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Dynamic Hedging with Transaction Costs

Dynamic Hedging with Transaction Costs. Outline: -Introduction -Adjusted path and Strategy -Leland model and its problem -Introducing New strategies -Concluding Comments. Topical issue. Internet brokerage and on-line trading at the core of financial strategy

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Dynamic Hedging with Transaction Costs

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  1. Dynamic Hedging with Transaction Costs Outline: -Introduction -Adjusted path and Strategy -Leland model and its problem -Introducing New strategies -Concluding Comments

  2. Topical issue • Internet brokerage and on-line trading at the core of financial strategy (e-trade, C. Schwab, Ameritrade …) (Instinet, Island, …) • Electronic markets (NASDAQ, EUREX vs LIFFE …) • New exotic products: digital, corridor or double barrier options, passport option..

  3. Derivatives challenge • Necessity of a dynamic hedge for Derivatives products • high leverage • can be OTC deals (digital, double barrier option) • principal size • dynamic hedging challenging • high transaction costs • specialised markets

  4. Introduction • Option Pricing based on Replication: Black Scholes (73) • no transaction costs • risk neutrality • utility-free approach • continuous time • Transaction costs: • loss / risk • imcompleteness • paradox: • Hankanson (79) Need and no need • Taleb (97) exploding volatility

  5. Adjusted sample path • Dynamic hedging • attempt to replication • positive/negative gamma hedging • Impact on portfolio • Boyle and Emmanuel (80) Leland (85), Gilster (90) : trade-off between variance and costs • Asymmetry of the position: • Buyer versus seller • Bid-ask spread

  6. Dynamic hedging policies • Three general policies: • Boyle and Emanuel (80) Leland (85) Lacoste (93): operator revises the portfolio at exogenously set of time increments • Boyle and Vorst (92): operator revises according to • utility based approach:Hodges and Neuberger (89),Davis Panas and Zariphoupoulou (93): second momentum of the portfolio

  7. Stylised facts • Transaction not negligible and high for illiquid markets: • Credit Derivatives : Default swap Spread options • Exotic: Binaries, barriers) • Dynamic hedge Non Markovian • Hedge revision according to Taleb (97) • Asymmetry between short/long Gamma (Gold Gamma)

  8. Leland(85) model • Adjusted volatility • Trade-off between dynamic hedging and transaction costs • Probabilistic framework with Lacoste(93) (using Wiener Chaos) • Extended by Whaley and Wilmot (93), Avellaneda and Paras (94) to convex functions. Deduce a PDE approach.

  9. Delta Hedging with Transaction Costs • Proportional spread at the middle point • Underlying as a Geometric Brownian motion • Dynamic hedge shares of the underlying • Change of the portfolio

  10. By Ito • Equaling the two terms give that re-hedging cost are equal to • Interpretation: Leland number measures the influence of transaction costs

  11. Modified volatility • Implications: • Convex Pay-off • or small transaction costs A<1 • Drawback if and A>1, static hedge

  12. New strategy? • Bid-ask spread important for Illiquid securities • Non convex derivatives like call spread … • Basic problem is to solve with

  13. New way of thinking • Transaction costs leads to incomplete markets (Pham&Touzi (97)) no unique price, no unique martingale measure, no replication Solution: super hedging, utility assumption or risk minimizing

  14. Introducing new hedging strategies • What are the solutions? • Obviously, convex functions are well-defined solutions. • For concave solutions, problem mathematically ill-posed • In terms of physics, problem known as obstacle problem • Solutions can be constructed as a piece-wise convex function

  15. Looking at discontinuous hedging strategy • Problem at the kink • Convex: hedging unavoidable since value of the portfolio declining • Concave: no need to dynamic hedge. Adopting a static hedge • Time decay and Option replication • At maturity, • implying an inequality

  16. Solution and implication • Let us define the obstacle problem as • and a.s. • At maturity

  17. Interest of the solution • Extend the result of Leland (85) Whaley and Wilmot (93) • Introduces new ways of dynamic hedging: Dynamic hedge now Non-Markovian • Trade-off between static hedge and dynamic hedge. • Reduction of transaction cost • Introduces strong similarities with American Option pricing: Therefore no closed formula for the general framework (solution PDE, Lattice method and Monte Carlo with estimation of the exercise frontier)

  18. Next Steps • Do simulations and Examine particular case (especially digital where high costs) • Problem with the revising rule. Taleb(97): Assuming a fixed capital at the beginning, a.s. the investor is going to be ruined. • Solution: using stopping times of a Brownian motion with a double barrier.

  19. Conclusion • This model includes the results of Leland (85) Whaley and Wilmot (93) • Consistent with • empirical fact : dynamic hedge often non Markovian • High transaction costs and negative gamma position • Drawback : No closed form in the general case • Need to be adapted to stochastic stopping time

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