Dynamic Hedging Portfolios for Derivative Securities in the Presence of Large Transaction Costs

We introduce a new class of strategies for hedging derivative securities in the presence of transaction costs assuming lognormal continuous time prices for the underlying asset We do not assume necessarily that the payo is convex as in Leland or that transaction costs are small compared to the price changes between portfolio adjustments as in Hoggard Whalley and Wilmott The type of hedging strategy to be used depends on the value of the Leland number A q k p t where k is the round trip transaction cost is the volatility of the underlying asset and t is the time lag between transactions If A it is possible to implement modi ed Black Scholes delta hedging strategies but not otherwise We propose new hedging strategies that can be used with A to control e ectively hedging risk and transaction costs These strategies are associated with the solution of a nonlinear obstacle problem for a di usion equation with volatility A p A In these strategies there are periods in which rehedging takes place after each interval t and other periods in which a static strategy is required The solution to the obstacle problem is simple to calculate and closed form solutions exist for many problems of practical interest Courant Institute of Mathematical Sciences New York University Mercer St New York N Y According to Black Scholes theory the value of an option is equal to the initial cost of a dynamic portfolio of traded securities that provides a similar cash ow at expiration i e it replicates the payo This theory explains to within reasonable approximation the prices of standard options and other derivative securities over moderate periods of time in markets in which transaction costs are negligible Under these ideal conditions the Black Scholes theory also provides essentially riskless strategies for hedging derivative products If bid ask spreads and other transaction costs are also taken into account each ad justment of the portfolio implies an additional cost and the replicating property of the Black Scholes hedge no longer holds Making frequent adjustments to maintain the theo retical hedge can increase costs considerably On the other hand if only few adjustments are made the Black Scholes exact hedge cannot be maintained due to movements in the price of the underlying asset between trades Therefore transaction costs cannot be ig nored without incurring in risk or loss This is an important practical problem especially in emerging markets where roundtrip transaction costs of and higher are not uncommon In Leland introduced a theory for pricing a call option with transaction costs Using an elegant argument he showed that the price of a call is given by the Black Scholes formula with an augmented volatility