In a market with transaction costs the option hedging is costly. The idea presented by Leland (1985) was to include the expected transaction costs in the cost of a replicating portfolio. The resulting Leland's pricing and hedging method is an adjusted Black-Scholes method where one uses a modified volatility in the Black-Scholes formulas for the option price and delta. The Leland's method has been criticized on different grounds. Despite the critique, the risk-return tradeoff of the Leland's strategy is often better than that of the Black-Scholes strategy even in the case when a hedger starts with the same initial value of a replicating portfolio. This implies that the Leland's modification of volatility does optimize somehow the Black-Scholes hedging strategy in the presence of transaction costs. In this paper we explain how the Leland's modified volatility works and show how the performance of the Leland's hedging strategy can be improved by finding the optimal modified volatility. It is not claimed that the Leland's hedging strategy is optimal. Rather, the optimization mechanism of the modified hedging volatility can be exploited to improve the risk-return tradeoffs of other well-known option hedging strategies in the presence of transaction 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 to the claim in Leland (1985).
In the presence of transaction costs the perfect option replication is impossible which invalidates the celebrated Black and Scholes (1973) model. In this chapter we consider some approaches to option pricing and hedging in the presence of transaction costs. The distinguishing feature of all these approaches is that the solution for the option price and hedging strategy is given by a nonlinear partial differential equation (PDE). We start with a review of the Leland (1985) approach which yields a nonlinear parabolic PDE for the option price, one of the first such in finance. Since the Leland's approach to option pricing has been criticized on different grounds, we present a justification of this approach and show how the performance of the Leland's hedging strategy can be improved. We extend the Leland's approach to cover the pricing and hedging of options on commodity futures contracts, as well as path-dependent and basket options. We also present examples of finite-difference schemes to solve some nonlinear PDEs. Then we proceed to the review of the most successful approach to option hedging with transaction costs, the utility-based approach pioneered by Hodges and Neuberger (1989). Judging against the best possible tradeoff between the risk and the costs of a hedging strategy, this approach seems to achieve excellent empirical performance. The asymptotic analysis of the option pricing and hedging in this approach reveals that the solution is also given by a nonlinear PDE. However, this approach has one major drawback that prevents the broad application of this approach in practice, namely, the lack of a closed-form solution. The numerical computations are cumbersome to implement and the calculations of the optimal hedging strategy are time consuming. Using the results of asymptotic analysis we suggest a simplified parameterized functional form of the optimal hedging strategy for either a single option or a portfolio of options and a method for finding the optimal parameters.
This thesis studies the problem of approximate hedging with constant proportional transaction costs in stochastic volatility models in different situations, using a simpler form for adjusted volatility in the Leland's algorithm. We show that asymptotic properties of hedging error are the same to those in deterministic volatility models and the rate of convergence can be impoved by controlling the model parameter. These can be extended to the case where transaction costs are defined by a general rule. We also show that jumps appear in asset price and/or in stochastic volatility do not affect asymptotic property of hedging error. In the next part, we consider the problem of approximate hedging in the presence of liquidity risks suggested by Cetin, Jarrow and Protter, of which proportional transaction costs models are a particular case. We show that liquidity costs due to smooth supply surves can be ignored using Leland's increasing volatility principle. In the third part, we study the case where the option is written on multiple risky assets. We demonstrate that approximately complete replication can be reached for exchange options using the same parameter suggested by Leland, but it is far from being obvious for other kinds of exotic options. Finally, we propose a simple method to reduce the option price which clearly approaches to the super hedging price in Leland's algorithm. whenever the seller accepts to take a risk defined by a given significance level.
Recognizing that robust decision making is vital in risk management, this book provides concepts and algorithms for computing the best decision in view of the worst-case scenario. The main tool used is minimax, which ensures robust policies with guaranteed optimal performance that will improve further if the worst case is not realized. The applications considered are drawn from finance, but the design and algorithms presented are equally applicable to problems of economic policy, engineering design, and other areas of decision making. Critically, worst-case design addresses not only Armageddon-type uncertainty. Indeed, the determination of the worst case becomes nontrivial when faced with numerous--possibly infinite--and reasonably likely rival scenarios. Optimality does not depend on any single scenario but on all the scenarios under consideration. Worst-case optimal decisions provide guaranteed optimal performance for systems operating within the specified scenario range indicating the uncertainty. The noninferiority of minimax solutions--which also offer the possibility of multiple maxima--ensures this optimality. Worst-case design is not intended to necessarily replace expected value optimization when the underlying uncertainty is stochastic. However, wise decision making requires the justification of policies based on expected value optimization in view of the worst-case scenario. Conversely, the cost of the assured performance provided by robust worst-case decision making needs to be evaluated relative to optimal expected values. Written for postgraduate students and researchers engaged in optimization, engineering design, economics, and finance, this book will also be invaluable to practitioners in risk management.
It's not hard to understand why options trading continues to growin popularity, especially among sophisticated investors with largestock portfolios. Options are a cheaper and therefore, inherentlyless risky way of speculating on the price movements of stocks orother under-lying goods, yet, due to their volatility, they providemore price action per dollar than do stocks. And, when traded inconjunction with stock portfolios, options can significantlyenhance an investor's ability to manipulate the risk and returncharacteristics of their entire investment. Yet, despite these andother advantages of options, many investors shy away from thishighly lucrative type of speculation because of the seemingimpenetrability of many of its underlying concepts and technicalprinciples. Now in a book that demystifies options for financial professionals,Professor Robert W. Kolb, one of the nation's leading authoritieson the subject, provides readers with a solid grounding in theprinciples and practices of options trading. An excellent resourcefor investors who need to quickly get up to speed in options,Understanding Options offers a balanced presentation that buildsswiftly from the most basic concepts and terms to advanced tradingstrategies and techniques. Written in plain English and filled withreal-life examples and case studies, it schools readers in: * All essential terms, concepts, principles, and practices * Popular trading techniques and their payoffs * Option strategies * Option hedging * Formal trading models, including the Binomial and Merton models * Options on stock indexes, foreign currency, and futures * Option pricing in both the American and European markets * The options approach to corporate securities * And much more Concise yet comprehensive, authoritative yet highly accessible,Understanding Options gives you everything you need to feel rightat home in the lucrative world of options. Comprehensive, practical, authoritative--the fastest, mostaccessible route to the lucrative world of options From the basics of what an option is to advanced techniques forprofiting from options in a variety of markets, UnderstandingOptions covers all the bases. Written by a leading internationalauthority on options trading, this practical, hands-on guide offersdetailed, step-by-step coverage of option trading techniques andtheir payoffs, option strategies, European and American optionpricing, option hedging, and much more. It also explores options onstock indexes, foreign currency, and futures, and takes a closelook at the options approach to corporate securities. A concise, yet comprehensive, introduction to options for financialprofessionals * Gets you quickly up and running with all the essential knowledgeyou need to break into the options markets * Featuring a balanced presentation that moves swiftly from basicterms and concepts to advanced trading models * Packed with easy-to-follow examples and case studies that lucidlyillustrate all points covered
