Utility Indifference Valuation Research Articles

Utility Indifference Valuation for Defaultable Corporate Bond with Credit Rating Migration

Credit risk modeling by debt pricing has been a popular theme in both academia and practice since the subprime crisis. In this paper, we devote our study to the indifferent price of a corporate bond with credit risk involving both default risk and credit rating migration risk in an incomplete market. The firm’s stock and a financial index on the market as tradable assets are introduced to hedge the credit risk, and the bond price is determined by the indifference of investors’ utilities with and without holding the bond. The models are established under the structural framework and result in Hamilton–Jacobi–Bellman (HJB) systems regarding utilities subject to default boundary and multiple migration boundaries. According to dynamic programming theory, closed-form solutions for pricing formulas are derived by implementing an inverted iteration program to overcome the joint effect of default and multiple credit rating migration. Therefore, with the derived explicit pricing formulas for the corporate bond, the models can be easily applied in practice, and investors can generate their strategies of hedging the credit risk by easily analyzing the impacts of the parameters on the bond price.

Pricing weather derivatives with the market price of risk extracted from the utility indifference valuation

In this paper, a PDE (partial differential equation) based approach is presented to price weather derivatives with the market price of risk extracted from the utility indifference valuation. Assuming that the underlying temperature follows an Ornstein–Uhlenbeck process, the PDEs associated with the utility indifference valuation are established and then solved numerically using a one-sided finite difference scheme. The solution procedure is validated through numerical experiments for the utility indifference futures prices, and then applied to price more complicated weather derivatives such as options.

Nash Equilibria for Game Contingent Claims with Utility-Based Hedging

Game contingent claims (GCCs) generalize American contingent claims in allowing the writer to recall the option as long as it is not exercised, at the price of paying some penalty. In incomplete markets, an appealing approach is to analyze GCCs like their European and American counterparts by solving option holder's and writer's optimal investment problems in the underlying securities. By this, partial hedging opportunities are taken into account. We extend results in the literature by solving the stochastic game corresponding to GCCs with both continuous time stopping and trading. Namely, we construct Nash equilibria by rewriting the game as a non-zero-sum stopping game in which players compare payoffs in terms of their exponential utility indifference values. As a by-product, we also obtain an existence result for the optimal exercise time of an American claim under utility indifference valuation by relating it to the corresponding nonlinear Snell envelope.

Utility indifference valuation of corporate bond with credit rating migration by structure approach

An indifference pricing model for corporate bond with rating migration risk is established in this article. Under the structural framework, the credit rating migration is modeled by the first attempt in an incomplete market, so far as we know. The model results in a HJB system. With the help of the dynamic programming theory, a closed form solution is derived by imposing a condition on the credit rating migration boundary. With the explicit migration boundary and closed form solution, the model is easy to be applied in practice. Based on the pricing formula, the impacts of the parameters on the indifference price are analyzed and relative financial explanations are discussed.

Exponential utility indifference value process in a general jump model based on random measures

This paper deals with the dynamic exponential utility indifferent value process of a contingent claim. We regard the risky asset pricing process as a semimartingale with general jumps based on a random measure, where the GKW (Galtchouk-Kunita-Watanabe) decomposition of the martingale part of the exponential utility indifferent value process's Doob-Meyer decomposition satisfies the Jacod decomposition. A backward stochastic differential equation (BSDE) is established from the fact that the exponential utility indifferent value process is a martingale for the optimal investment strategy under the minimal entropy measure. The uniqueness of the solution to the equation is proved by converting the BSDE to bounded mean oscillation (BMO) martingales under a new probability measure. The division of the equation's generator into [ΔA = 0] and [ΔA≠ 0] proves the existence of an optimal investment strategy. As a result, the exponential utility indifference value process of the contingent claim is the solution of the BSDE.

Utility Indifference Valuation for Non-smooth Payoffs with an Application to Power Derivatives

We consider the problem of exponential utility indifference valuation under the simplified framework where traded and nontraded assets are uncorrelated but where the claim to be priced possibly depends on both. Traded asset prices follow a multivariate Black and Scholes model, while nontraded asset prices evolve as generalized Ornstein---Uhlenbeck processes. We provide a BSDE characterization of the utility indifference price (UIP) for a large class of non-smooth, possibly unbounded, payoffs depending simultaneously on both classes of assets. Focusing then on Vanilla claims and using the Gaussian structure of the model allows us to employ some BSDE techniques (in particular, a Malliavin-type representation theorem due to Ma and Zhang, Ann Appl Probab 12:1390---1418, 2002) to prove the regularity of Z and to characterize the UIP for possibly discontinuous Vanilla payoffs as a viscosity solution of a suitable PDE with continuous space derivatives. The optimal hedging strategy is also identified essentially as the delta hedging strategy corresponding to the UIP. Since there are no closed-form formulas in general, we also obtain asymptotic expansions for prices and hedging strategies when the risk aversion parameter is small. Finally, our results are applied to pricing and hedging power derivatives in various structural models for energy markets.

Utility indifference valuation of corporate bond with rating migration risk

A pricing model for a corporate bond with rating migration risk is established in this article. With the technology of utility-indifference valuation under the Markov-modulated framework, we analyze the price of a multi-rating bond and obtain closed formulae in a three-rating case. Based on the pricing formulae, the impacts of the parameters on the indifference price are analyzed and some reasonable financial explanations are provided as well.

Indifference pricing and hedging in a multiple-priors model with trading constraints

This paper considers utility indifference valuation of derivatives under model uncertainty and trading constraints, where the utility is formulated as an additive stochastic differential utility of both intertemporal consumption and terminal wealth, and the uncertain prospects are ranked according to a multiple-priors model of Chen and Epstein (2002). The price is determined by two optimal stochastic control problems (mixed with optimal stopping time in the case of American option) of forward-backward stochastic differential equations. By means of backward stochastic differential equation and partial differential equation methods, we show that both bid and ask prices are closely related to the Black-Scholes risk-neutral price with modified dividend rates. The two prices will actually coincide with each other if there is no trading constraint or the model uncertainty disappears. Finally, two applications to European option and American option are discussed.

The exp-UIV for Markets with Partial Information and Complete Information

We consider an incomplete market with two information structures, complete and partial information and , respectively. The dynamics of the market are given by a risky asset driven by a m-dimensional Brownian motion W = (W1, …, Wm)′ as well as an integer-valued random measure μ(du, dy). To study the values with respect to the different information filtrations, we introduce the concept of dynamic < ![CDATA[exp]] >-utility indifference value (UIV) of with respect to denoted by Ct and the concept of dynamic < ![CDATA[exp]] >-UIV of the contingent claim H denoted by Ct(H), and we describe the dynamics of Ct and Ct(H) by BSDEs.

Explicit solutions to quadratic BSDEs and applications to utility maximization in multivariate affine stochastic volatility models

Over the past few years quadratic Backward Stochastic Differential Equations (BSDEs) have been a popular field of research. However there are only very few examples where explicit solutions for these equations are known. In this paper we consider a class of quadratic BSDEs involving affine processes and show that their solution can be reduced to solving a system of generalized Riccati ordinary differential equations. In other words we introduce a rich and flexible class of quadratic BSDEs which are analytically tractable, i.e. explicit up to the solution of an ODE. Our results also provide analytically tractable solutions to the problem of utility maximization and indifference pricing in multivariate affine stochastic volatility models. This generalizes univariate results of Kallsen and Muhle-Karbe (2010) and some results in the multivariate setting of Leippold and Trojani (2010) by establishing the full picture in the multivariate affine jump-diffusion setting. In particular we calculate the interesting quantity of the power utility indifference value of change of numeraire. Explicit examples in the Heston, Barndorff-Nielsen–Shephard and multivariate Heston setting are calculated.

Pseudo linear pricing rule for utility indifference valuation

This paper considers exponential utility indifference pricing for a multidimensional non-traded assets model, and provides two linear approximations for the utility indifference price. The key tool is a probabilistic representation for the utility indifference price by the solution of a functional differential equation, which is termed pseudo linear pricing rule. We also provide an alternative derivation of the quadratic BSDE representation for the utility indifference price.

A pricing defaultable claim model based on exponential utility indifference value process in an incomplete market

We study the dynamics of exponential utility indifference value process C t ( B , α ) for a defaultable claim in an incomplete market. Other than most of papers dealing with defautable claim, we do not assume the hypothesis of invariance of martingale that is H hypothesis. We firstly transform the default sensitive assets S to be a local martingale through progressively enlargement of filtration and changes of measure. Then, we introduce a specific backward stochastic differential equation (BSDE) and prove the existence and uniqueness of the solution. Finally, we prove the exponential utility indifference value process C t ( B , α ) is the solution to the BSDE based on the fact that C t ( B , α ) is a supermartingale for investment strategies and a martingale for optimal strategy under the minimal entropy martingale measure.

Accounting for risk aversion in derivatives purchase timing

We study the problem of optimal timing to buy/sell derivatives by a risk-averse agent in incomplete markets. Adopting the exponential utility indifference valuation, we investigate this timing flexibility and the associated delayed purchase premium. This leads to a stochastic control and optimal stopping problem that combines the observed market price dynamics and the agent’s risk preferences. Our results extend recent work on indifference valuation of American options, as well as the authors’ first paper (Leung and Ludkovski, SIAM J. Fin. Math., 2011). In the case of Markovian models of contracts on non-traded assets, we provide analytical characterizations and numerical studies of the optimal purchase strategies, with applications to both equity and credit derivatives.

Convergence results for the indifference value based on the stability of BSDEs

We study the exponential utility indifference value h for a contingent claim H in an incomplete market driven by two Brownian motions. The claim H depends on a non-tradable asset variably correlated with the traded asset available for hedging. We provide an explicit sequence that converges to h, complementing the structural results for h known from the literature. Our study is based on a convergence result for quadratic backward stochastic differential equations. This convergence result, which we prove in a general continuous filtration under weak conditions, also yields that the indifference value in a setting with trading constraints enjoys a continuity property in the constraints.

Accounting for Risk Aversion in Derivatives Purchase Timing

We study the problem of optimal timing to buy/sell derivatives by a risk-averse agent in incomplete markets. Adopting the exponential utility indifference valuation, we investigate this timing flexibility and the associated delayed purchase premium. This leads to a stochastic control and optimal stopping problem that combines the observed market price dynamics and the agent's risk preferences. Our results extend recent work on indifference valuation of American options, as well as the authors' first paper (Leung and Ludkovski, SIAM J. Fin. Math., 2011). In the case of Markovian models of contracts on non-traded assets, we provide analytical characterizations and numerical studies of the optimal purchase strategies, with applications to both equity and credit derivatives.

Optimal Exponential Utility in a Jump Bond Market

We consider the optimal exponential utility in a bond market with jumps basing on a model similar to Björk et al. [4], which is arbitrage free. Similar to the normalized integral with respect to the cylindrical martingale first introduced in Mikulevicius and Rozovskii [13], we introduce the (𝕄, Q 0)-normalized martingale and local (𝕄, Q 0)-normalized martingale. For a given maturity T 0 ∈ [0, T*], we describe the minimal entropy martingale (MEM) based on [T 0, T*] by a backward semimartingale equation (BSE) w.r.t. the (𝕄, Q 0)-normalized martingale. Then we give an explicit form of the optimal approximate wealth to the optimal exp-utility problem by making use of the solution of the BSE. Finally, we describe the dynamics of the exp utility indifference valuation of a bounded contingent claim H ∈ L ∞(ℱ T 0 ) by another BSE under the minimal entropy martingale measure in the incomplete market.

ROBUST EXPONENTIAL HEDGING AND INDIFFERENCE VALUATION

We discuss the problem of exponential hedging in the presence of model uncertainty expressed by a set of probability measures. This is a robust utility maximization problem with a contingent claim. We first consider the dual problem which is the minimization of penalized relative entropy over a product set of probability measures, showing the existence and variational characterizations of the solution. These results are applied to the primal problem. Then we consider the robust version of exponential utility indifference valuation, giving the representation of indifference price using a duality result.

Utility indifference valuation for jump risky assets

We discuss utility maximization problems with exponential preferences in an incomplete market where the risky asset dynamics is described by a pure jump process driven by two independent Poisson processes. This includes results on portfolio optimization under an additional European claim. Value processes of the optimal investment problems, optimal hedging strategies and the indifference price are represented in terms of solutions to backward stochastic equations driven by the Poisson martingales. Via a duality result, the solution to the dual problems is derived. In particular, an explicit expression for the density of the minimal martingale measure is provided. The Markovian case is also discussed. This includes either asset dynamics dependent on a pure jump stochastic factor or claims written on a correlated non tradable asset.

A Modified Structural Model for Credit Risk: Utility Indifference Valuation

This paper modifies the classical structural models for credit risk by embedding them into the framework of optimal portfolio problems in an incomplete market. The price of corporate bonds is derived based on the indifference between the investor's two utility maximization problems. Besides the uncertainty of the firm's asset value, we introduce another diffusion process, which comes from the firm's stock value, to drive credit risk. This results in the different behaviors of the credit spread from classical structural models. Two new parameters are introduced into the model, namely the investor's risk aversion parameter and the correlation coefficient between the firm's assets and the stocks it issues, which results in the nonlinearity of the pricing rule. Under the Markovian framework, default occurring at the maturity and default occurring at the first-passage-time are considered, and the corresponding closed formulae are derived by solving the HJB equation Cauchy problem and the HJB equation Dirichlet problem respectively.

Exponential utility indifference valuation in two Brownian settings with stochastic correlation

We study the exponential utility indifference valuation of a contingent claimBin an incomplete market driven by two Brownian motions. The claim depends on a nontradable asset stochastically correlated with the traded asset available for hedging. We use martingale arguments to provide upper and lower bounds, in terms of bounds on the correlation, for the valueVBof the exponential utility maximization problem with the claimBas random endowment. This yields an explicit formula for the indifference valuebofBat any time, even with a fairly general stochastic correlation. Earlier results with constant correlation are recovered and extended. The reason why all this works is that, after a transformation to the minimal martingale measure, the valueVBenjoys a monotonicity property in the correlation between tradable and nontradable assets.