingent claims, resulting in a double-penalty iteration. We also propose boundary conditions and their discretization for the XVA PDE problem in the cases of call and put options, as well as the case of a forward contract. Numerical results demonstrate the effectiveness of our methods.

Penalty methods for bilateral XVA pricing in European and American contingent claims by a partial differential equation model

Accounting for default risk in the valuation of financial derivatives has become increasingly important, especially since the 2007–8 financial crisis. Under some assumptions, the valuation of financial derivatives, including a value adjustment to account for default risk (the so-called XVA), gives rise to a nonlinear partial differential equation (PDE). We propose numerical methods for handling the nonlinearity in this PDE, the most efficient of which are the discrete penalty iteration methods. We first formulate a penalty iteration method for the case of European contingent claims and study its convergence. We then extend the method to the case of American contingent claims, which results in a double-penalty iteration. We also propose boundary conditions and their discretization for the XVA PDE problem in the case of a call option, a put option and a forward contract. Numerical results demonstrate the effectiveness of our methods.

Markov chains under nonlinear expectation

AbstractIn this paper, we consider continuous‐time Markov chains with a finite state space under nonlinear expectations. We define so‐called Q‐operators as an extension of Q‐matrices or rate matrices to a nonlinear setup, where the nonlinearity is due to model uncertainty. The main result gives a full characterization of convex Q‐operators in terms of a positive maximum principle, a dual representation by means of Q‐matrices, time‐homogeneous Markov chains under convex expectations, and a class of nonlinear ordinary differential equations. This extends a classical characterization of generators of Markov chains to the case of model uncertainty in the generator. We further derive an explicit primal and dual representation of convex semigroups arising from Markov chains under convex expectations via the Fenchel–Legendre transformation of the generator. We illustrate the results with several numerical examples, where we compute price bounds for European contingent claims under model uncertainty in terms of the rate matrix.

A functional analysis approach to the static replication of European options

The replication of any European contingent claim by a static portfolio of calls and puts with strikes forming a continuum, formally proven by Carr and Madan [Towards a theory of volatility trading. In Volatility: New Estimation Techniques for Pricing Derivatives, edited by R.A. Jarrow, Vol. 29, pp. 417–427, 1998 (Risk books)], is part of the more general theory of integral equations. We use spectral decomposition techniques to show that exact payoff replication may be achieved with a discrete portfolio of special options. We discuss applications for fast pricing of vanilla options that may be suitable for large option books or high frequency option trading, and for model pricing when the characteristic function of the underlying asset price is known.

Static Replication of European Standard Dispersion Options

The replication of any European contingent claim by a static portfolio of calls and puts with strikes forming a continuum, formally proven by Carr and Madan (1998), extends to options written on the Euclidean norm of a vector of n asset performances. With the help of integral equation techniques we derive replicating portfolios for calls, puts and indeed any claim contingent on standard dispersion using vanilla basket calls whose basket weights span an n-dimensional continuum. Consequently multi-asset standard dispersion options admit a model-free price enforced by arbitrage, just as single-asset European claims do.

The value of insider information for super-replication with quadratic transaction costs

We study super-replication of European contingent claims in an illiquid market with insider information. Illiquidity is captured by quadratic transaction costs and insider information is modeled by an investor who can peek into the future. Our main result describes the scaling limit of the super-replication prices when the number of trading periods increases to infinity. Moreover, the scaling limit gives us the asymptotic value of being an insider.

Approximate XVA for European claims

We consider the problem of computing the Value Adjustment of European contingent claims when default of either party is considered, possibly including also funding and collateralization requirements. As shown in Brigo et al. (\cite{BLPS}, \cite{BFP}), this leads to a more articulate variety of Value Adjustments ({XVA}) that introduce some nonlinear features. When exploiting a reduced-form approach for the default times, the adjusted price can be characterized as the solution to a possibly nonlinear Backward Stochastic Differential Equation (BSDE). The expectation representing the solution of the BSDE is usually quite hard to compute even in a Markovian setting, and one might resort either to the discretization of the Partial Differential Equation characterizing it or to Monte Carlo Simulations. Both choices are computationally very expensive and in this paper we suggest an approximation method based on an appropriate change of numeraire and on a Taylor's polynomial expansion when intensities are represented by means of affine processes correlated with the asset's price. The numerical discussion at the end of this work shows that, at least in the case of the CIR intensity model, even the simple first-order approximation has a remarkable computational efficiency.

Quantile hedging in models with dividends and application to equity-linked life insurance contracts

The paper demonstrates the effect of the dividends on pricing and hedging the European contingent claims under a budget constraint and presents insurance applications. Explicit formulae for the quantile pricing and hedging of the European call option are derived assuming the jump-diffusion model of the financial market. These results are used to determine the premium of the pure endowment with fixed guarantee equity-linked life insurance contract as well as the survival probability of the insured. A numerical example is given to illustrate the role of dividends in valuation and risk management of such insurance contracts.

A Functional Analysis Approach to Static Replication of European Options

The replication of any European contingent claim by a static portfolio of calls and puts with strikes forming a continuum, formally proven by Carr and Madan (1998), is part of the more general theory of integral equations. We apply spectral decomposition techniques to show that replication may also be achieved with a discrete portfolio of special options. We propose a numerical application for fast pricing of vanilla options that may be suitable for large option books or high frequency option trading, and we use a reflected Brownian motion model to show how pricing formulas for the special options may be obtained.

The CEV model and its application to financial markets with volatility uncertainty

We survey the financial markets whose risks are caused by uncertain volatilities. The financial markets focus on the assets which are effectively allocated in one risk-free asset and one risky asset, whose price process is governed by the constant elasticity of variance ( C E V for short) model which contains the G -Brownian motion rather than the classical Brownian motion. Such the C E V model which includes the G -Brownian motion utilized to financial markets is the extension of the classical C E V model. Applying the concept of arbitrage and the properties of G -expectation, we consider stock price dynamics which exclude arbitrage opportunities. Moreover, the interval of no-arbitrage price for the general European contingent claims is found in the Markovian case.

An Implicit-Explicit Computational Method Based on Time Semi-Discretization for Pricing Financial Derivatives with Jumps

This paper considers pricing European options under the well-known of SVJ model of Bates and related computational methods. According to the no-arbitrage principle, we first derive a partial differential equation that the value of any European contingent claim should satisfy, where the asset price obeys the SVJ model. This equation is numerically solved by using the implicit- explicit backward difference method and time semi-discretization. In order to explain the validity of our method, the stability of time semi-discretization scheme is also proved. Finally, we use a simulation example to illustrate the efficiency of the method.