1. FMG PH.D. seminar (21 October 1999) slide 1 Dynamic Hedging with Transaction Costs Outline: -Introduction -Adjusted path and Strategy -Leland model and its problem -Introducing New strategies -Concluding Comments

2. FMG PH.D. seminar (21 October 1999) slide 2 Topical issue n Internet brokerage and on-line trading at the core of financial strategy (e-trade, C. Schwab, Ameritrade …) (Instinet, Island, …) n Electronic markets (NASDAQ, EUREX vs LIFFE …) n New exotic products: digital, corridor or double barrier options, passport option..

3. FMG PH.D. seminar (21 October 1999) slide 3 Derivatives challenge n Necessity of a dynamic hedge for Derivatives products F high leverage F can be OTC deals (digital, double barrier option) F principal size n dynamic hedging challenging F high transaction costs F specialised markets

4. FMG PH.D. seminar (21 October 1999) slide 4 Introduction n Option Pricing based on Replication: Black Scholes (73) u no transaction costs u risk neutrality u utility-free approach u continuous time n Transaction costs: u loss / risk u imcompleteness u paradox: F Hankanson (79) Need and no need F Taleb (97) exploding volatility

5. FMG PH.D. seminar (21 October 1999) slide 5 Adjusted sample path n Dynamic hedging u attempt to replication u positive/negative gamma hedging n Impact on portfolio u Boyle and Emmanuel (80) Leland (85), Gilster (90) : trade-off between variance and costs n Asymmetry of the position: u Buyer versus seller u Bid-ask spread

6. FMG PH.D. seminar (21 October 1999) slide 6 Dynamic hedging policies n Three general policies: u Boyle and Emanuel (80) Leland (85) Lacoste (93): operator revises the portfolio at exogenously set of time increments u Boyle and Vorst (92): operator revises according to u utility based approach:Hodges and Neuberger (89),Davis Panas and Zariphoupoulou (93): second momentum of the portfolio

7. FMG PH.D. seminar (21 October 1999) slide 7 Stylised facts n Transaction not negligible and high for illiquid markets: –Credit Derivatives : Default swap Spread options –Exotic: Binaries, barriers) n Dynamic hedge Non Markovian n Hedge revision according to Taleb (97) n Asymmetry between short/long Gamma (Gold Gamma)

8. FMG PH.D. seminar (21 October 1999) slide 8 Leland(85) model n Adjusted volatility n Trade-off between dynamic hedging and transaction costs n Probabilistic framework with Lacoste(93) (using Wiener Chaos) n Extended by Whaley and Wilmot (93), Avellaneda and Paras (94) to convex functions. Deduce a PDE approach.

9. FMG PH.D. seminar (21 October 1999) slide 9 Delta Hedging with Transaction Costs n Proportional spread at the middle point n Underlying as a Geometric Brownian motion n Dynamic hedge shares of the underlying n Change of the portfolio

10. FMG PH.D. seminar (21 October 1999) slide 10 n By Ito n Equaling the two terms give that re- hedging cost are equal to n Interpretation: Leland number measures the influence of transaction costs

11. FMG PH.D. seminar (21 October 1999) slide 11 n Modified volatility n Implications: u Convex Pay-off u or small transaction costs A<1 n Drawback if and A>1, static hedge

12. FMG PH.D. seminar (21 October 1999) slide 12 New strategy? n Bid-ask spread important for Illiquid securities n Non convex derivatives like call spread … n Basic problem is to solve with

13. FMG PH.D. seminar (21 October 1999) slide 13 New way of thinking n 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. FMG PH.D. seminar (21 October 1999) slide 14 Introducing new hedging strategies n What are the solutions? u Obviously, convex functions are well-defined solutions. u For concave solutions, problem mathematically ill-posed n In terms of physics, problem known as obstacle problem u Solutions can be constructed as a piece-wise convex function

15. FMG PH.D. seminar (21 October 1999) slide 15 Looking at discontinuous hedging strategy n Problem at the kink u Convex: hedging unavoidable since value of the portfolio declining u Concave: no need to dynamic hedge. Adopting a static hedge n Time decay and Option replication u At maturity, u implying an inequality

16. FMG PH.D. seminar (21 October 1999) slide 16 Solution and implication n Let us define the obstacle problem as n and a.s. n At maturity

17. FMG PH.D. seminar (21 October 1999) slide 17 Interest of the solution n Extend the result of Leland (85) Whaley and Wilmot (93) n Introduces new ways of dynamic hedging: Dynamic hedge now Non-Markovian Trade-off between static hedge and dynamic hedge. Reduction of transaction cost n 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. FMG PH.D. seminar (21 October 1999) slide 18 Next Steps n Do simulations and Examine particular case (especially digital where high costs) n Problem with the revising rule. Taleb(97): Assuming a fixed capital at the beginning, a.s. the investor is going to be ruined. n Solution: using stopping times of a Brownian motion with a double barrier.

19. FMG PH.D. seminar (21 October 1999) slide 19 Conclusion n This model includes the results of Leland (85) Whaley and Wilmot (93) n Consistent with u empirical fact : dynamic hedge often non Markovian u High transaction costs and negative gamma position n Drawback : No closed form in the general case n Need to be adapted to stochastic stopping time