Hedging Error Research Articles

Rational Bounds and the Robust Risk Management of Derivatives

The risk management of derivative portfolios is vulnerable to model error. This paper explores risk management strategies based on no-arbitrage bounds, which are independent of any model. In particular, we determine the bounds on the price of a general barrier option given the price of a set of European call options and identify the hedging strategy that enforces the bounds. The strategy puts a floor on the maximum loss that can be incurred by the writer of the barrier option. We show how the strategy can be made dynamic and the floor raised over time. The distribution of hedge errors under the strategy is compared with that under alternative strategies.

Optimal No‐Arbitrage Bounds on S&P 500 Index Options and the Volatility Smile

AbstractThis article shows that the volatility smile is not necessarily inconsistent with the Black–Scholes analysis. Specifically, when transaction costs are present, the absence of arbitrage opportunities does not dictate that there exists a unique price for an option. Rather, there exists a range of prices within which the option's price may fall and still be consistent with the Black–Scholes arbitrage pricing argument. This article uses a linear program (LP) cast in a binomial framework to determine the smallest possible range of prices for Standard & Poor's 500 Index options that are consistent with no arbitrage in the presence of transaction costs. The LP method employs dynamic trading in the underlying and risk‐free assets as well as fixed positions in other options that trade on the same underlying security. One‐way transaction‐cost levels on the index, inclusive of the bid–ask spread, would have to be below six basis points for deviations from Black–Scholes pricing to present an arbitrage opportunity. Monte Carlo simulations are employed to assess the hedging error induced with a 12‐period binomial model to approximate a continuous‐time geometric Brownian motion. Once the risk caused by the hedging error is accounted for, transaction costs have to be well below three basis points for the arbitrage opportunity to be profitable two times out of five. This analysis indicates that market prices that deviate from those given by a constant‐volatility option model, such as the Black–Scholes model, can be consistent with the absence of arbitrage in the presence of transaction costs. © 2001 John Wiley & Sons, Inc. Jrl Fut Mark 21:1151–1179, 2001

Dynamic hedging with a deterministic local volatility function model

We compare the dynamic hedging performance of the deterministic local volatility function approach with the implied/constant volatility method. Using an example in which the underlying price follows an absolute diffusion process, we illustrate that hedge parameters computed from the implied/constant volatility method can have significant error even though the implied volatility method is able to calibrate the current option prices of different strikes and maturities. In particular the delta hedge parameter produced by the implied/constant volatility method is consistently significantly larger than that of the exact delta when the underlying price follows an absolute diffusion. In order to compute a better hedge parameter, accurate estimation of the local volatility function in a region surrounding the current asset price is crucial. We illustrate that a suitably implemented volatility function method can estimate this local volatility function sufficiently accurately to generate more accurate hedge parameters. Hedging using this volatility function for the absolute diffusion example leads to a smaller average absolute hedging error when compared with using the implied/constant volatility rate. When comparing the hedging performance in the S&P 500 index option market as well as the S&P 500 futures option market, we similarly observe that the delta hedge parameter from the implied/constant volatility method is typically greater than that using the volatility function approach. Examination of the hedging error reveals that using a larger delta factor greater than that of the true volatility yields more positive

Pricing and hedging derivative securities with neural networks: Bayesian regularization, early stopping, and bagging

We study the effectiveness of cross validation, Bayesian regularization, early stopping, and bagging to mitigate overfitting and improving generalization for pricing and hedging derivative securities with daily S&P 500 index daily call options from January 1988 to December 1993. Our results indicate that Bayesian regularization can generate significantly smaller pricing and delta-hedging errors than the baseline neural-network (NN) model and the Black-Scholes model for some years. While early stopping does not affect the pricing errors, it significantly reduces the hedging error (HE) in four of the six years we investigated. Although computationally most demanding, bagging seems to provide the most accurate pricing and delta hedging. Furthermore, the standard deviation of the MSPE of bagging is far less than that of the baseline model in all six years, and the standard deviation of the average HE of bagging is far less than that of the baseline model in five out of six years. We conclude that they be used at least in cases when no appropriate hints are available.

Leland's Approach to Option Pricing: The Evolution of a Discontinuity

A claim of Leland (1985) states that in the presence of transaction costs a call option on a stock S, described by geometric Brownian motion, can be perfectly hedged using Black–Scholes delta hedging with a modified volatility. Recently Kabanov and Safarian (1997) disproved this claim, giving an explicit (up to an integral) expression of the limiting hedging error, which appears to be strictly negative and depends on the path of the stock price only via the stock price at expiry ST. We prove in this paper that the limiting hedging error, considered as a function of ST, exhibits a removable discontinuity at the exercise price. Furthermore, we provide a quantitative result describing the evolution of the discontinuity: Hedging errors, plotted over the price at expiry, show a peak near the exercise price. We determine the rate at which that peak becomes narrower (producing the discontinuity in the limit) as the lengths of the revision intervals shrink.

Rational Bounds on the Prices of Exotic Options

The paper explores the bounds which can be placed on the price of exotic options while making minimal assumptions about the price process. In particular, we identify the bounds on the price of a general barrier option given the price of a set of European call options. We show the hedging strategies which enforce those bounds. The hedging strategies are robust in that, after inception, trading occurs only when the barrier is breached. The hedge strategies put a floor on the maximum loss. The distribution of hedge errors under the strategy is compared with that under alternative strategies.

Hedging long-term exposures with multiple short-term futures contracts

The paper analyzes the problem facing an agent who has a long term commodity supply commitment, and who wishes to hedge that commitment using short maturity commodity futures contracts. As time evolves, the agent has to roll the hedge as old futures contracts mature and new futures contracts are listed. This gives rise to hedge errors. The optimal hedging strategy is characterized in a world where contracts of several different maturities coexist. The strategy is independent both of the agent's risk aversion and, under certain conditions, of beliefs about expected returns from holding futures contracts. The methodology is compared with approaches based on dynamic models of the term structure. It is tested on data from the oil futures market.

Hedging Long-Term Exposures with Multiple Short-Term Futures Contracts

This article analyzes the problem facing an agent who has a long-term commodity supply commitment and who wishes to hedge that commitment using short-maturity commodity futures contracts. As time evolves, the agent has to roll the hedge as old futures contracts mature and new futures contracts are listed. This gives rise to hedge errors. The optimal hedging strategy is characterized in a world where contracts of several different maturities coexist. The strategy is independent both of the agent's risk aversion and, under certain conditions, of beliefs about expected returns from holding futures contracts. The methodology is compared with approaches based on dynamic models of the term structure. It is tested on data from the oil futures market. Article published by Oxford University Press on behalf of the Society for Financial Studies in its journal, The Review of Financial Studies.

Exploding hedging errors for digital options

In the complete market model of geometric Brownian motion, all kinds of exotic options can be priced and hedged perfectly using a delta hedging strategy which duplicates the option's payoff. If trading takes place in a frictionless market, this delta hedging strategy is said to eliminate the option writer's risk completely. It will be shown that for certain contingent claims, for example digital options, the hedge can fail completely if the underlying risky asset does not follow the assumed geometric Brownian motion. Indeed, the hedging error may diverge and delta hedging can actually increase the risk of the option writer.

A Portfolio Approach to Risk Reduction in Discretely Rebalanced Option Hedges

This paper analyses the accumulated hedging errors generated by discretely rebalanced option hedges. We show that simple generalizations of the prior research can underestimate the variance of the accumulated hedging errors and that even with daily rebalancing, these accumulated hedging errors can introduce substantial risk in arbitrage strategies suggested by the Black-Scholes option pricing model. We also show that the correlation between the accumulated hedging errors for different options can be quite high, so that the risk of arbitrage due to hedging errors can be substantially reduced by optimally combining options into portfolios. The results also suggest that tests of market pricing of traded options which are based on employing a portfolio approach are likely to be much better specified than the standard tests that focus on individual options.

On Leland's strategy of option pricing with transactions costs

We compute the limiting hedging error of the Leland strategy for the approximate pricing of the European call option in a market with transactions costs. It is not equal to zero in the case when the level of transactions costs is a constant, in contradiction with the claim in Leland (1985).

Multi-period minimax hedging strategies

Option pricing and hedging under transaction costs are of major importance to marketmakers and investors. In this paper we present the basic minimax strategy which determines the optimum number of shares that minimizes the worst-case potential hedging error under transaction costs for the next period. We present two extensions of this strategy. The first extension is the two-period minimax where the worst case is defined over a two-period setting. The objective function of the basic minimax strategy is augmented to include the hedging error for the second period. The second extension is the variable minimax strategy where early rebalancing is triggered by the minimax hedging error. Simulation results suggest that the basic minimax strategy and its two extensions are superior in performance to delta hedging and that the variable minimax strategy is superior to both the basic and the two-period strategies. This result is due to the opportunity provided by the variable minimax strategy to rebalance early. The greatest amount of business for traded options is done for at-the-money options; in this paper, we have concluded that the minimax strategies are particularly suitable for managing the risk of such options. In the Appendix, we present the minimax algorithm used for the implementation of these strategies.

On the Mean-Variance Tradeoff in Option Replication with Transactions Costs

This paper analyzes the tradeoff between cost and risk of discretely rebalanced option hedges in the presence of transactions costs. I present closed form solutions for expected hedging error, transactions costs, and variance of the cash flow from a time-based hedging strategy similar to that analyzed by Leland (1985). Furthermore, I characterize the cost and risk of a move-based hedging strategy without resorting to Monte Carlo simulations. All results are sufficiently general to accommodate the use of a transactions costs adjusted hedging volatility and an asset rate of return that differs from the risk-free rate of return.

Hedging Strategies of Financial Intermediaries: Pricing Options with a Bid‐Ask Spread

AbstractThis paper uses a model similar to the Boyle‐Vorst and Ritchken‐Kuo arbitrage‐free models for the valuation of options with transactions costs to determine the maximum price to be charged by the financial intermediary writing an option in a non‐auction market. Earlier models are extended by recognizing that, in the presence of transactions costs, the price‐taking intermediary devising a hedging portfolio faces a tradeoff: to choose a short trading interval with small hedging errors and high transactions costs, or a long trading interval with large hedging errors and low transactions costs. The model presented here also recognizes that when transactions costs induce less frequent portfolio adjustments, investors are faced with a multinomial distribution of asset returns rather than a binomial one. The price upper bound is determined by selecting the trading frequency that will equalize the marginal gain from decreasing hedging errors and the marginal cost of transactions.

Minimax hedging strategy

We present several variants of a robust risk management strategy based on minimax for the writer of a European call option on a stock and show that it performs at least as well as the standard hedging strategy, delta hedging. When using the minimax strategy, the hedger specifies a worst case scenario in terms of the price of the underlying stock. The minimax strategy recommends the number of shares in the underlying stock the hedger should hold in order to minimize the hedging error against the worst case occurring. The minimax hedging error may correspond to an extreme point of the price range being considered or to a mid-range solution. Simulation and empirical results suggest that the minimax strategy is particularly powerful for hedging the risk of writing an option when the price of the underlying stock is both highly volatile and crosses over the exercise frequently.