We introduce a new class of strategies for hedging derivative securities taking into account transaction costs, assuming lognormal continuous-time prices for the underlying asset. We do not assume that the payoff is convex as in Leland (J of Finance, 1985), or that the transaction costs are small compared to the price changes between portfolio adjustments, as in Hoggard, Whalley and Wilmott (Adv. in Futures and Options Res., 1993). The Leland number, A, which is proportional to the ratio of the round-trip tansaction cost over the typical price movement during the period between transactions, is a measure of the importance of transaction costs versus hedging risk. If A is greater than or equal to one, standard delta-hedging methods fail unless the payoff of the derivative security is a convex function of the price of the underlying asset. In contrast, our new strategies can be used effectively in the presence of large transaction costs to control simultaneously hedge-slippage as well as hedging costs. These strategies are associated with the solution an quot;obstacle problemquot; for a Black-Scholes diffusion equation with Leland's quot;augmentedquot; volatility, a parameter which depends on the volatility of the underlying asset as well as on A. The new strategies are such that the frequency for rebalancing the portfolio is variable. There are periods in which rehedging takes place often to control gamma-risk and other periods, which can be relatively long, when no transactions are needed. Moreover, instead of replicating exactly the final payoff, the strategies can yield a positive cash flow at expiration, according to the price history of the underlying security. The solution to the quot;obstacle problemquot; is often simple to calculate. There exist closed-form solutions for various securities of practical interest, such as digital options.

We investigate the optimal strategy over a finite time horizon for a portfolio of stock and bond and a derivative in an multiplicative Markovian market model with transaction costs (friction). The optimization problem is solved by a Hamilton-Bellman-Jacobi equation, which by the verification theorem has well-behaved solutions if certain conditions on a potential are satisfied. In the case at hand, these conditions simply imply arbitrage-free (Black-Scholes) pricing of the derivative. While pricing is hence not changed by friction allow a portfolio to fluctuate around a delta hedge. In the limit of weak friction, we determine the optimal control to essentially be of two parts: a strong control, which tries to bring the stock-and-derivative portfolio towards a Black-Scholes delta hedge; and a weak control, which moves the portfolio by adding or subtracting a Black-Scholes hedge. For simplicity we assume growth-optimal investment criteria and quadratic friction.

Multi-period guarantees are often embedded in life insurance contracts. In this paper we consider the problem of hedging these multi-period guarantees in the presence of transaction costs. We derive the hedging strategies for the cheapest hedge portfolio for a multi-period guarantee that with certainty makes the insurance company able to meet the obligations from the insurance policies it has issued. We find that by imposing transaction costs, the insurance company reduces the rebalancing of the hedge portfolio. The cost of establishing the hedge portfolio also increases as the transaction cost increases. For the multi-period guarantee there is a rather large rebalancing of the hedge portfolio as we go from one period to the next. By introducing transaction costs we find the size of this rebalancing to be reduced. Transaction costs may therefore be one possible explanation for why we do not see the insurance companies performing a large rebalancing of their investment portfolio at the end of each year.

We derive a nonlinear parabolic partial differential equation for the value of portfolios of options in the presence of proportional transaction costs. This assumes a Leland world of transacting after each time interval, which is of fixed length. The equation reduces to the modified variance case described by Leland in the case of a single option. We demonstrate the nonlinear nature of option portfolios and give results for several simple combinations of options.

The book is the first monograph on this highly important subject.

An introduction to general theories of stochastic processes and modern martingale theory. The volume focuses on consistency, stability and contractivity under geometric invariance in numerical analysis, and discusses problems related to implementation, simulation, variable step size algorithms, and random number generation.