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).     

Hedging errors with Leland's option model in the presence of transaction costs

Nonzero transaction costs invalidate the Black–Scholes [1973. Journal of Political Economy 81, 637–654] arbitrage argument based on continuous trading. Leland [1985. Journal of Finance 40, 1283–1301] developed a hedging strategy which modifies the Black–Scholes hedging strategy with a volatility adjusted by the length of the rebalance interval and the rate of the proportional transaction cost. Kabanov and Safarian [1997. Finance and Stochastics 1, 239–250] calculated the limiting hedging error of the Leland strategy and pointed out that it is nonzero for the approximate pricing of an European call option, in contradiction to Leland's claim. As a further contribution, we first identify the mathematical flaw in the argument of Leland's claim and then quantify the expected percentage of hedging losses in terms of the hedging frequency and the level of the option strike price.     

Asymptotic replication with modified volatility under small transaction costs

We consider the dynamic hedging of a European option under a general local volatility model with small proportional transaction costs. Extending the approach of Leland, we introduce a class of continuous strategies of finite cost that asymptotically (super-)replicate the payoff. An associated central limit theorem for the hedging error is proved. We also obtain an explicit trading strategy minimizing the asymptotic error variance.     

Arbitrage-free interval and dynamic hedging in an illiquid market

This paper derives two pricing PDEs for a general European option under liquidity risk. We provide two modified hedges: one hedge replicates a short option and the other replicates a long option inclusive of liquidity costs under continuous rebalancing. We identify an arbitrage-free interval by calculating the costs of the two hedges. Unlike in a setting with infinite overall transaction costs, the overall liquidity cost in our model is proved to be finite even under continuous rebalancing. Numerical results on option pricing and the moments of hedge errors of Black–Scholes and our modified hedges are also presented.     

An endogenous volatility approach to pricing and hedging call options with transaction costs

Standard delta hedging fails to exactly replicate a European call option in the presence of transaction costs. We study a pricing and hedging model similar to the delta hedging strategy with an endogenous volatility parameter for the calculation of delta over time. The endogenous volatility depends on both the transaction costs and the option strike prices. The optimal hedging volatility is calculated using the criterion of minimizing the weighted upside and downside replication errors. The endogenous volatility model with equal weights on the up and down replication errors yields an option premium close to the Leland [J. Finance, 1985 Leland, HE. 1985. Option pricing and replication with transaction costs. J. Finance, 40: 1283–1301. [Crossref], [Web of Science ®] , [Google Scholar], 40, 1283–1301] heuristic approach. The model with weights being the probabilities of the option's moneyness provides option prices closest to the actual prices. Option prices from the model are identical to the Black–Scholes option prices when transaction costs are zero. Data on S&P 500 index cash options from January to June 2008 illustrate the model.     

Almost-sure hedging with permanent price impact

We consider a financial model with permanent price impact. Continuous-time trading dynamics are derived as the limit of discrete rebalancing policies. We then study the problem of superhedging a European option. Our main result is the derivation of a quasilinear pricing equation. It holds in the sense of viscosity solutions. When it admits a smooth solution, it provides a perfect hedging strategy.     

Option pricing in a lognormal securities market with discrete trading

This paper derives a call option valuation equation assuming discrete trading in securities markets where the underlying asset and market returns are bivariate lognormally distributed and investors have increasing, concave utility functions exhibiting skewness preference. Since the valuation does not require the continouus time riskfree hedging of Black and Scholes, nor the discrete time riskfree hedging of Cox, Ross and Rubinstein, market effects are introduced into the option valuation relation. The new option valuation seems to correct for the systematic mispricing of well-in and well-out of the money options by the Black and Scholes option pricing formula.     

Pricing and Hedging of Discrete Dynamic Guaranteed Funds

We derive a risk‐neutral pricing model for discrete dynamic guaranteed funds with geometric Gaussian underlying security price process. We propose a dynamic hedging strategy by adding a gamma factor to the conventional delta. Simulation results demonstrate that, when hedging discretely, the risk‐neutral gamma‐adjusted‐delta strategy outperforms the dynamic delta hedging strategy by reducing the expected hedging error, lowering the hedging error variability, and improving the self‐financing possibility. The discrete dynamic delta‐only hedging not only causes potential overcharge to clients but also could be costly to the issuers. We show that a naive application of continuous‐time hedging formula to a discrete‐time hedging setting tends to worsen these possibilities.     

Pricing options on illiquid assets with liquid proxies using utility indifference and dynamic-static hedging

This work addresses the problem of optimal pricing and hedging of a European option on an illiquid asset Z using two proxies: a liquid asset S and a liquid European option on another liquid asset Y. We assume that the S-hedge is dynamic while the Y-hedge is static. Using the indifference pricing approach, we derive a Hamilton–Jacobi–Bellman equation for the value function. We solve this equation analytically (in quadrature) using an asymptotic expansion around the limit of perfect correlation between assets Y and Z. While in this paper we apply our framework to an incomplete market version of Merton’s credit-equity model, the same approach can be used for other asset classes (equity, commodity, FX, etc.), e.g. for pricing and hedging options with illiquid strikes or illiquid exotic options.     

Hedging credit: Equity liquidity matters

We theorize and confirm a new channel by means of which liquidity costs are embedded in CDS spreads. We show that credit default swap (CDS) spreads are directly related to equity market liquidity in the Merton [Merton, R.C., 1974. On the pricing of corporate debt: The risk structure of interest rates. J. Finance 29, 449–470] model via hedging. We confirm this relationship empirically using a sample of 1452 quarterly CDS spreads over 2001–2005. In the model, this relationship is monotone increasing when credit quality worsens. These results are robust to alternative measures of equity liquidity and other possible determinants of CDS spreads.     

THE ECONOMIC VALUE OF USING REALIZED VOLATILITY IN FORECASTING FUTURE IMPLIED VOLATILITY

We examine the economic benefits of using realized volatility to forecast future implied volatility for pricing, trading, and hedging in the S&P 500 index options market. We propose an encompassing regression approach to forecast future implied volatility, and hence future option prices, by combining historical realized volatility and current implied volatility. Although the use of realized volatility results in superior performance in the encompassing regressions and out-of-sample option pricing tests, we do not find any significant economic gains in option trading and hedging strategies in the presence of transaction costs.     

American-type basket option pricing: a simple two-dimensional partial differential equation

We consider the pricing of American-type basket derivatives by numerically solving a partial differential equation (PDE). The curse of dimensionality inherent in basket derivative pricing is circumvented by using the theory of comonotonicity. We start with deriving a PDE for the European-type comonotonic basket derivative price, together with a unique self-financing hedging strategy. We show how to use the results for the comonotonic market to approximate American-type basket derivative prices for a basket with correlated stocks. Our methodology generates American basket option prices which are in line with the prices obtained via the standard Least-Square Monte-Carlo approach. Moreover, the numerical tests illustrate the performance of the proposed method in terms of computation time, and highlight some deficiencies of the standard LSM method.     

Optimal discrete hedging of American options using an integrated approach to options with complex embedded decisions

In order to solve the problem of optimal discrete hedging of American options, this paper utilizes an integrated approach in which the writer’s decisions (including hedging decisions) and the holder’s decisions are treated on equal footing. From basic principles expressed in the language of acceptance sets we derive a general pricing and hedging formula and apply it to American options. The result combines the important aspects of the problem into one price. It finds the optimal compromise between risk reduction and transaction costs, i.e. optimally placed rebalancing times. Moreover, it accounts for the interplay between the early exercise and hedging decisions. We then perform a numerical calculation to compare the price of an agent who has exponential preferences and uses our method of optimal hedging against a delta hedger. The results show that the optimal hedging strategy is influenced by the early exercise boundary and that the worst case holder behavior for a sub-optimal hedger significantly deviates from the classical Black–Scholes exercise boundary.     

Pricing and Hedging Discount Bond Options in the Presence of Model Risk

This paper focuses on pricing and hedging options on a zero-coupon bond in a Heath–Jarrow–Morton (1992) framework when the value and/or functional form of forward interest rates volatility is unknown, but is assumed to lie between two fixed values. Due to the link existing between the drift and the diffusion coefficients of the forward rates in the Heath, Jarrow and Morton framework, this is equivalent to hedging and pricing the option when the underlying interest rate model is unknown. We show that a continuous rangeof option prices consistent with no arbitrage exist. This range is bounded by the smallest upper-hedging strategy and the largest lower-hedging strategy prices, which are characterized as the solutions of two non-linear partial differential equations. We also discuss several pricing and hedging illustrations.     

The pricing of basket-spread options

Since the pioneering paper of Black and Scholes was published in 1973, enormous research effort has been spent on finding a multi-asset variant of their closed-form option pricing formula. In this paper, we generalize the Kirk [Managing Energy Price Risk, 1995] approximate formula for pricing a two-asset spread option to the case of a multi-asset basket-spread option. All the advantageous properties of being simple, accurate and efficient are preserved. As the final formula retains the same functional form as the Black–Scholes formula, all the basket-spread option Greeks are also derived in closed form. Numerical examples demonstrate that the pricing and hedging errors are in general less than 1% relative to the benchmark results obtained by numerical integration or Monte Carlo simulation with 10 million paths. An implicit correction method is further applied to reduce the pricing errors by factors of up to 100. The correction is governed by an unknown parameter, whose optimal value is found by solving a non-linear equation. Owing to its simplicity, the computing time for simultaneous pricing and hedging of basket-spread option with 10 underlying assets or less is kept below 1 ms. When compared against the existing approximation methods, the proposed basket-spread option formula coupled with the implicit correction turns out to be one of the most robust and accurate methods.     

Who benefits in a crisis? Evidence from hedge fund stock and option holdings

We use a unique data set of hedge fund long equity and equity option positions to investigate a significant lockup-related premium earned during the tech bubble (1999–2001) and financial crisis (2007–2009). Net fund flows are significantly greater among lockup funds during crisis and noncrisis periods. Managers of hedge funds with locked-up capital trade opportunistically against flow-motivated trades of non-lockup managers, consistent with a hypothesis of rent extraction in providing crisis era liquidity. The success of this opportunistic trading is concentrated during periods of high borrowing costs, in less liquid stock markets, and is enhanced by hedging in the equity option market.     

Pricing and managing risks of European-style options in a Markovian regime-switching binomial model

We discuss the pricing and risk management problems of standard European-style options in a Markovian regime-switching binomial model. Due to the presence of an additional source of uncertainty described by a Markov chain, the market is incomplete, so the no-arbitrage condition is not sufficient to fix a unique pricing kernel, hence, a unique option price. Using the minimal entropy martingale measure, we determine a pricing kernel. We examine numerically the performance of a simple hedging strategy by investigating the terminal distribution of hedging errors and the associated risk measures such as Value at Risk and Expected Shortfall. The impacts of the frequency of re-balancing the hedging portfolio and the transition probabilities of the modulating Markov chain on the quality of hedging are also discussed.     

Model uncertainty,recalibration, and the emergence of delta–vega hedging

We study option pricing and hedging with uncertainty about a Black–Scholes reference model which is dynamically recalibrated to the market price of a liquidly traded vanilla option. For dynamic trading in the underlying asset and this vanilla option, delta–vega hedging is asymptotically optimal in the limit for small uncertainty aversion. The corresponding indifference price corrections are determined by the disparity between the vegas, gammas, vannas and volgas of the non-traded and the liquidly traded options.     

Hedging of Asian options under exponential Lévy models: computation and performance

In this paper we consider the problem of hedging an arithmetic Asian option with discrete monitoring in an exponential Lévy model by deriving backward recursive integrals for the price sensitivities of the option. The procedure is applied to the analysis of the performance of the delta and delta–gamma hedges in an incomplete market; particular attention is paid to the hedging error and the impact of model error on the quality of the chosen hedging strategy. The numerical analysis shows the impact of jump risk on the hedging error of the option position, and the importance of including traded options in the hedging portfolio for the reduction of this risk.     

A moment computation algorithm for the error in discrete dynamic hedging

This paper develops a computational approach to determining the moments of the distribution of the error in a dynamic hedging or payoff replication strategy under discrete trading. In particular, an algorithm is developed for portfolio affine trading strategies, which lead to portfolio dynamics that are affine in the portfolio variable. This structure can be exploited in the computation of moments of the hedging error of such a strategy, leading to a lattice based backward recursion similar in nature to lattice based pricing techniques, but not requiring the portfolio variable. We use this algorithm to analyze the performance of portfolio affine hedging strategies under discrete trading through the moments of the hedging error.