Static Hedging of American Barrier Options and Applications San-Lin Chung, Pai-Ta Shih and Wei-Che...

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

OutlinesOutlines

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

Conclusions2蔡維哲 @ National Taiwan University

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)


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.


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.


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).


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).]


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).


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.


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.



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.



For example, at time t(n-1), three conditions imply that


1

1 1 1 1 1

0 , , , , ,

ˆ, , , , , , , , , , ,

En

E En n n n n

P H X r q T t

w P H B r q T t w C H H r q T t

1 1 1

1 1 1 1 1 1 1

, , , , ,

ˆ, , , , , , , , , , ,

En n n

E En n n n n n n

X B P B X r q T t

w P B B r q T t w C B H r q T t

1 1

1 1 1 1 1 1 1

1 , , , , ,

ˆ, , , , , , , , , ,

Ep n n

E En p n n n n c n n

B X r q T t

w B B r q T t w B H r q T t


Figure 1. The static hedge portfolio for an AUOP option



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:


iW ˆiW iB it

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.


Panel A. Static Hedge an American Up-and-Out Put Option using Nonstandard StrikesQuantity of

European Call Strike Quantity of European Put Strike Expiration

(months)Value for S0 =

100

-0.004415 110 0.046494 80.238013 2 -0.002381

-0.011848 110 0.047374 80.810245 4 -0.015407

-0.026374 110 0.048418 81.621783 6 -0.057712

-0.057201 110 0.052649 82.799137 8 -0.182052

-0.125989 110 0.007167 84.558963 10 -0.588633

-0.109803 110 0.166623 88.061471 12 -0.276521

1 100 12 6.003998

Net 4.881292

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.


0S0S

Panel B. Static Hedge an American Up-and-Out Put Option using Standard StrikesQuantity of

European Call Strike Quantity of European Put Strike Expiration

(months)Value for S0 =

100

-0.004400 110 0.042348 80 2 -0.002426

-0.011806 110 0.045227 80 4 -0.016477

-0.026556 110 0.032733 80 6 -0.066060

-0.057673 110 0.215716 80 8 -0.132589

-0.125408 110 -0.267288 80 10 -0.749253

-0.109651 110 0.315565 85 12 -0.158067

1 100 12 6.003998

Net 4.879124

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.


Hedging Performance - Mismatch Values on the Barrier and the EEB using Standard Strikes


The accurate early exercise boundary is calculated from the Black-Scholes model of Ritchken (1995) with 52,000 time steps per year.


Hedging Performance - The profit-and-loss distribution (6-SHP)


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).


Numerical results - Hedging performance of SHP and DHP under the BS model


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.


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:


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).


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.


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.


T-t Benchmark Value SHP Difference (%)0 45.0000 45.0000 0.0000%

1/52 42.6295 42.7451 0.2712%1/26 41.9174 41.8710 -0.1107%3/52 41.4324 41.4148 -0.0424%1/13 41.0384 41.0268 -0.0284%5/52 40.7337 40.7064 -0.0670%3/26 40.4305 40.4312 0.0018%7/52 40.2149 40.1893 -0.0635%2/13 40.0000 39.9730 -0.0676%9/52 39.7859 39.7770 -0.0222%5/26 39.6152 39.5980 -0.0434%

11/52 39.4449 39.4330 -0.0301%3/13 39.3175 39.2802 -0.0949%1/4 39.1481 39.1380 -0.0260%

7/26 39.0214 39.0049 -0.0422%15/52 38.8950 38.8803 -0.0378%4/13 38.7688 38.7630 -0.0149%

17/52 38.6848 38.6525 -0.0836%

. . . .

. . . .

. . . .1/2 37.8932 37.8836 -0.0253%3/4 37.2350 37.2154 -0.0527%1 36.8275 36.8276 0.0002%

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.


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.


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


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.


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.


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.



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.

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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).


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).

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蔡維哲 @ National Taiwan University

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).

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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!

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