Dewen Xiong

Mean Variance Hedging in a General Jump Model

Abstract We consider the mean-variance hedging of a contingent claim H when the discounted price process S is an -valued quasi-left continuous semimartingale with bounded jumps. We relate the variance-optimal martingale measure (VOMM) to a backward semimartingale equation (BSE) and show that the VOMM is equivalent to the original measure P if and only if the BSE has a solution. For a general contingent claim, we derive an explicit solution of the optimal strategy and the optimal cost of the mean-variance hedging by means of another BSE and an appropriate predictable process δ

Change of filtrations and mean–variance hedging

We consider the mean–variance hedging (MVH) problem (under measure P) of two kinds of investors for two different levels of information, described by two filtrations and such that . Under the assumption that there exists a measure such that all -martingales are -martingales, we give the variance-optimal martingale measure (VOMM) with respect to and through a couple of stochastic Riccati equation (SRE)s, which can be viewed as the same SRE with differential terminal value under . Then we derive an explicit form of the optimal mean–variance strategy and the optimal costs with respect to and . We describe the concept of -no-value-to-investment in the means of mean–variance, and for a given contingent claim , we compare their optimal costs with respect to and .

The Minimal Entropy and the Convergence of the p-Optimal Martingale Measures in a General Jump Model

Abstract We first consider the minimal entropy martingale measure in a general jump model introduced in Kohlmann and Xiong (International Journal of Pure and Applied Mathematics, 37(3):321–348) and give a description of this measure (MEMM) as the solution of a backward martingale equation (BME). To relate the (MEMM) to the p-optimal martingale measure (p-OMM) we consider the convergence of the solution of the BME associated with the (p-OMM) to the solution of the BME associated with the MEMM. Under some assumptions, we prove the convergence of the p-OMM to the MEMM both in entropy and strongly in L 1(P). As an application, we consider the exp-optimal utility of an investor with utility function U exp(x) = −exp(− k 0 x), and as q↑∞, we show that the q-optimal terminal wealth of an investor with utility converges to the exp-optimal terminal wealth of an investor with utility function U exp(x) strongly in L r (P) for a large enough r.

Defaultable Bond Markets with Jumps

We construct a model for the term structure in a market of defaultable bonds with jumps {p d (t, T); t ≤ T}, T ∈ (0, T*]. We derive the instantaneous defaultable forward rate f d (t, T) defined by in the real world probability. We are also given default-free bonds {p(t, T); t ≤ T}, T ∈ (0, T*] and we establish the market consisting of both the defaultable and the nondefaultable bonds. In this market, we study the common equivalent martingale measure and in this arbitrage free market we derive the relationship between the forward rates f(t, T) and f d (t, T) associated with the two sorts of bonds. Especially, it is proved that in a parameterized market with common equivalent martingale measure where f(t, T) can be described by (1.1) the defaultable forward rate f d (t, T) can be reconstructed from the special form of the default-free forward rate f(t, T) if a certain system of BSDEs has a solution. Finally, we extend the results to a market with recovery rate and give examples where the system of BSDEs has a solution.

The Mean-Variance Hedging in a Bond Market with Jumps

We construct a market of bonds with jumps driven by a general marked point process as well as by a ℝ n -valued Wiener process based on Björk et al. [6], in which there exists at least one equivalent martingale measure Q 0. Then we consider the mean-variance hedging of a contingent claim H ∈ L 2(ℱ T 0 ) based on the self-financing portfolio based on the given maturities T 1,…, T n with T 0 < T 1 < … <T n ≤ T*. We introduce the concept of variance-optimal martingale (VOM) and describe the VOM by a backward semimartingale equation (BSE). By making use of the concept of ℰ*-martingales introduced by Choulli et al. [8], we obtain another BSE which has a unique solution. We derive an explicit solution of the optimal strategy and the optimal cost of the mean-variance hedging by the solutions of these two BSEs.

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.

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.

The Dynamic Convex Valuation Related to the Price Process in a Market with General Jumps

Abstract We consider an incomplete market with general jumps in the given price process S of a risky asset. We define the S-related dynamic convex valuation (S-related DCV) which is time-consistent. We discuss the representation for a given S-related DCV C in terms of a ‘penalty functional’ α and give some characteristics of α, which are the sufficient conditions for a given C to be an S-related DCV. Finally, we give two special forms of α satisfying those conditions to describe the dynamics of the corresponding S-related DCV by a backward semimartingale equation.

Utility maximization under g*-expectation

ABSTRACT In this article, we introduce a nonlinear expectation, called g*-expectation, based on g-expectation and consider the optimal utility under g*-expectation in the market with a risk-free bond and d risky stocks in finite trading interval [0, T]. We construct a stochastic family by taking advantage of the comparison theorem of backward stochastic differential equations and the g*-martingale. We generalize the results of Hu et al. (Annals of Applied Probability 28(2):1691–1712, 2005), and obtain the explicit forms of the optimal trading strategies both for exp -utility and the power utility, when g(t, z) = βt|z|2 + γtz.

Modeling the forward CDS spreads with jumps

We consider the forward CDS in the framework of stochastic interest rates whose term structures are modeled in the sense of the Heath–Jarrow–Morton model with jumps adapted to a filtration 𝔽 (see [2]). Under the assumption that the density process of the default is a bounded 𝔽-predictable process, we obtain a quadratic-exponential type system of BSDEs similar to [2], which always has a unique solution (X, θ, ϑ). By the solution of such a system of BSDEs, we will describe the dynamics of the the pre-default values of the defaultable bond, the defaultable forward Libor rates and the restricted defaultable forward measure (see in [6]) explicitly. Then we introduce another quadratic-exponential type system of BSDEs (called adjoint system of BSDEs), which also always has a unique solution, and, using this solution, we describe the dynamic of the fair spread of the forward CDS with the tenor structure 𝕋 = {a = T 0 < T 1 < …