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Static Hedging and Pricing American Exotic Options

Static Hedging and Pricing American Exotic Options. San-Lin Chung, Pai-Ta Shih*, and Wei-Che Tsai Presenter: Pai-Ta Shih National Taiwan University. Outlines. Introduction Formulation of the static hedging portfolio

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Static Hedging and Pricing American Exotic Options

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  1. Static Hedging and Pricing American Exotic Options San-Lin Chung, Pai-Ta Shih*, and Wei-Che Tsai Presenter: Pai-Ta Shih National Taiwan University

  2. Outlines Introduction Formulation of the static hedging portfolio The hedge performance of static versus dynamic hedging of an American up-and-out put option Efficiency of the static hedge portfolio for pricing American exotic options Conclusions

  3. Introduction Pricing and hedging American-style exotic options is an important yet difficult problem in the finance literature. Lattice methods: Boyle and Lau (1994), Ritchken (1995), Cheuk and Vorst (1996), and Chung and Shih (2007) Analytical approximation formulae: Gao, Huang, and Subrahmanyam (2000), AitSahlia, Imhof, and Lai (2003), and Chang, Kang, Kim, and Kim (2007)

  4. Introduction Extend the seminal works of Derman, Ergener, and Kani (1995) and Carr, Ellis, and Gupta (1998) to several ways: Beyond the Black-Scholes model: Andersen, Andreasen, and Eliezer (2002), Fink (2003), Nalholm and Poulsen (2006a), and Takahashi and Yamazaki (2009) Several replication methods: the optimal static-dynamic hedge method of Ilhan and Sircar (2006) the risk-minimizing method of Siven and Poulsen (2009) the line segments method of Liu (2010)

  5. Introduction Hedging performance or model risk: Toft and Xuan (1998), Nalholm and Poulsen (2006b), and Engelmann, Fengler, Nalholm, and Schwendner (2006). Beyond the barrier options: American options (Chung and Shih (2009)) Asian options (Albrecher, Dhaene, Goovaerts, and Schoutens (2005)) Installment options (Davis, Schachermayer, and Tompkins (2001))

  6. Introduction This paper utilizes the static hedge portfolio (SHP) approach to price and/or hedge American exotic options. At least three advantages: The proposed method is applicable for more general stochastic processes, but the existing numerical or analytical approximation methods may be difficult to extend them to other stochastic processes, e.g. the constant elasticity of variance (CEV) model of Cox (1975).

  7. Introduction The hedge ratios, such as delta and theta, are automatically derived at the same time when the static hedge portfolio is formed. When the stock price and/or time to maturity instantaneously change, the recalculation of the prices and hedge ratios for the American exotic options under the proposed method is quicker than the initial computational time because there is no need to solve the static hedge portfolio again.

  8. Introduction We contribute to the static hedge literature in three ways. We show how to construct static hedge portfolios for American barrier options and floating strike lookback options due to the complexity of boundary conditions of the option. We investigate the hedging performance of the proposed method and compare with that of the dynamic hedge strategy. This article analyzes the efficiency of the proposed method for pricing American barrier options and floating strike lookback options.

  9. Formulation of the static hedging portfolio We demonstrate how to construct the SHP for an American up-and-out put option and the SHP for an American floating strike lookback put option under the Black-Scholes model. for an American up-and-out put option: The options used in the SHP are standard European options traded in the option exchange. for an American floating strike lookback put option: We utilizes hypothetic European options, whose underlying asset is a non-tradable asset, to form the SHP.

  10. The static hedge portfolio for an American up-and-out put

  11. The static hedge portfolio for an American up-and-out put Forming a static hedge portfolio is not a trivial question due to the complexity of boundary conditions. To do so, one has to carefully choose the types of standard European options (call or put) and their strike prices. 11

  12. The static hedge portfolio for an American up-and-out put Two boundary conditions: One is the knock-out boundary and the other one is the early exercise boundary. The first boundary implies that the SHP value must be zero when the stock price equals the barrier. To match the second boundary, we adopt the method of Chung and Shih (2009) by applying the value-matching and smooth-pasting conditions on the early exercise boundary to solve the SHP.

  13. The static hedge portfolio for an American up-and-out put Our static hedge portfolio starts with one unit of the corresponding European option to match the terminal condition of the AUOP. At time t(n-1), three conditions imply that

  14. The static hedge portfolio for an American up-and-out put Figure 1. The static hedge portfolio for an AUOP option

  15. The static hedge portfolio for an American up-and-out put 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:

  16. The static hedge portfolio for an American up-and-out put Why are both call and put options used? 16

  17. The static hedge portfolio for an American floating strike lookback put We consider a floating strike lookback put option with the maturity payoff , where , is the initial time, and T is the option’s maturity date. According to Babbs (2000), the American floating strike lookback put price, satisfies the following partial differential equation under the Black-Scholes model:

  18. The static hedge portfolio for an American floating strike lookback put Using the standard European options to form an SHP for the American floating strike lookback put is difficult, if not impossible, because the early exercise boundary depends on and thus is path-dependent. 18

  19. The static hedge portfolio for an American floating strike lookback put We use the underlying asset as the numeraire and express the price of an American floating strike lookback put option.

  20. The hedge performance of static versus dynamic hedging of an American up-and-out put option The hedge effectiveness of the SHP is of great concern in the literature, e.g. see Toft and Xuan (1998), Fink (2003), and Engelmann, Fengler, Nalholm, and Schwendner (2006). We consider two types of SHPs. The first SHP is formed by exactly following the previous procedure and we call this portfolio is termed “SHP with nonstandard strikes”. The second portfolio utilizes European put options with standard strikes and thus is called “SHP with standard strikes”.

  21. Table 1. The Static Hedge Portfolio (SHP) for an American Up-and-Out Put 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.

  22. Figure 2. The Mismatch Values on the Boundary for the Static Hedge Portfolios 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.

  23. Figure 3. The profit and loss distributions of two SHPs and the dynamic hedging strategy 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).

  24. Table 2. Hedging Performance of the SHPs and the DHPs Parameters: S0 = 100, X = 100, H = 110, r = 4%, q = 0, sigma = 0.2, and T = 1. We adopt four risk measures used by Siven and Poulsen (2009) to evaluate the hedging performance. For example, n=130 means that we dynamically adjust delta-hedged portfolios every 400/52000 time.

  25. Efficiency of the static hedge portfolio for pricing American exotic options In this section, we will evaluate the numerical efficiency of the SHP approach for pricing American barrier options and American lookback options, respectively. In the former case, we consider pricing AUOP options under the CEV model of Cox (1975). In the latter case, we consider valuing American floating strike lookback put options under the BS model to demonstrate that the proposed method is applicable for other types of exotic options beyond barrier options.

  26. Efficiency of the static hedge portfolio for pricing American exotic options Pricing barrier options under the CEV model is still a difficult task in the literature, even for the European-style options. Davydov and Linetsky (2001) successfully derive closed-form solutions for the European barrier option prices under the CEV model. However, to the best of our knowledge, no attempt has been done for American barrier options.

  27. Efficiency of the static hedge portfolio for pricing American exotic options The proposed method (the SHP value) provides an alternative tool for pricing these options.

  28. Figure 4. The Convergence of the SHP Values to the AUOP Price under CEV model 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. We set , and take beta parameter (β=4/3) from Schroder (1989). 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.

  29. Table 3. The Early Exercise Boundary of the AUOP under 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 ( n ) in the SHP method is 52 per year.

  30. Table 4. The Valuation of American Up-and-Out Put Options 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.

  31. Table 4. The Valuation of American Up-and-Out Put Options 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.

  32. Table 4. The Valuation of American Up-and-Out Put Options under the CEV 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. Table 4 indicates that the numerical efficiency of our SHP method is comparable to the tree method of Boyle and Tian (1999). One advantage of the SHP method is the recalculation of the American exotic option prices.

  33. Pricing American floating strike lookback put options under the Black-Scholes model The numerical results are based on the parameter setting of Chang, Kang, Kim, and Kim (2007) and are reported in Figure 5 and Table 5. • 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.

  34. Table 5. The Valuation of American Floating Strike Lookback Put Options under the Black-Scholes Model 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.

  35. Table 5. The Valuation of American Floating Strike Lookback Put Options under the Black-Scholes Model 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.

  36. Conclusions - Main contributions • We successfully construct a static hedge portfolio to match the terminal and boundary conditions of American barrier options and lookback options. • We show that while the average profit and loss values of all hedge portfolios are similar, the SHPs are far less risky than the DHPs no matter which risk measure is used. • We conduct detailed efficiency analyses and show that the proposed method is as efficient as the numerical methods for pricing American barrier options under the CEV model and American lookback options under the Black-Scholes model.

  37. Conclusions - The SHP method is attractive • The proposed method is applicable for more general models or option types. • The hedge ratios, such as delta and theta, can be easily computed at the same time when the static hedge portfolio is found. • The recalculation of the prices and hedge ratios for the American exotic options is also easy and quick when the stock price and/or time to maturity are changed.

  38. Static Hedging and Pricing American Exotic Options • 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!

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