American Options under Proportional Transaction Costs: Pricing, Hedging and Stopping Algorithms for Long and Short Positions

American options are studied in a general discrete market in the presence of proportional transaction costs, modelled as bid-ask spreads. Pricing algorithms and constructions of hedging strategies, stopping times and martingale representations are presented for short (seller’s) and long (buyer’s) positions in an American option with an arbitrary payoff. This general approach extends the special cases considered in the literature concerned primarily with computing the prices of American puts under transaction costs by relaxing any restrictions on the form of the payoff, the magnitude of the transaction costs or the discrete market model itself. The largely unexplored case of pricing, hedging and stopping for the American option buyer under transaction costs is also covered. The pricing algorithms are computationally efficient, growing only polynomially with the number of time steps in a recombinant tree model. The stopping times realising the ask (seller’s) and bid (buyer’s) option prices can differ from one another. The former is generally a so-called mixed (randomised) stopping time, whereas the latter is always a pure (ordinary) stopping time.     

具限制投资组合和交易费用的博弈未定权益的保值

博弈期权是由kifer(2000)提出的,但就其本质而言,仍是美式期权的一种,只是增加了卖方中止合约的权利.本文主要对连续市场模型中具交易费用和限制投资组合的博弈未定权益的保值问题进行了研究,给出了买卖双方的保值价格和一个无套利区间.     

Hedging of the European option in discrete time under proportional transaction costs

In the paper hedging of the European option in a discrete time financial market with proportional transaction costs is studied. It is shown that for a certain class of options the set of portfolios which allow to hedge an option in a discrete time model with a bounded set of possible changes in a stock price is the same as the set of such portfolios, under assumption that the stock price evolution is given by a suitable CRR model.     

带交易费用和红利的欧式期权定价的数值方法

本文在 L eland的带交易费用的欧式期权定价模型基础上 ,先推导出一般费用模型的定价公式 ,然后用二叉树图法给出了带有交易费用和红利的欧式看涨期权定价的数值方法 ,并比较了多头和空头的不同价值。     

American option valuation in a stochastic volatility model with transaction costs

In the present paper we analyse the American option valuation problem in a stochastic volatility model when transaction costs are taken into account. We shall show that it can be formulated as a singular stochastic optimal control problem, proving the existence and uniqueness of the viscosity solution for the associated Hamilton–Jacobi–Bellman partial differential equation. Moreover, after performing a dimensionality reduction through a suitable choice of the utility function, we shall provide a numerical example illustrating how American options prices can be computed in the present modelling framework.     

基于CEV过程带交易费的脆弱期权定价的数值算法

主要研究基于CEV过程且支付交易费的脆弱期权定价的数值计算问题.首先通过构造无风险投资组合,导出了基于CEV过程且支付交易费用的脆弱期权定价的偏微分方程模型;其次应用有限差分方法将定价模型离散化,并设计数值算法;最后以看跌期权为例进行数值试验,分析各定价参数对看跌期权价值的影响.     

面对成比例交易成本的情况下外汇期权的定价

The purpose of present work is to examine the financial problem of finding the universal reservation prices of a European call option written on exchange rate when there is proportional transaction costs of trading foreign currency in the market. An approach is suggested to compute the reservation bid-ask price of foreign currency call option based on maximizing the investor's expected utility. Option prices are determined from the investor's basic portfolio selection problem, without the need to solve a more complex optimization problem involving the insertion of the option payoffs into the terminal value function. Option prices are computed numerically in a Markov chain approximation for the case of exponential utility.Numerical results show that the option price bounds are almost independent of the alternative risk aversion parameter, but the bounds of NT region becomes narrower and the range of values of the initial holding for which the fair price lies within the bid-ask spread is shifted to a lower value when the risk aversion parameter increases.     

分形布朗运动下有交易成本的外汇期权定价

研究了有交易成本的分形Black-Scholes外汇期权定价问题.基于汇率的分形布朗运动分布假设,运用分形布朗运动的性质和随机微积分方法,得到了欧式外汇期权价格所满足的偏微分方程.最后,建立离散时间条件下的非线性期权定价模型,并且通过解期权价格的偏微分方程给出了有交易成本的欧式外汇期权定价公式.     

FOREIGN CURRENCY OPTION PRICING WITH PROPORTIONAL TRANSACTION COSTS

The purpose of present work is to examine the financial problem of finding the universal reservation prices of a European call option written on exchange rate when there is proportional transaction costs of trading foreign currency in the market. An approach is suggested to compute the reservation bid-ask price of foreign currency call option based on maximizing the investor＇s expected utility. Option prices are determined from the investor＇s basic portfolio selection problem, without the need to solve a more complex optimization problem involving the insertion of the option payoffs into the terminal value function. Option prices are computed numerically in a Markov chain approximation for the case of exponential utility. Numerical results show that the option price bounds are almost independent of the alternative risk aversion parameter, but the bounds of NT region becomes narrower and the range of values of the initial holding for which the fair price lies within the bid-ask spread is shifted to a lower value when the risk aversion parameter increases.     

支付交易费的回望期权定价

以Black-Scholes模型为基础,通过对回望期权的研究,结合有交易费的欧式期权的定价公式,运用证券组合技术与无套利原理,建立了支付交易费的回望期权定价模型.通过对方程化简和分析,运到PDE相关方法化为Cauchy问题,得出定价公式.     

有交易成本的回望期权定价研究

基于标的资产价格的几何布朗运动假设，Black—Seholes模型运用连续交易保值策略成功解决了完全市场下的欧式期权定价问题。然而，在实际的金融市场中，存在着数量可观的交易成本。本文主要研究了在不完全市场下有交易成本的回望期权的定价问题，并且利用Ito公式，得到了在该模型下期权价格所满足的微分方程。     

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.     

个人投资与消费模型的期望效用最大化

研究了具有初始财富的投资者如何最大化终端资产和消费的期望效用,首先通过交易费用函数建立带交易费的连续时间投资与消费模型,然后运用鞅分析和对偶理论证明了：在有效市场中,如果投资者积极交易,则只会降低终端财富的期望值,并得到了最优投资消费组合过程和终端资产.     

Computational methods for incentive option valuation

Employing stochastic programming, we provide a general framework for option pricing based on marginal bid/ask price valuation. It is applied to numerical analysis of options with European and American style exercise using a double binary tree. Incentive options are valued considering hedging restrictions and other market frictions, such as transaction and short position costs, and different borrowing and lending rates. The framework also includes correlated labor income. The possibility of partial sales is analyzed using ask price functions. Without friction costs and labor income, our model is the discrete-time equivalent of Ingersoll (J Bus 79:453–487, 2006). When labor income and/or market frictions are present, or a fraction of options is sold, the option values are materially different compared to Ingersoll (J Bus 79:453–487, 2006).

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Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.      

Optimal trading strategy for European options with transaction costs

The European option with transaction costs is studied. The cost of making a transaction is taken to be proportional by a factor λ to the value (in dollars) of stock traded. When there are no transaction costs (i.e. when λ=0) the well-known Black-Scholes strategy tells how to hedge the option. Since no non-trivial perfect hedging strategy exists when λ>0 (see (Ann. Appl. Probab. 5(2) (1995) 327)), we instead try to maximize the expected utility attainable. We seek to understand the effect transaction costs have on the maximum attainable expected utility over all strategies, when λ is small but non-zero. It turns out that transaction costs diminish the expected utility by an amount which has the order of magnitude λ^2/3. We will compute that correction explicitly modulo an error which is small compared to λ^2/3. We will exhibit an explicit strategy whose expected utility differs from the maximum attainable expected utility by an error small in comparison to λ^2/3.     

Numerical Solutions for Option Pricing Models Including Transaction Costs and Stochastic Volatility

The option pricing problem when the asset is driven by a stochastic volatility process and in the presence of transaction costs leads to solving a nonlinear partial differential equation. The nonlinear term in the PDE reflects the presence of transaction costs. Under a particular market completion assumption we derive the nonlinear PDE whose solution may be used to find the price of options. In this paper under suitable conditions, we give an algorithmic scheme to obtain the solution of the problem by an iterative method and provide numerical solutions using the finite difference method.     

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 a European Basket Option in the Presence of Proportional Transaction Costs

A crucial assumption in the Black–Scholes theory of options pricing is the no transaction costs assumption. However, following such a strategy in the presence of transaction costs would lead to immediate ruin. This paper presents a stochastic control approach to the pricing and hedging of a European basket option, dependent on primitive assets whose prices are modelled as lognormal diffusions, in the presence of costs proportional to the size of the transaction. Under certain assumptions on the individual preferences, it is able to reduce the dimensionality of the resulting control problem. This facilitates considerably the study of the value function and the characterisation of the optimal trading policy. For solution of the problem a perturbation analysis scheme is utilized to derive a non‐trivial, asymptotically optimal result. The findings reveal that this result can be expressed by means of a small correction to the corresponding solution of the frictionless Black–Scholes type problem, resembling a multi‐dimensional ‘bandwidth’ around the vanilla case, which, moreover, is readily tractable.     

连续支付红利及有交易成本的领子期权定价模型

在无风险利率r(t)和波动率σ(t)均为时间t的函数及市场无套利假设下,分别考虑了连续红利率q(t)和有交易成本情况下的领子期权定价,通过建立相应定价模型,得到了领子期权不同的定价公式.     

A Black-Scholes option pricing model with transaction costs

We consider a boundary value problem for a nonlinear differential equation which arises in an option pricing model with transaction costs. We apply the method of upper and lower solutions in order to obtain solutions for the stationary problem. Moreover, we give conditions for the existence of solutions of the general evolution equation.