Perfect Hedging Research Articles

Option pricing and perfect hedging on correlated stocks

We develop a theory for option pricing with perfect hedging in an inefficient market model where the underlying price variations are autocorrelated over a time τ⩾0. This is accomplished by assuming that the underlying noise in the system is derived by an Ornstein-Uhlenbeck, rather than from a Wiener process. With a modified portfolio consisting in calls, secondary calls and bonds we achieve a riskless strategy which results in a closed and exact expression for the European call price which is always lower than Black-Scholes price. We obtain the same price and a modified delta hedging if we start from an effective one-dimensional market model. We compare these strategies and study the sensitivity of the call price to several parameters where the correlation effects are also observed.

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.

Local Expected Shortfall-Hedging in Discrete Time *

This paper proposes a self-financing trading strategy that minimizes the expected shortfall locally when hedging a European contingent claim. A positive shortfall occurs if the hedger is not willing to follow a perfect hedging or a superhedging strategy. In contrast to the classical variance criterion, the expected shortfall criterion depends only on undesirable outcomes where the terminal value of the written option exceeds the terminal value of the hedge portfolio. Searching a strategy which minimizes the expected shortfall is equivalent to the iterative solution of linear programs whose number increases exponentially with respect to the number of trading dates. Therefore, we partition this complex overall problem into several one-period problems and minimize the expected shortfall only locally, i.e., only over the next trading period. This approximation is quite accurate and the number of linear programs to be solved increases only linearly with respect to the number of trading dates.

Market Incompleteness in Regional Electricity Transmission. Part II: The Forward and Real Time Markets

The paper extends our previous analysis of market incompleteness to the case where there exist both a forward and real time market. The discussion is conducted with reference to a two-stage framework commonly used in stochastic programming and finance, but also familiar to the power industry. The paper considers two market designs. The first organization assumes that physical transmission services are auctioned in the forward market. The second organization supposes a financial market of transmission services. We recover the known perfect hedging property of nodal pricing and show that the flowgate model does not share this property. A standard proposition in finance is that market completeness requires that the number of tradable instruments is equal to or exceeds the number of risk factors. We show that none of the nodal or flowgate models satisfy this property. Other instruments, of the option type, should thus be introduced in these systems in order to better trade risk and complete the market.

Consistent pricing and hedging for a modified constant elasticity of variance model

Abstract This paper considers a modification of the well known constant elasticity of variance model where it is used to model the growth optimal portfolio (GOP). It is shown that, for this application, there is no equivalent risk neutral pricing measure and therefore the classical risk neutral pricing methodology fails. However, a consistent pricing and hedging framework can be established by application of the benchmark approach. Perfect hedging strategies can be constructed for European style contingent claims, where the underlying risky asset is the GOP. In this framework, fair prices for contingent claims are the minimal prices that permit perfect replication of the claims. Numerical examples show that these prices may differ significantly from the corresponding ‘risk neutral’ prices.

Consistent pricing and hedging for a modified constant elasticity of variance model

This paper considers a modification of the well known constant elasticity of variance model where it is used to model the growth optimal portfolio (GOP). It is shown that, for this application, there is no equivalent risk neutral pricing measure and therefore the classical risk neutral pricing methodology fails. However, a consistent pricing and hedging framework can be established by application of the benchmark approach. Perfect hedging strategies can be constructed for European style contingent claims, where the underlying risky asset is the GOP. In this framework, fair prices for contingent claims are the minimal prices that permit perfect replication of the claims. Numerical examples show that these prices may differ significantly from the corresponding ‘risk neutral’ prices.

Partial Hedging in a Stochastic Volatility Environment

We consider the problem of partial hedging of derivative risk in a stochastic volatility environment. It is related to state-dependent utility maximization problems in classical economics. We derive the dual problem from the Legendre transform of the associated Bellman equation and interpret the optimal strategy as the perfect hedging strategy for a modified claim. Under the assumption that volatility is fast mean-reverting and using a singular perturbation analysis, we derive approximate value functions and strategies that are easy to implement and study. The analysis identifies the usual mean historical volatility and the harmonically averaged long-run volatility as important statistics for such optimization problems without further specification of a stochastic volatility model. The approximation can be improved by specifying a model and can be calibrated for the leverage effect from the implied volatility skew. We study the effectiveness of these strategies using simulated stock paths.

PARTIAL HEDGING IN A STOCHASTIC VOLATILITY ENVIRONMENT

We consider the problem of partial hedging of derivative risk in a stochastic volatility environment. It is related to state‐dependent utility maximization problems in classical economics. We derive the dual problem from the Legendre transform of the associated Bellman equation and interpret the optimal strategy as the perfect hedging strategy for a modified claim. Under the assumption that volatility is fast mean‐reverting and using a singular perturbation analysis, we derive approximate value functions and strategies that are easy to implement and study. The analysis identifies the usual mean historical volatility and the harmonically averaged long‐run volatility as important statistics for such optimization problems without further specification of a stochastic volatility model. The approximation can be improved by specifying a model and can be calibrated for the leverage effect from the implied volatility skew. We study the effectiveness of these strategies using simulated stock paths.

Weak Convergence of Hedging Strategies of Contingent Claims

This paper presents results on the convergence for hedging strategies in the setting of incomplete financial markets. We examine the convergence of the so-called locally risk-minimizing strategy. It is proved that such a choice for the trading strategy, when perfect hedging of contingent claims is infeasible, is robust under weak convergence. Several fundamental examples, such as trinomial trees and stochastic volatility models, extracted from the financial modeling literature illustrate this property for both deterministic and random time intervals shrinking to zero.

The Optimal Allocation of Risks Under Prospect Theory

This paper deals with the optimal allocation of risks for an agent whose preferences may be represented with prospect theory (Tversky and Kahneman, 1992). A simple setting is considered with an identically distributed and symmetric sources of risk. Under expected utility, equal diversification of risks is optimal in this setting (do not put your eggs in the same basket). Conversely, under prospect theory, provided that the subjective probability of obtaining a perfect hedge is negligible, risk concentration is optimal (do put your eggs in the same basket). The intuitive reason behind this result is that a prospect theory agent is risk-seeking over losses, with the consequence that the property of diversification of averaging downside risks is welfare-reducing rather than welfare-improving.

Futures Hedging Under Disappointment Aversion

AbstractThis article considers optimal futures hedging decisions when the hedger is disappointment‐averse (Gul, 1991). When the futures contract is a perfect hedge instrument, a disappointment‐averse hedger always holds a position closer to the full hedge than a nondisappointment‐averse hedger. In the presence of basis risk, the optimal futures position is either a partial hedge or a full hedge. Neither Texas hedge nor overhedge could be optimal. The effects of different degrees of disappointment aversion on futures trading are also analyzed. © 2001 John Wiley & Sons, Inc. Jrl Fut Mark 21:1029–1042, 2001

Shortfall risk minimization under model uncertainty in the binomial case: adaptive and robust approaches

We consider the problem of minimizing the shortfall risk when the aim is to hedge a contingent claim in a binomial market model and the initial capital is insufficient for a perfect hedge. This problem has been solved under complete information on the underlying model in [3]. We present two possible solutions to the same problem in the case of incomplete information, namely when the underlying probability measure is unknown. The results obtained can also be applied to other classical problems, such as VaR minimization or maximum loss minimization.

Local Expected Shortfall-Hedging

This paper proposes a self-financing trading strategy that minimizes the expected shortfall locally when hedging a European contingent claim. A positive shortfall occurs if the hedger is not willing to follow a perfect hedging or a superhedging strategy. In contrast to the classical variance criterion, the expected shortfall criterion depends only on undesirable outcomes where the terminal value of the written option exceeds the terminal value of the hedge portfolio. Searching a strategy which minimizes the expected shortfall is equivalent to the iterative solution of linear programs whose number increases exponentially with respect to the number of trading dates. Therefore, we partition this complex overall problem into several one-period problems and minimize the expected shortfall only locally, i.e. only over the next trading period. This approximation is quite accurate and the number of linear programs to be solved increases only linearly with respect to the number of trading dates.

The impact of inflation and stock prices in two SADC countries

Economic events and key economic variables affect stock markets on a daily basis. Inflation is one such economic variable that influences share prices to some extent. This article focuses on two SADC countries, South Africa and Namibia, and measures the impact of inflation on share market prices in these countries. It can be concluded from the study that neither South African nor Namibian companies can offer investors a perfect hedge against inflation.

The application of financial theory to the pricing of upward-only rent reviews

Traditional and discounted cash flow (DCF) methods of property appraisal do not deal adequately with the option nature of upward-only rent reviews. DCF techniques typically value the higher of the expected market level of rents after future rent reviews and the current rent. However, the expected level of income after a review is higher than either of these quantities because, assuming no voids, the distribution of rents is truncated at the current rent until the lease comes to an end. This feature exhibits the same financial characteristics as a fixed-income security with a call option to exchange the fixed income for the market rent if that is higher. An option pricing approach is developed for pricing this option and then an adjusted DCF method is developed. The option-pricing approach is an improvement on those previously published in the literature because it does not need to make the unrealistic assumption of perfect hedging. The DCF approach requires the input of a risk discount rate, and the option pricing approach requires a parameter which has to be estimated from other investment market data.

Integrating real and financial options in demand-side electricity contracts

In a competitive electricity market traditional demand-side management (DSM) options offering customers curtailable service at reduced rates are replaced by voluntary customer responses to electricity spot prices. In this new environment, customers wishing to ensure a fixed electricity price while taking advantage of their flexibility to curtail loads can do so by purchasing a forward electricity contract bundled with a financial option that provides a hedge against price risk and reflects the “real options” available to the customer. This paper describes a particular financial instrument referred to as a “double-call” option and derives the value of that option under the assumption that forward electricity prices behave as a geometric Brownian motion process. It is shown that a forward contract bundled with an appropriate double-call option provides a “Perfect hedge” for customers, which can curtail loads in response to high spot prices and can mitigate their curtailment losses when the curtailment decision is made with sufficient lead time.

Option pricing and replication with transaction costs and dividends

This paper derives optimal perfect hedging portfolios in the presence of transaction costs within the binomial model of stock returns , for a market maker that establishes bid and ask prices for American call options on stocks paying dividends prior to expiration. It is shown that, while the option holder's optimal exercise policy at the ex-dividend date varies according to the stock price, there are intervals of values for such a price where the optimal policy would depend on the holder's preferences. Nonetheless, the perfect hedging assumption still allows the derivation of optimal hedging portfolios for both long and short positions of a market maker on the option.

Local Volatility Changes in the Black-Scholes Model

In this paper we address a problem arising in risk management; namely the study of price variations of different contingent claims in the Black-Scholes model due to anticipating future events. The method we propose to use is an extension of the classical Vega index, i.e. the price derivative with respect to the constant volatility, in the sense that we perturb the volatility in different directions. This directional derivative, which we denote the local Vega index, will serve as the main object in the paper and one of the purposes is to relate it to the classical Vega index. We show that for all contingent claims studied in this paper the local Vega index can be expressed as a weighted average of the perturbation in volatility. In the particular case where the interest rate and the volatility are constant and the perturbation is deterministic, the local Vega index is an average of this perturbation multiplied by the classical Vega index. We also study the well-known goal problem of maximizing the probability of a perfect hedge and show that the speed of convergence is in fact dependent of the local Vega index.

Efficient hedging: Cost versus shortfall risk

An investor faced with a contingent claim may eliminate risk by (super-) hedging in a financial market. As this is often quite expensive, we study partial hedges which require less capital and reduce the risk. In a previous paper we determined quantile hedges which succeed with maximal probability, given a capital constraint. Here we look for strategies which minimize the shortfall risk defined as the expectation of the shortfall weighted by some loss function. The resulting efficient hedges allow the investor to interpolate in a systematic way between the extremes of no hedge and a perfect (super-) hedge, depending on the accepted level of shortfall risk.

Optimal Shortfall Hedging of Credit Risk

In this paper we examine the problem of partially hedging a given credit risk exposure. We derive hedges which satisfy certain optimality criteria: For a given investment into the hedge they minimize the remaining risk, or vice versa. This is motivated by the fact that it is a core business of financial intermediaries to carry risks, and that therefore they do not want to hedge their risks completely. In contrast to the usual mean-variance criterion, our hedging strategies try to minimize either the shortfall probability (SP) or the expected shortfall (ES). In complete markets this allows the investor to save money by hedging only part of the claim, while taking a certain (minimal) risk that the hedge does not cover the claim completely. In incomplete markets, a perfect hedge is not always available, and this methodology introduces a new way to find a hedging strategy which minimizes the shortfall risk. We apply this to a credit risk model, where default occurs at the first jump time of a Poisson process. The write-down after default is stochastic and independent of the time of default. In this stylized model we compare hedging strategies for defaultable bonds and credit default swaps which minimize either the SP (Quantile Hedging) or the ES. We consider first a complete market where the martingale measure is unique and derive explicit results. Hedging strategies for both objectives are compared. In the incomplete markets setting, we consider two situations: In the first, we assume that the default risk premium is unknown from the beginning, and therefore we have to select the worst-case martingale measure from the set of possible martingale measures. In the second, the market is complete at the beginning, but at a future time point the default risk parameter will change randomly, for example because of a rating change, and this makes the market incomplete. Strategies for both situations are developed.