Mean-variance Hedging for Pricing European-type Contingent Claims with Transaction Costs

In this paper,a European-type contingent claim pricing problem with transaction costs is considered by a mean-variance hedging argument.The investor has to pay transaction costs which areproportional to the amount of stock transacted.The writer‘‘s hedging object is to minimize the hedgingrisk,defined as the variance of hedging error at expiration,with a proper expected excess return level.At first, we consider the mean-variance hedging problem:for initial hedging wealth f,maximizing the excess expected return under the minimum hedging risk level V0.On the other hand,we consider a mean-variance portfolio problem,which is to maximize the expected return with initial wealth 0 under the same risk level V0.The minimum initial hedging wealth f,which can offset the difference of the maximum expected return of these two problems,is the writer‘s price.     

A numerically efficient closed-form representation of mean-variance hedging for exponential additive processes based on Malliavin calculus

ABSTRACT

We focus on mean-variance hedging problem for models whose asset price follows an exponential additive process. Some representations of mean-variance hedging strategies for jump-type models have already been suggested, but none is suited to develop numerical methods of the values of strategies for any given time up to the maturity. In this paper, we aim to derive a new explicit closed-form representation, which enables us to develop an efficient numerical method using the fast Fourier transforms. Note that our representation is described in terms of Malliavin derivatives. In addition, we illustrate numerical results for exponential Lévy models.     

Dual Representation of the Cost of Designing a Portfolio Satisfying Multiple Risk Constraints

ABSTRACT

We consider, within a Markovian complete financial market, the problem of finding the least expensive portfolio process meeting, at each payment date, three different types of risk criterion. Two of them encompass an expected utility-based measure and a quantile hedging constraint imposed at inception on all the future payment dates, while the other one is a quantile hedging constraint set at each payment date over the next one. The quantile risk measures are defined with respect to a stochastic benchmark and the expected utility-based constraint is applied to random payment dates. We explicit the Legendre-Fenchel transform of the pricing function. We also provide, for each quantile hedging problem, a backward dual algorithm allowing to compute their associated value function by backward recursion. The algorithms are illustrated with a numerical example.     

The Mean-Variance Hedging of a Defaultable Option with Partial Information

Abstract

We consider the mean-variance hedging of a defaultable claim in a general stochastic volatility model. By introducing a new measure Q ⁰, we derive the martingale representation theorem with respect to the investors' filtration . We present an explicit form of the optimal-variance martingale measure by means of a stochastic Riccati equation (SRE). For a general contingent claim, we represent the optimal strategy and the optimal cost of the mean-variance hedging by means of another backward stochastic differential equation (BSDE). For the defaultable option, especially when there exists a random recovery rate we give an explicit form of the solution of the BSDE.     

Risk minimizing hedging for a partially observed high frequency data model

Risk-minimizing hedging strategies for contingent claims are studied in a general model for intraday stock price movements in the case of partial information. The dynamics of the risky asset price is described throught a marked point process Y, whose local characteristics depend on some unobservable hidden state variable X. In the model presented the processes Y and X may have common jump times, which means that the trading activity may affect the law of X and could be also related to the presence of catastrophic events. The hedger is restricted to observing past asset prices. Thus, we are in presence not only of an incomplete market situation but also of partial information. Considering the case where the price of the risky asset is modeled directly under a martingale measure, the computation of the risk-minimizing hedging strategy under this partial information is obtained by using a projection result (M. Schweizer, Risk minimizing hedging strategies under restricted information, Mathematical Finance 4 (1994) 327–342). This approach leads to a filtering problem with marked point process observations whose solution, obtained via the Kushner-Stratonovich equation, allows us to provide a complete solution to the heding problem.     

Some results on quadratic hedging with insider trading

We consider the hedging problem in an arbitrage-free incomplete financial market, where there are two kinds of investors with different levels of information about the future price evolution, described by two filtrations F and G=F∨σ(G) where G is a given r.v. representing the additional information. We focus on two types of quadratic approaches to hedge a given square-integrable contingent claim: local risk minimization (LRM) and mean-variance hedging (MVH). By using initial enlargement of filtrations techniques, we solve the hedging problem for both investors and compare their optimal strategies under both approaches.

In particular, for LRM, we show that for a large class of additional non trivial r.v.s G both investors will pursue the same locally risk minimizing portfolio strategy and the cost process of the ordinary agent is just the projection on F of that of the insider. For the MVH approach, we study also some general stochastic volatility model, including Hull and White, Heston and Stein and Stein models. In this more specific setting and for r.v.s G which are measurable with respect to the filtration generated by the volatility process, we obtain an expression for the insider optimal strategy in terms of the ordinary agent optimal strategy plus a process admitting a simple feedback-type representation.     

Cooperative hedging in the complete market under <Emphasis Type="Italic">g</Emphasis>-expectation constraint

The paper studies the muiti-agent cooperative hedging problem of contingent claims in the complete market when the g-expected shortfall risks are bounded. We give the optimal cooperative hedging strategy explicitly by the Neyman-Pearson lemma under g-probability.     

Quadratic hedging for sequential claims with random weights in discrete time

We study a quadratic hedging problem for a sequence of contingent claims with random weights in discrete time. We obtain the optimal hedging strategy explicitly in a recursive representation, without imposing the non-degeneracy (ND) condition on the model and square integrability on hedging strategies. We relate the general results to hedging under random horizon and fair pricing in the quadratic sense. We illustrate the significance of our results in an example in which the ND condition fails.     

Optimal Cross Hedging of Insurance Derivatives

Abstract

We consider insurance derivatives depending on an external physical risk process, for example, a temperature in a low dimensional climate model. We assume that this process is correlated with a tradable financial asset. We derive optimal strategies for exponential utility from terminal wealth, determine the indifference prices of the derivatives, and interpret them in terms of diversification pressure. Moreover, we check the optimal investment strategies for standard admissibility criteria. Finally, we compare the static risk connected with an insurance derivative to the reduced risk due to a dynamic investment into the correlated asset. We show that dynamic hedging reduces the risk aversion in terms of entropic risk measures by a factor related to the correlation.     

离散条件下标的资产带跳的德尔塔对冲误差研究

讨论了离散条件下的德尔塔对冲以及含泊松跳跃的布莱克—休斯模型下期权的定价问题.在布莱克—休斯模型中对冲被假设为连续发生的,当应用于离散的交易时,对冲误差就产生了.考虑到对冲误差,得出一种离散条件下标的资产带泊松跳跃的修正的布莱克—休斯方程和依赖再对冲区间长度的更精确的德尔塔值.     

基于动态条件相关系数模型的石油市场套期保值比估计

以WTI和Brent两地的原油现货市场和期货市场为研究对象,选择对角化的动态条件相关(DCC)模型估计了市场间的动态条件相关系数,求解了WTI市场、Brent市场及跨市的动态套期保值比,评价了各种市场组合的套期保值效果.得到如下几点结论:第一,WTI市场的一体化程度高于Brent市场;第二,两个月期货的套期保值比高于1个月期货的套期保值比,WTI相应市场组合的套期保值比要高于Brent市场;第三,采取Brent期货对WTI现货进行对冲时,其套期保值比要高于用WTI期货对Brent现货对冲时的情形,也高于Brent市场的套期保值比;第四,套期保值比越高,套期保值效果越好.     

Discrete–time delta hedging and the Black–Scholes model with transaction costs

The paper deals with the problem of discrete–time delta hedging and discrete-time option valuation by the Black–Scholes model. Since in the Black–Scholes model the hedging is continuous, hedging errors appear when applied to discrete trading. The hedging error is considered and a discrete-time adjusted Black–Scholes–Merton equation is derived. By anticipating the time sensitivity of delta in many cases the discrete-time delta hedging can be improved and more accurate delta values dependent on the length of the rebalancing intervals can be obtained. As an application the discrete-time trading with transaction costs is considered. Explicit solution of the option valuation problem is given and a closed form delta value for a European call option with transaction costs is obtained.     

基于债券久期的国债期货套期保值模型分析

使用久期的方法在中国国债期货市场上进行套期保值是否有效?使用久期的方法研究国债期货套期保值的效率问题在国外已经很多,然而这种方法是否适合于目前中国的国债市场,相关研究还不多见,还有待进一步的证实.为此借鉴国外相关理论,采用比较研究的方法,以国债期货上市后2013年9月到2014年5月初,国债现货和国债期货的数据为样本,以基于久期的最优套期保值比率模型为主,其他模型为辅,比较出最优套期保值效率.研究结果表明,基于久期的套期保值方法在目前中国的国债市场效果一般.     

Quadratic Hedging Methods for Defaultable Claims

We apply the local risk-minimization approach to defaultable claims and we compare it with intensity-based evaluation formulas and the mean-variance hedging. We solve analytically the problem of finding respectively the hedging strategy and the associated portfolio for the three methods in the case of a default put option with random recovery at maturity.     

Semi-static hedging of variable annuities

This paper focuses on hedging financial risk in variable annuities with guarantees. We show that insurers should incorporate the specificity of the periodic payment of variable annuities fees to best hedge embedded guarantees and should focus on hedging the net liability. We develop a new hedging strategy based on semi-static hedging techniques, which takes into account the periodically collected fees, and confirm that it is more effective than delta-hedging with same rebalancing dates, as well as traditional semi-static hedging strategies that do not consider the specificity of the payments of fees in their optimization. It is also verified that short-selling or using put options as hedging instruments allows more effective hedging.     

Convergence and optimality of BS-type discrete hedging strategy under stochastic interest rate

We focus on the asymptotic convergence behavior of the hedging errors of European stock option due to discrete hedging under stochastic interest rates. There are two kinds of BS-type discrete hedging differ in hedging instruments: one is the portfolio of underlying stock, zero coupon bond, and the money market account (Strategy BSI); the other is the underlying stock, zero coupon bond (Strategy BSII). Similar to the results of the deterministic interest rate case, we show that convergence speed of the disco...     

Approximate Hedging of Contingent Claims under Transaction Costs for General Pay-offs

Abstract

In 1985 Leland suggested an approach to price contingent claims under proportional transaction costs. Its main idea is to use the classical Black–Scholes formula with a suitably enlarged volatility for a periodically revised portfolio whose terminal value approximates the pay-off h(S _?T )?=?(S _?T ???K)⁺ of the call option. In subsequent studies, Lott, Kabanov and Safarian, and Gamys and Kabanov provided a rigorous mathematical analysis and established that the hedging portfolio approximates this pay-off in the case where the transaction costs decrease to zero as the number of revisions tends to infinity. The arguments used heavily the explicit expressions given by the Black–Scholes formula leaving open the problem whether the Leland approach holds for more general options and other types of price processes. In this paper we show that for a large class of the pay-off functions Leland's method can be successfully applied. On the other hand, if the pay-off function h(x) is not convex, then this method does not work.     

Pricing and hedging derivative securities in markets with uncertain volatilities

We present a model for pricing and hedging derivative securities and option portfolios in an environment where the volatility is not known precisely, but is assumed instead to lie between two extreme values σ_minand σ_max. These bounds could be inferred from extreme values of the implied volatilities of liquid options, or from high-low peaks in historical stock- or option-implied volatilities. They can be viewed as defining a confidence interval for future volatility values. We show that the extremal non-arbitrageable prices for the derivative asset which arise as the volatility paths vary in such a band can be described by a non-linear PDE, which we call the Black-Scholes-Barenblatt equation. In this equation, the ‘pricing’ volatility is selected dynamically from the two extreme values, σ_min, σ_max, according to the convexity of the value-function. A simple algorithm for solving the equation by finite-differencing or a trinomial tree is presented. We show that this model captures the importance of diversification in managing derivatives positions. It can be used systematically to construct efficient hedges using other derivatives in conjunction with the underlying asset.