OPTIMAL STATIC–DYNAMIC HEDGES FOR BARRIER OPTIONS

We study optimal hedging of barrier options, using a combination of a static position in vanilla options and dynamic trading of the underlying asset. The problem reduces to computing the Fenchel–Legendre transform of the utility-indifference price as a function of the number of vanilla options used to hedge. Using the well-known duality between exponential utility and relative entropy, we provide a new characterization of the indifference price in terms of the minimal entropy measure, and give conditions guaranteeing differentiability and strict convexity in the hedging quantity, and hence a unique solution to the hedging problem. We discuss computational approaches within the context of Markovian stochastic volatility models.     

Convex duality for Epstein–Zin stochastic differential utility

This paper introduces a dual problem to study a continuous‐time consumption and investment problem with incomplete markets and Epstein–Zin stochastic differential utilities. Duality between the primal and dual problems is established. Consequently, the optimal strategy of this consumption and investment problem is identified without assuming several technical conditions on market models, utility specifications, and agent's admissible strategies. Meanwhile, the minimizer of the dual problem is identified as the utility gradient of the primal value and is economically interpreted as the “least favorable” completion of the market.     

MULTI‐ASSET STOCHASTIC LOCAL VARIANCE CONTRACTS

Variance swaps now trade actively over‐the‐counter (OTC) on both stocks and stock indices. Also trading OTC are variations on variance swaps which localize the payoff in time, in the underlying asset price, or both. Given that the price of the underlying asset evolves continuously over time, it is well known that there exists a semirobust hedge for these localized variance contracts. Remarkably, the hedge succeeds even though the stochastic process describing the instantaneous variance is never specified. In this paper, we present a generalization of these results to the case of two or more underlying assets.     

A FIRST‐ORDER BSPDE FOR SWING OPTION PRICING

We study an optimal control problem related to swing option pricing in a general non‐Markovian setting in continuous time. As a main result we uniquely characterize the value process in terms of a first‐order nonlinear backward stochastic partial differential equation and a differential inclusion. Based on this result we also determine the set of optimal controls and derive a dual minimization problem.     

Super‐replication in fully incomplete markets

In this work, we introduce the notion of fully incomplete markets. We prove that for these markets, the super‐replication price coincides with the model‐free super‐replication price. Namely, the knowledge of the model does not reduce the super‐replication price. We provide two families of fully incomplete models: stochastic volatility models and rough volatility models. Moreover, we give several computational examples. Our approach is purely probabilistic.     

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, 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 interacting subsidiaries; optimal risk sharing for ‐tolerant g‐risk measures, and risk contribution for coherent g‐risk measures are investigated. Insurance g‐risk measure and other ways to induce g‐risk measures are also studied at the end of the paper.     

Analytical approximations of local‐Heston volatility model and error analysis

This paper studies the expansion of an option price (with bounded Lipschitz payoff) in a stochastic volatility model including a local volatility component. The stochastic volatility is a square root process, which is widely used for modeling the behavior of the variance process (Heston model). The local volatility part is of general form, requiring only appropriate growth and boundedness assumptions. We rigorously establish tight error estimates of our expansions, using Malliavin calculus. The error analysis, which requires a careful treatment because of the lack of weak differentiability of the model, is interesting on its own. Moreover, in the particular case of call–put options, we also provide expansions of the Black–Scholes implied volatility that allow to obtain very simple formulas that are fast to compute compared to the Monte Carlo approach and maintain a very competitive accuracy.     

OPTIMAL DIVIDEND PAYMENTS WHEN CASH RESERVES FOLLOW A JUMP‐DIFFUSION PROCESS

We consider a model in which a firm faces two types of liquidity risks: a Brownian risk and a Poisson risk. The firm chooses a dividend policy to maximize shareholder value. We characterize the optimal firm value and we show that the optimal dividend policy is a barrier strategy: the firm keeps cash inside when the cash reserves level is less than a critical threshold and pays cash in excess of this threshold. We also analyze the problem of insurance against the Poisson risk. We find that it is optimal for the firm to buy full insurance when its cash reserves are above a critical threshold and not to insure otherwise.     

ARBITRAGE‐FREE MULTIFACTOR TERM STRUCTURE MODELS: A THEORY BASED ON STOCHASTIC CONTROL

We present an alternative approach to the pricing of bonds and bond derivatives in a multivariate factor model for the term structure of interest rates that is based on the solution of an optimal stochastic control problem. It can also be seen as an alternative to the classical approach of computing forward prices by forward measures and as such can be extended to other situations where traditionally a change of measure is involved based on a change of numeraire. We finally provide explicit formulas for the computation of bond options in a bivariate linear‐quadratic factor model.     

A FIRST‐ORDER BSPDE FOR SWING OPTION PRICING: CLASSICAL SOLUTIONS

In a companion paper, we studied a control problem related to swing option pricing in a general non‐Markovian setting. The main result there shows that the value process of this control problem can uniquely be characterized in terms of a first‐order backward stochastic partial differential equation (BSPDE) and a pathwise differential inclusion. In this paper, we additionally assume that the cash flow process of the swing option is left‐continuous in expectation. Under this assumption, we show that the value process is continuously differentiable in the space variable that represents the volume in which the holder of the option can still exercise until maturity. This gives rise to an existence and uniqueness result for the corresponding BSPDE in a classical sense. We also explicitly represent the space derivative of the value process in terms of a nonstandard optimal stopping problem over a subset of predictable stopping times. This representation can be applied to derive a dual minimization problem in terms of martingales.     

Arbitrage‐free XVA

We develop a framework for computing the total valuation adjustment (XVA) of a European claim accounting for funding costs, counterparty credit risk, and collateralization. Based on no‐arbitrage arguments, we derive backward stochastic differential equations associated with the replicating portfolios of long and short positions in the claim. This leads to the definition of buyer's and seller's XVA, which in turn identify a no‐arbitrage interval. In the case that borrowing and lending rates coincide, we provide a fully explicit expression for the unique XVA, expressed as a percentage of the price of the traded claim, and for the corresponding replication strategies. In the general case of asymmetric funding, repo, and collateral rates, we study the semilinear partial differential equations characterizing buyer's and seller's XVA and show the existence of a unique classical solution to it. To illustrate our results, we conduct a numerical study demonstrating how funding costs, repo rates, and counterparty risk contribute to determine the total valuation adjustment.     

STOCK LIQUIDATION VIA STOCHASTIC APPROXIMATION USING NASDAQ DAILY AND INTRA-DAY DATA

By focusing on computational aspects, this work is concerned with numerical methods for stock selling decision using stochastic approximation methods. Concentrating on the class of decisions depending on threshold values, an optimal stopping problem is converted to a parametric stochastic optimization problem. The algorithms are model free and are easily implementable on-line. Convergence of the algorithms is established, second moment bound of estimation error is obtained, and escape probability from a neighborhood of the true parameter is also derived. Numerical examples using both daily closing prices and intra-day data are provided to demonstrate the performance of the algorithms.     

MOMENT EXPLOSIONS AND LONG‐TERM BEHAVIOR OF AFFINE STOCHASTIC VOLATILITY MODELS

We consider a class of asset pricing models, where the risk‐neutral joint process of log‐price and its stochastic variance is an affine process in the sense of Duffie, Filipovic, and Schachermayer. First we obtain conditions for the price process to be conservative and a martingale. Then we present some results on the long‐term behavior of the model, including an expression for the invariant distribution of the stochastic variance process. We study moment explosions of the price process, and provide explicit expressions for the time at which a moment of given order becomes infinite. We discuss applications of these results, in particular to the asymptotics of the implied volatility smile, and conclude with some calculations for the Heston model, a model of Bates and the Barndorff‐Nielsen–Shephard model.     

EXPLICIT IMPLIED VOLATILITIES FOR MULTIFACTOR LOCAL‐STOCHASTIC VOLATILITY MODELS

We consider an asset whose risk‐neutral dynamics are described by a general class of local‐stochastic volatility models and derive a family of asymptotic expansions for European‐style option prices and implied volatilities. We also establish rigorous error estimates for these quantities. Our implied volatility expansions are explicit; they do not require any special functions nor do they require numerical integration. To illustrate the accuracy and versatility of our method, we implement it under four different model dynamics: constant elasticity of variance local volatility, Heston stochastic volatility, three‐halves stochastic volatility, and SABR local‐stochastic volatility.     

TUG‐OF‐WAR,MARKET MANIPULATION,AND OPTION PRICING

We develop an option pricing model based on a tug‐of‐war game. This two‐player zero‐sum stochastic differential game is formulated in the context of a multidimensional financial market. The issuer and the holder try to manipulate asset price processes in order to minimize and maximize the expected discounted reward. We prove that the game has a value and that the value function is the unique viscosity solution to a terminal value problem for a parabolic partial differential equation involving the nonlinear and completely degenerate infinity Laplace operator.     

BILATERAL COUNTERPARTY RISK UNDER FUNDING CONSTRAINTS—PART II: CVA

The correction in value of an over‐the‐counter derivative contract due to counterparty risk under funding constraints is represented as the value of a dividend‐paying option on the value of the contract clean of counterparty risk and excess funding costs. This representation allows one to analyze the structure of this correction, the so‐called Credit Valuation Adjustment (CVA for short), in terms of replacement cost/benefits, credit cost/benefits, and funding cost/benefits. We develop a reduced‐form backward stochastic differential equations (BSDE) approach to the problem of pricing and hedging the CVA. In the Markov setup, explicit CVA pricing and hedging schemes are formulated in terms of semilinear partial differential equations.     

CVaR HEDGING USING QUANTIZATION‐BASED STOCHASTIC APPROXIMATION ALGORITHM

In this paper, we investigate a method based on risk minimization to hedge observable but nontradable source of risk on financial or energy markets. The optimal portfolio strategy is obtained by minimizing dynamically the conditional value‐at‐risk (CVaR) using three main tools: a stochastic approximation algorithm, optimal quantization, and variance reduction techniques (importance sampling and linear control variable), as the quantities of interest are naturally related to rare events. As a first step, we investigate the problem of CVaR regression, which corresponds to a static portfolio strategy where the number of units of each tradable assets is fixed at time 0 and remains unchanged till maturity. We devise a stochastic approximation algorithm and study its a.s. convergence and weak convergence rate. Then, we extend our approach to the dynamic case under the assumption that the process modeling the nontradable source of risk and financial assets prices is Markovian. Finally, we illustrate our approach by considering several portfolios in connection with energy markets.     

AN EXACT FORMULA FOR DEFAULT SWAPTIONS’ PRICING IN THE SSRJD STOCHASTIC INTENSITY MODEL

We develop and test a fast and accurate semi‐analytical formula for single‐name default swaptions in the context of a shifted square root jump diffusion (SSRJD) default intensity model. The model can be calibrated to the CDS term structure and a few default swaptions, to price and hedge other credit derivatives consistently. We show with numerical experiments that the model implies plausible volatility smiles.     

Pricing Stock Options in a Jump‐Diffusion Model with Stochastic Volatility and Interest Rates: Applications of Fourier Inversion Methods

Fast closed form solutions for prices on European stock options are developed in a jump‐diffusion model with stochastic volatility and stochastic interest rates. The probability functions in the solutions are computed by using the Fourier inversion formula for distribution functions. The model is calibrated for the S and P 500 and is used to analyze several effects on option prices, including interest rate variability, the negative correlation between stock returns and volatility, and the negative correlation between stock returns and interest rates.     

Analytical valuation of Asian options with counterparty risk under stochastic volatility models

In this paper, we consider Asian options with counterparty risk under stochastic volatility models. We propose a simple way to construct stochastic volatility models through the market factor channel. In the proposed framework, we obtain an explicit pricing formula of Asian options with counterparty risk and illustrate the effects of systematic risk on Asian option prices. Specially, the U-shaped and inverted U-shaped curves appear when we keep the total risk of the underlying asset and the issuer's assets unchanged, respectively.