1. Static Hedging of American Barrier Options and Applications San-Lin Chung, Pai-Ta Shih and Wei-Che Tsai Presenter: Wei-Che Tsai 1 蔡維哲 @ National Taiwan University

2. Outlines  Introduction  Formulation of the static hedging portfolio for an American barrier option under Black-Scholes model  The hedging performance for an American barrier option  Pricing American barrier option under CEV model and American lookback options under Black-Scholes model  Conclusions 2 蔡維哲 @ National Taiwan University

3. Introduction  Non-standard or exotic options are widely used today by banks, corporations and institutional investors, in their management of risk.  The use of non-standard options may not only fit the risk to be hedged better, but also lower the hedging cost, in such cases.  Barrier options are among the most common exotic options that are used in the foreign exchange, interest rate and equity options markets. (Hsu, 1997) 3 蔡維哲 @ National Taiwan University

4. Introduction  Derman, Ergener, and Kani (1995) develop an approach to static hedge European barrier options by using a standard European option to match the boundary at maturity of the barrier option and a continuum of standard European options with different maturities but with the strike price equaling the boundary value before maturity to match boundary before maturity of the barrier option.  Carr (1998) uses standard options with different strike prices for static hedging exotic options. 4 蔡維哲 @ National Taiwan University

5. Introduction  The focus of this paper is on the static hedging of American barrier options which are not known in the literature.  On the other hand, pricing exotic options are also important in the literature.  Merton (1973) first derives the price of a continuously monitored down-and-out European call in the literature and Rubinstein and Reiner (1991) further provide pricing formulas for various standard European barrier options under Black-Scholes model. 5 蔡維哲 @ National Taiwan University

6. Introduction  Davydov and Linetsky (2001) derive the European barrier option pricing formula under CEV model in closed form.  However, pricing American barrier options is an important yet difficult problem in the finance research especially when the underlying asset follows other stochastic processes instead of geometric Brownian motion, e.g. the constant elasticity of variance (CEV) model of Cox (1975). 6 蔡維哲 @ National Taiwan University

7. Introduction  In the literature, researchers have developed various numerical methods to discuss this problem. [Hull and White (1993), Ritchken (1995), Cheuk and Vorst (1996), Boyle and Tian (1999), Babbs (2000), Gao, Huang, and Subrahmanyam (2000), Zvan, Vetzal, and Forsyth (2000), AitSahlia, Imhof, and Lai (2003), Lai and Lim (2004), Tebaldi (2005), and Chang, Kang, Kim, and Kim (2007).] 7 蔡維哲 @ National Taiwan University

8. Introduction - three advantages of static hedging  First, static hedging may be considerably cheaper than dynamic hedging when the transaction cost is large.  Second, static hedging method has vegas and gammas close to those of the target option being hedged.  Finally, several articles have documented that static hedging is less sensitive to the model risk such as volatility misspecification, see for example Thomsen (1998) and Tompkins (2002). 8 蔡維哲 @ National Taiwan University

9. Introduction  In this paper, we also show that our static hedging approach serves as a good pricing method for American exotic options.  We extend the idea of static hedging to price American barrier options under CEV model, and to price American floating lookback option under Black-Scholes model.  The results indicate that our static hedging approach is comparable to the tree methods of Boyle and Tian (1999) and Babbs (2000) respectively. 9 蔡維哲 @ National Taiwan University

10. Static hedging of American barrier options  Following the seminal work of Gao, Huang, and Subrahmanyam (2000), we focus on American up- and-out put options written on non-dividend paying underlying assets.  One particular difficulty of statically hedging the American barrier option is that it involves a free boundary problem, i.e. the early exercise boundary has to be determined at the same time when the static hedge portfolio is formulated. 10 蔡維哲 @ National Taiwan University

11. Static hedging of American barrier options  We solve this problem by using value-matching condition on the barrier boundary H and two well- known the value-matching and smooth-pasting conditions on the early exercise boundary.  Similar to the lattice method, we work backward to determine the number of the standard European options and their strike prices for the above n-point static hedge portfolio. 11 蔡維哲 @ National Taiwan University

12. Static hedging of American barrier options  For example, at time t(n-1), three conditions imply that 12 蔡維哲 @ National Taiwan University

13. Static hedging of American barrier options Figure 1. The static hedge portfolio for an AUOP option 13 蔡維哲 @ National Taiwan University

14. Static hedging of American barrier options  Using similar procedures, we work backward to determine the number of units of the European option,,, and the early exercise price at time (i=n-2, n-3,…, 0).  Finally, the value of the n-point static hedge portfolio at time 0 is obtained as follows: 14 蔡維哲 @ National Taiwan University

15. Hedging Performance - The SHP of an AUOP under the BS model  Parameters: S0 = 100, X = 100, H = 110, r = 4%, q = 0, sigma = 0.2, and T = 1.  The benchmark value 4.890921 uses Ritchken’s trinomial lattice method with 52,000 time steps. 15 蔡維哲 @ National Taiwan University Panel A. Static Hedge an American Up-and-Out Put Option using Nonstandard Strikes Quantity of European Call Strike Quantity of European Put Strike Expiration (months) Value for S0 = 100 -0.0044151100.04649480.2380132-0.002381 -0.0118481100.04737480.8102454-0.015407 -0.0263741100.04841881.6217836-0.057712 -0.0572011100.05264982.7991378-0.182052 -0.1259891100.00716784.55896310-0.588633 -0.1098031100.16662388.06147112-0.276521 1100126.003998 Net4.881292

16. Hedging Performance - The SHP of an AUOP under the BS model  Parameters: S0 = 100, X = 100, H = 110, r = 4%, q = 0, sigma = 0.2, and T = 1.  The benchmark value 4.890921 uses Ritchken’s trinomial lattice method with 52,000 time steps. 16 蔡維哲 @ National Taiwan University Panel B. Static Hedge an American Up-and-Out Put Option using Standard Strikes Quantity of European Call Strike Quantity of European Put Strike Expiration (months) Value for S0 = 100 -0.0044001100.042348802-0.002426 -0.0118061100.045227804-0.016477 -0.0265561100.032733806-0.066060 -0.0576731100.215716808-0.132589 -0.125408110-0.2672888010-0.749253 -0.1096511100.3155658512-0.158067 1100126.003998 Net4.879124

17. Hedging Performance - Mismatch Values on the Barrier and the EEB using Nonstandard Strikes  Parameters: S0 = 100, X = 100, H = 110, r = 4%, q = 0, sigma = 0.2, and T = 1.  The accurate early exercise boundary is calculated from the Black-Scholes model of Ritchken (1995) with 52,000 time steps per year. 17 蔡維哲 @ National Taiwan University

18. Hedging Performance - Mismatch Values on the Barrier and the EEB using Standard Strikes  Parameters: S0 = 100, X = 100, H = 110, r = 4%, q = 0, sigma = 0.2, and T = 1.  The accurate early exercise boundary is calculated from the Black-Scholes model of Ritchken (1995) with 52,000 time steps per year. 18 蔡維哲 @ National Taiwan University

19. Hedging Performance - The profit-and-loss distribution (6-SHP)  Parameters: S0 = 100, X = 100, H = 110, r = 4%, q = 0, sigma = 0.2, and T = 1.  We run 50,000 Monte Carlo simulations with time discretized to 52,000 time steps per year to construct the profit-and-loss distribution.  At every knock-out stock price time point or early exercise decision, we liquidate the static hedging portfolio (SHP). 19 蔡維哲 @ National Taiwan University

20. Numerical results - Hedging performance of SHP and DHP under the BS model  Parameters: S0 = 100, X = 100, H = 110, r = 4%, q = 0, sigma = 0.2, and T = 1.  For example, n=130 means that we dynamically adjust delta- hedged portfolios every 400/52000 time.  We adopt four risk measures used by Siven and Poulsen (2009) to evaluate the hedging performance of the two static hedge portfolios. 20 蔡維哲 @ National Taiwan University

21. Pricing of American barrier options using the static hedging approach under CEV model  In this section, we present the flexible of the static hedging approach even if we consider the constant elasticity variance model proposed by Cox (1975).  In the CEV model, the stock price satisfies the following diffusion process: 21 蔡維哲 @ National Taiwan University

22. Pricing of American barrier options using the static hedging approach under CEV model  In the following numerical analysis, we refer to the parameter setting of Gao, Huang, and Subrahmanyam (2000) and Chang, Kang, Kim, and Kim (2007) with the following parameters: S = 40, X = 45, H = 50, r = 4.48%, σ = 0.683990, and q = 0.  The beta parameter (β=4/3) is taken from Schroder (1989) and Chung and Shih (2009). 22 蔡維哲 @ National Taiwan University

23. Numerical results - The convergence pattern of an AUOP under the CEV model  The benchmark accurate American up-and-out put option price which uses Boyle and Tian’s method with 52,000 time steps is about 5.358829. 23 蔡維哲 @ National Taiwan University

24. Numerical results - The EEB of an AUOP under the CEV model  The “Benchmark Value” is obtained by the Boyle and Tian method with 52,000 time steps of the early exercise price of an AUOP option.  The number of nodes matched on the early exercise boundary in the SHP method is 52 per year. 24 蔡維哲 @ National Taiwan University T-tBenchmark ValueSHPDifference (%) 045.0000 0.0000% 1/5242.629542.74510.2712% 1/2641.917441.8710-0.1107% 3/5241.432441.4148-0.0424% 1/1341.038441.0268-0.0284% 5/5240.733740.7064-0.0670% 3/2640.430540.43120.0018% 7/5240.214940.1893-0.0635% 2/1340.000039.9730-0.0676% 9/5239.785939.7770-0.0222% 5/2639.615239.5980-0.0434% 11/5239.444939.4330-0.0301% 3/1339.317539.2802-0.0949% 1/439.148139.1380-0.0260% 7/2639.021439.0049-0.0422% 15/5238.895038.8803-0.0378% 4/1338.768838.7630-0.0149% 17/5238.684838.6525-0.0836%............ 1/2 37.893237.8836-0.0253% 3/4 37.235037.2154-0.0527% 1 36.827536.82760.0002%

25. Numerical results - American up-and-out put option prices under the CEV model  Parameters: X = 45, H = 50, r = 4.48%, q = 0.  The “Benchmark Value” shows the numerical results of option values from the Boyle and Tian method with 52,000 time steps. 25 蔡維哲 @ National Taiwan University

26. Numerical results - American up-and-out put option prices under the BS model  Parameters: X = 45, H = 50, r = 4.48%, q = 0.  The “Benchmark Value” shows the numerical results of option values from the Boyle and Tian method with 52,000 time steps. 26 蔡維哲 @ National Taiwan University

27. Numerical results - American up-and-out put option prices under the BS model  Parameters: X = 45, H = 50, r = 4.48%, q = 0.  The root-mean-squared absolute error (RMSE) and the root-mean- squared relative error (RMSRE) are presented in this Table.  the advantage of recalculation using the static hedging approach 27 蔡維哲 @ National Taiwan University

28. Pricing of American lookback options using the static hedging approach under BS model  Similar to Chang, Kang, Kim, and Kim (2007), we focus on American floating-strike lookback put options. 28 蔡維哲 @ National Taiwan University

29. Static hedging of American lookback options  Following Babbs’s suggestion, we can change the PDE into a one-state-variable PDE by changing its numéraire. 29 蔡維哲 @ National Taiwan University

30. Numerical results - American floating strike lookback put option prices (y0 = 1.02) 30 蔡維哲 @ National Taiwan University

31. Numerical results - American floating strike lookback put option prices (y0 = 1.02)  Parameters: S0 = 50, y0 = M0/S0 = 1.02.  The “Benchmark Value” shows the numerical results of option values from the Babb’s method with 52,000 time steps. 31 蔡維哲 @ National Taiwan University

32. Numerical results - American floating strike lookback put option prices (y0 = 1.02)  Parameters: S0 = 50, y0 = M0/S0 = 1.02.  The “Benchmark Value” shows the numerical results of option values from the Babb’s method with 52,000 time steps. 32

33. Conclusions  This paper proposes a static hedging method to the pricing and hedging of the American barrier options.  The static hedging approach for an American barrier option is to formulate by simultaneously matching the boundary conditions of the PDE.  The static hedging approach compared to dynamic hedging also improves hedging performance significantly for an American up-and-out put option under four risk measures suggested by Siven and Poulsen (2009). 33 蔡維哲 @ National Taiwan University

34. Conclusions  We apply the idea of static hedging to price American barrier option under the constant elasticity of variance (CEV) model of Cox (1975) and American floating lookback option under Black- Scholes model.  The numerical results indicate that our static hedging approach is comparable to the trinomial lattice method of Boyle and Tian (1999) and the binomial lattice approach of Babbs (2000). 34 蔡維哲 @ National Taiwan University

35. Conclusions  The recalculation of the American exotic option price in the future is very easy because there is no need to solve the problem again and static hedge approach is flexible to extend to other stochastic processes, e.g. the trending Ornstein-Uhlenbeck process of Lo and Wang (1995) and the deterministic volatility function option valuation model of Dumas, Fleming, and Whaley (1998). 35 蔡維哲 @ National Taiwan University

36. Static Hedging of American Barrier Options and Applications  We thank Farid AitSahlia, Te-Feng Chen, Paul Dawson, Tze Leung Lai, and seminar participants at National Taiwan University for comments and suggestions.  Thank you for listening! 36 蔡維哲 @ National Taiwan University