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Author: Marco Avellaneda Publisher: ISBN: Category : Languages : en Pages :
Book Description
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.
Author: Marco Avellaneda Publisher: ISBN: Category : Languages : en Pages :
Book Description
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.
Author: Erik Aurell Publisher: ISBN: Category : Languages : en Pages : 17
Book Description
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.
Author: Stein-Erik Fleten Publisher: ISBN: Category : Languages : en Pages : 17
Book Description
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.
Author: Paul Wilmott Publisher: ISBN: Category : Languages : en Pages :
Book Description
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.
Author: David C. Heath Glen Swindle Publisher: American Mathematical Soc. ISBN: 9780821867624 Category : Investments Languages : en Pages : 184
Book Description
The foundation for the subject of mathematical finance was laid nearly 100 years ago by Bachelier in his fundamental work, Theorie de la speculation. In this work, he provided the first treatment of Brownian motion. Since then, the research of Markowitz, and then of Black, Merton, Scholes, and Samuelson brought remarkable and important strides in the field. A few years later, Harrison and Kreps demonstrated the fundamental role of martingales and stochastic analysis in constructing and understanding models for financial markets. The connection opened the door for a flood of mathematical developments and growth. Concurrently with these mathematical advances, markets have grown, and developments in both academia and industry continue to expand. This lively activity inspired an AMS Short Course at the Joint Mathematics Meetings in San Diego (CA). The present volume includes the written results of that course. Articles are featured by an impressive list of recognized researchers and practitioners. Their contributions present deep results, pose challenging questions, and suggest directions for future research. This collection offers compelling introductory articles on this new, exciting, and rapidly growing field.
Author: D. Kannan Publisher: CRC Press ISBN: 1482294702 Category : Mathematics Languages : en Pages : 808
Book Description
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.
Author: Ioannis Karatzas Publisher: American Mathematical Soc. ISBN: 0821809091 Category : Business & Economics Languages : en Pages : 163
Book Description
In this text, the author discusses the main aspects of mathematical finance. These include, arbitrage, hedging and pricing of contingent claims, portfolio optimization, incomplete and/or constrained markets, equilibrium, and transaction costs. The book outlines advances made possible during the last fifteen years due to the methodologies of stochastic analysis and control. Readers are presented with current research, and open problems are suggested. This tutorial survey of the rapidly expanding field of mathematical finance is addressed primarily to graduate students in mathematics. Familiarity is assumed with stochastic analysis and parabolic partial differential equations. The text makes significant use of students' mathematical skills, but always in connection with interesting applied problems.
Author: Marco Avellaneda
Author: Marco Avellaneda
Author: Erik Aurell
Author: Stein-Erik Fleten
Author: Paul Wilmott
Author: Yuri Kabanov
Author: David C. Heath Glen Swindle
Author: Stein-Erik Fleten
Author: D. Kannan
Author: Ioannis Karatzas
Author: Mohsen Mazaheri