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.     

基于风险测度CaR_k的最优投资组合(英文)

本文定义一种k阶在险资本值(CaRk)来度量风险,并研究在经典Black-Scholes市场中的均值-CaRk最优投资组合问题,给出了CaRk的显示表达式,并得到了均值-CaRk最优投资组合问题的最优策略及相应的最优财富值.     

Comparison of Two Methods for Superreplication

Abstract

We compare two methods for superreplication of options with convex pay-off functions. One method entails the overestimation of an unknown covariance matrix in the sense of quadratic forms. With this method the value of the superreplicating portfolio is given as the solution of a linear Black–Scholes BS-type equation. In the second method, the choice of quadratic form is made pointwise. This leads to a fully non-linear equation, the so-called Black–Scholes–Barenblatt (BSB) equation, for the value of the superreplicating portfolio. In general, this value is smaller for the second method than for the first method. We derive estimates for the difference between the initial values of the superreplicating strategies obtained using the two methods.     

On minimizing drawdown risks of lifetime investments

Drawdown measures the decline of portfolio value from its historic high-water mark. In this paper, we study a lifetime investment problem aiming at minimizing the risk of drawdown occurrences. Under the Black–Scholes framework, we examine two financial market models: a market with two risky assets, and a market with a risk-free asset and a risky asset. Closed-form optimal trading strategies are derived under both models by utilizing a decomposition technique on the associated Hamilton–Jacobi–Bellman (HJB) equation. We show that it is optimal to minimize the portfolio variance when the fund value is at its historic high-water mark. Moreover, when the fund value drops, the proportion of wealth invested in the asset with a higher instantaneous rate of return should be increased. We find that the instantaneous return rate of the minimum lifetime drawdown probability (MLDP) portfolio is never less than the return rate of the minimum variance (MV) portfolio. This supports the practical use of drawdown-based performance measures in which the role of volatility is replaced by drawdown.     

A note on convergence of an approximate hedging portfolio with liquidity risk

When the underlying asset price depends on activities of traders, hedging errors include costs due to the illiquidity of the underlying asset and the size of this cost can be substantial. Cetin et al. (2004), Liquidity risk and arbitrage pricing theory, Finance and Stochastics, 8(3), 311-341, proposed a hedging strategy that approximates the classical Black–Scholes hedging strategy and produces zero liquidity costs. Here, we compute the rate of convergence of the final value of this hedging portfolio to the option payoff in case of a European call option; i.e. we see how fast its hedging error converges to zero. The hedging strategy studied here is meaningful due to its simple liquidity cost structure and its smoothness relative to the classical Black–Scholes delta.     

Option Pricing for Log-Symmetric Distributions of Returns

We derive an option pricing formula on assets with returns distributed according to a log-symmetric distribution. Our approach is consistent with the no-arbitrage option pricing theory: we propose the natural risk-neutral measure that keeps the distribution of returns in the same log-symmetric family reflecting thus the specificity of the stock’s returns. Our approach also provides insights into the Black–Scholes formula and shows that the symmetry is the key property: if distribution of returns X is log-symmetric then 1/X is also log-symmetric from the same family. The proposed options pricing formula can be seen as a generalization of the Black–Scholes formula valid for lognormal returns. We treat an important case of log returns being a mixture of symmetric distributions with the particular case of mixtures of normals and show that options on such assets are underpriced by the Black–Scholes formula. For the log-mixture of normal distributions comparisons with the classical formula are given.      

Optimal investment to minimize the probability of drawdown

We determine the optimal investment strategy in a Black–Scholes financial market to minimize the so-called probability of drawdown, namely, the probability that the value of an investment portfolio reaches some fixed proportion of its maximum value to date. We assume that the portfolio is subject to a payout that is a deterministic function of its value, as might be the case for an endowment fund paying at a specified rate, for example, at a constant rate or at a rate that is proportional to the fund’s value.     

Mean–variance optimal portfolios in the presence of a benchmark with applications to fraud detection

We first study mean–variance efficient portfolios when there are no trading constraints and show that optimal strategies perform poorly in bear markets. We then assume that investors use a stochastic benchmark (linked to the market) as a reference portfolio. We derive mean–variance efficient portfolios when investors aim to achieve a given correlation (or a given dependence structure) with this benchmark. We also provide upper bounds on Sharpe ratios and show how these bounds can be useful for fraud detection. For example, it is shown that under some conditions it is not possible for investment funds to display a negative correlation with the financial market and to have a positive Sharpe ratio. All the results are illustrated in a Black–Scholes market.     

Optimal portfolio strategies benchmarking the stock market

The paper investigates the impact of adding a shortfall risk constraint to the problem of a portfolio manager who wishes to maximize his utility from the portfolios terminal wealth. Since portfolio managers are often evaluated relative to benchmarks which depend on the stock market we capture risk management considerations by allowing a prespecified risk of falling short such a benchmark. This risk is measured by the expected loss in utility. Using the Black–Scholes model of a complete financial market and applying martingale methods, explicit analytic expressions for the optimal terminal wealth and the optimal portfolio strategies are given. Numerical examples illustrate the analytic results.     

Arbitrage and the tax code

We provide a detailed characterization of arbitrage-free asset prices in the presence of capital gains and income taxes. The distinguishing feature of our analysis is that we impose on the model two important features of the tax code: the limited use of capital losses and the inability to wash sell. We show that under remarkably mild conditions, the lack of pre-tax arbitrage implies the lack of post-tax arbitrage with the limited use of capital losses. The conditions are that the risk free interest rate be positive and that tax rates on interest income exceed capital gains tax rates. The result also holds when only a wash sale constraint is imposed and no investor holds a portfolio with a large capital loss. We allow investors to face different tax rates and have different bases for the calculation of capital gains taxes. The characterizations we provide have important implications for both asset pricing and portfolio choice. Our results imply that models that use arbitrage-free pre-tax models continue for derivative pricing and hedging are also arbitrage free in a world with taxes. Similarly, portfolio choice models with taxes typically specify pre-tax arbitrage free price processes and then analyze portfolio choice in the presence of taxes. In these models, it is unclear if portfolio recommendations are based on risk-return tradeoffs or on the arbitrage opportunities present in the model. Our results imply that if the above features of the tax code are modeled explicitly, then we can isolate the post-tax risk-return tradeoffs.     

Hedging Large Portfolios of Options in Discrete Time*

The problem studied is that of hedging a portfolio of options in discrete time where underlying security prices are driven by a combination of idiosyncratic and systematic risk factors. It is shown that despite the market incompleteness introduced by the discrete time assumption, large portfolios of options have a unique price and can be hedged without risk. The nature of the hedge portfolio in the limit of large portfolio size is substantially different from its continuous time counterpart. Instead of linearly hedging the total risk of each option separately, the correct portfolio hedge in discrete time eliminates linear as well as second and higher order exposures to the systematic risk factors only. The idiosyncratic risks need not be hedged, but disappear through diversification. Hedging portfolios of options in discrete time thus entails a trade‐off between dynamic and cross‐sectional hedging errors. Some computations are provided on the outcome of this trade‐off in a discrete‐time Black–Scholes world.     

Optimal investment with deferred capital gains taxes

We solve the optimal portfolio problem of an investor in a complete market who is liable to deferred taxes due on capital gains, irrespective of their origin. In a Brownian framework we explicitly determine optimal strategies. Our analysis is based on a modification of the standard martingale method applied to the after-tax utility function, which exhibits a kink at the level of initial wealth, and Clark’s formula. Numerical results show that the Merton strategy is close to optimal under taxation.     

基于排序预测的带交易费用在线投资组合策略

在线投资组合决策过程中频繁调整资产头寸会产生较多的交易费用。本文提出了一个综合考虑预期收益和交易费用的在线投资组合策略。通过预测资产的排序计算组合的预期收益,利用相对熵距离衡量交易费用,构造了一个极大化预期收益和极小化交易费用的优化模型,从而得到了一个在线投资组合更新策略。然后,从理论上证明了该策略具有BH泛证券性,即该策略与离线的最优购买并持有策略具有相同的渐近平均指数收益率。最后,采用中美股票市场实际数据,对该策略进行了数值分析。结果表明,该策略的表现优于已有的在线投资组合策略,且对模型的参数不敏感。     

Wick–Itô formula for regular processes and applications to the Black and Scholes formula

We derive a Wick–Itô formula, that is, an Itô-type formula based on Wick integration. We derive it in the context of regular Gaussian processes which include Brownian motion and fractional Brownian motion with Hurst parameter greater than 1/2. We then consider applications to the Black and Scholes formula for the pricing of a European call option. It has been shown that using Wick integration in this context is problematic for economic reasons. We show that it is also problematic for mathematical reasons because the resulting Black and Scholes formula depends only on the variance of the process and not on its dependence structure.     

Option Pricing on Multiple Assets

Since 1973, the Black–Scholes formula has been used in financial markets to price financial derivatives such as options. In the classical Black–Scholes model for the market, the following type of mix is assumed or postulated: in the simplest case, it consists of an essentially riskless bond and a single risky asset. Hence, certainty mixed with uncertainty: safe vs risky! Here we consider more complex products where each component in a portfolio entails several variables with constraints. This leads to elegant models based on multivariable stochastic integration, and describing several securities simultaneously [Etheridge, A Course in Financial Calculus, Cambridge University Press, UK (2002), Jiang, Mathematical Modeling and Methods of Option Pricing, Higher Education, Beijing, China (2003)] and [Broadie, Detemple, Math. Financ. 7:241–286 (1997)]. We derive a general asymptotic solution in a short time interval using the heat kernel expansion on a Riemannian metric. We then use our formula to predict the better price of options on multiple underlying assets. We then apply our method to the case known as the two-color rainbow option, i.e., the special case of the model with two underlying assets. This asymptotic solution is important, as it explains hidden effects in a class of financial models.This paper is dedicated to the memory of the first named author, Professor Thomas P. Branson (1953–2006).     

Mean-risk portfolio management with bankruptcy prohibition

In accordance with Solvency II, the commonly tightened government regulation on insurance cooperations, they have been obligated to take conservative investment strategies such as those ruling out the possibility of bankruptcy. With this in mind, in this article, we aim to continue our work (Wong et al., 2017a,b) . First, we study the solvability of mean-risk portfolio optimization problem with bankruptcy prohibition, in the complete market in which the investor aims to maximize the expected payoff and to minimize the deviation risk simultaneously, which is of great use in the insurance paradigm. Secondly, we also provide the original weak convergence result of the optimal terminal wealth of a sequence of approximate markets to that of the limiting market through their corresponding pricing kernels. As a result, we establish an effective numerical algorithm calibrating the optimal terminal wealth under Black–Scholes models by that of binomial tree models. The results of our numerical simulations indicate that the downside risk of the optimal payoff can be effectively reduced by imposing the bankruptcy prohibition.     

For what trading strategies is the tax payment stream of infinite variation?

For an Itô asset price process and under quite mild structural assumptions, we show that the accumulated payments of a linear tax on trading gains are of infinite variation if the quadratic covariation of the trading strategy and the asset price is negative. By contrast, if the strategy is a smooth function of the asset price and some finite variation processes with positive partial derivative with respect to the price variable, then accumulated tax payments are of finite variation. An interesting example are constant proportion portfolio insurance (CPPI) strategies which we extend to models with capital gains taxes. The associated tax payment stream is of finite variation if the tax-adjusted constant multiple of the cushion which is invested in the risky asset is bigger or equal to one. Otherwise, it is of infinite variation.     

Risk-neutral valuation of participating life insurance contracts in a stochastic interest rate environment

Over the last years, the valuation of life insurance contracts using concepts from financial mathematics has become a popular research area for actuaries as well as financial economists. In particular, several methods have been proposed of how to model and price participating policies, which are characterized by an annual interest rate guarantee and some bonus distribution rules. However, despite the long terms of life insurance products, most valuation models allowing for sophisticated bonus distribution rules and the inclusion of frequently offered options assume a simple Black–Scholes setup and, more specifically, deterministic or even constant interest rates.We present a framework in which participating life insurance contracts including predominant kinds of guarantees and options can be valuated and analyzed in a stochastic interest rate environment. In particular, the different option elements can be priced and analyzed separately. We use Monte Carlo and discretization methods to derive the respective values.The sensitivity of the contract and guarantee values with respect to multiple parameters is studied using the bonus distribution schemes as introduced in [Bauer, D., Kiesel, R., Kling, A., Ruß, J., 2006. Risk-neutral valuation of participating life insurance contracts. Insurance: Math. Econom. 39, 171–183]. Surprisingly, even though the value of the contract as a whole is only moderately affected by the stochasticity of the short rate of interest, the value of the different embedded options is altered considerably in comparison to the value under constant interest rates. Furthermore, using a simplified asset portfolio and empirical parameter estimations, we show that the proportion of stock within the insurer’s asset portfolio substantially affects the value of the contract.     

Martingales and stochastic integrals in the theory of continuous trading

This paper develops a general stochastic model of a frictionless security market with continuous trading. The vector price process is given by a semimartingale of a certain class, and the general stochastic integral is used to represent capital gains. Within the framework of this model, we discuss the modern theory of contingent claim valuation, including the celebrated option pricing formula of Black and Scholes. It is shown that the security market is complete if and only if its vector price process has a certain martingale representation property. A multidimensional generalization of the Black-Scholes model is examined in some detail, and some other examples are discussed briefly.     

Optimal portfolio selection for general provisioning and terminal wealth problems

In Dhaene et al. (2005), multiperiod portfolio selection problems are discussed, using an analytical approach to find optimal constant mix investment strategies in a provisioning or a savings context. In this paper we extend some of these results, investigating some specific, real-life situations. The problems that we consider in the first section of this paper are general in the sense that they allow for liabilities that can be both positive or negative, as opposed to Dhaene et al. (2005), where all liabilities have to be of the same sign. Secondly, we generalize portfolio selection problems to the case where a minimal return requirement is imposed. We derive an intuitive formula that can be used in provisioning and terminal wealth problems as a constraint on the admissible investment portfolios, in order to guarantee a minimal annualized return. We apply our results to optimal portfolio selection.