Hedging barrier options : current methods and alternatives

This paper applies to the static hedge of barrier options a technique mean square hedging designed to minimize the size of the hedging error when perfect replication is not possible It introduces an extension of this technique which preserves the computational e ciency of mean square hedging while being consistent with any prior pricing model or with any linear constraint on the hedging residual This improves on current static hedging methods which aim at exactly replicating barrier options and rely on strong assumptions on the availability of traded options with certain strikes or maturities or on the distribution of the underlying asset Derivatives markets have become an essential part of the global economic and nancial system in the past decade Growth in trading activity has been particulary fast in over the counter OTC markets which have become the dominant force in the industry One factor in the success of OTC markets has been their ability to o er new exotic derivatives customized to customers needs Among the most successful OTC option products are barrier options The payo of a barrier option depends on whether the The author thanks Casper de Vries Terry Cheuk Ton Vorst and seminar partici pants at Eurandom Groningen University and Erasmus University The usual disclaimer applies As of December OTC transactions accounted for around percent trillion of the total notional value of derivatives contracts exchanges for about percent trillion source IMF International Capital Markets Sept price of the underlying asset crosses a given threshold the barrier before maturity The simplest barrier options are knock in options which come into existence when the price of the underlying asset touches the barrier and knock out options which come out of existence in that case For example an up and out call has the same payo as a regular plain vanilla call if the price of the underlying asset remains below the barrier over the life of the option but becomes worthless as soon as the price of the underlying asset crosses the barrier Barrier options ful ll di erent economic needs and have been widely used in foreign exchange and xed income derivatives markets since the mid s Steinherr and Taleb discuss issues linked with using barrier options Barrier options reduce the cost of modifying one s exposure to risk because they are cheaper than their plain vanilla counterparts They help traders who place directional bets enhance their leverage and investors who accept to keep some residual risk on their books reduce their hedging costs More generally barrier options allow market participants to tailor their trading strategies to their speci c market views Traders who believe in a market upswing of limited amplitude prefer to buy up and out calls rather than more expensive plain vanilla calls In contrast fund managers who use options to insure only against a limited downswing prefer down and out puts Using barrier options investors can easily translate chartist views into their trading strategies Clarke presents detailed applications of these instruments to technical trading in foreign exchange markets Options with more complex barrier features are also traded Naturally traders who use barrier options to express their directional views on the market do not want to hedge their positions they use such options to gain leveraged exposure to risk In contrast their counterparties may not be directional players with opposite views but rather dealers who provide liquidity by running barrier option books and hedge their positions Barrier options are di cult to hedge because they combine features of plain vanilla options and of a bet on whether the underlying asset price hits the barrier Merely adding a barrier provision to a regular option signi cantly impacts the sensitivity of the option value to changes in the price or the volatility of the underlying asset The Greeks of barrier options behave very di erently from those of plain vanilla options Figure The discon tinuity in the payo of barrier options complicates hedging especially those that are in the money when they come out of existence such as up and out calls and down and out puts Delta hedging these options is particularly unpractical For example when the price nears the barrier and the option is about to expire the Delta and the Gamma of an up and out call takes large negative values because the option payo turns into a spike in this re gion Vega also turns negative when the price of the underlying asset is close to the barrier because a volatility pickup near the barrier increases the like The Greeks are the sensitivities of the option price to changes in the parameters and are used to assess risk They include Delta the rst order derivative of the option value with respect to the price of the underlying asset Gamma its second order derivative with respect to the same parameter and Vega the derivative of the option value with respect to the volatility lihood of the price passing through it In contrast the Delta Gamma and Vega of regular options are always positive and well behaved functions Despite the di culties associated with hedging barrier options banks write large amounts of those instruments to accomodate customer demand and typically prefer to hedge their positions Moreover when devising hedg ing strategies banks have to face the limitations of real markets like trans action costs discrete trading and lack of liquidity One way of addressing these concerns is to limit the frequency of trading and in the limit to solely consider static hedging To compensate the associated loss of exibility one may prefer to hedge barrier options with regular options instead of with the underlying asset and a riskless bond because regular options are more closely related to barrier options Derman Ergener and Kani there after DEK and Carr Ellis and Gupta thereafter CEG model static hedging of barrier options with regular options on the same underlying as set They are able to achieve perfect replication of the barrier option but to achieve this result they need strong assumptions on the availibility of reg ular options with certain strikes or maturities or on the distribution of the underlying asset Their assumptions are not all satis ed in actual markets and DEK and CEG do not present a structured approach on how to adapt their techniques in this case The paper presents an alternative methodology Instead of aiming at perfectly replicating a barrier option we choose to approximate it The metric used to gauge the t of the hedge is the mean of the square of the hedging residual the di erence between the payo of the claim and that of the hedging portfolio leading to the mean square hedging method introduced in Du e and Richardson and Schweizer We also extend the mean square hedging methodology to make the hedging portfolio consistent with linear constraints For example we can incorporate the requirement that the value of the portfolio coincide with the value that a given pricing model a ects to the barrier option Financial economists have developed other hedging methods in incom plete markets The super hedging strategy of El Karoui and Quenez satis es investors who want to avoid any shortfall risk the risk that the value of the hedging portfolio be below that of the asset it is designed to hedge For investors willing to hedge only partially quantile hedging achieves the lowest replication cost for a given probability of shortfall and ts well the popular value at risk VaR approach F ollmer and Leukert How ever Vorst indicates that portfolio optimization under VaR constraint can lead to unattractive risk pro les An important di erence between the present paper and the approaches cited above or the current literature on mean square hedging is that this paper aims at providing a framework that investors can use to implement mean square hedging to a case of practical importance barrier options To do so it uses only discrete time techniques and can be adapted to any pricing model In contrast current research focuses on the theoretical issues of introducing mean square or quantile hedging in continuous time models where jumps or stochastic volatility render markets incomplete The remainder of the paper is organized as follows Section brie y reviews hedging principles and summarizes the contributions of DEK and CEG Section introduces the theory of mean square hedging and section applies this technique to hedging an up and out call in an environment similar to the one used in DEK Section presents our own extension to mean square hedging and section applies it to hedging an up and out call under some constraints on the residual Hedging Barrier Options Financial modelers typically rely on the assumption that markets are com plete and compute the prices of derivative assets via hedging arguments In contrast practitioners of nance face the limitations of real markets and the impossibility of perfect hedging Hedging and Pricing in Complete and Incomplete Mar kets Pricing derivatives securities typically uses a hedging argument based on the assumption that markets are complete or dynamically complete that is that any claim can be replicated by trading marketable assets because there are enough instruments or because trading can take place su ciently often Under these conditions the no arbitrage price of any derivatives product is the cost of the replicating portfolio and securities can be valued indepen dently of investors preferences Black and Scholes derive the price of plain vanilla options and Merton that of barrier options and other derivatives by showing how continuous trading in the underlying stock and in a riskless bond replicates the option s payo in every state of the world In contrast when claims cannot be perfectly replicated by trading mar ketable assets any hedging strategy leaves some residual risk and investors attitudes toward risk a ect the pricing of the claims For example Hull and White intro