No ANALYTIC AMERICAN OPTION PRICING AND APPLICATIONS. By A. Sbuelz. July 2003 ISSN

Transcription

1 No ANALYTIC AMERICAN OPTION PRICING AND APPLICATIONS By A. Sbuelz July 23 ISSN

2 Analytic American Option Pricing and Applications Alessandro Sbuelz First Version: June 3, 23 This Version: July 13, 23 Abstract I use a convenient value breakdown in order to obtain analytic solutions for finitematurity American option prices. Such a barrier-option-based breakdown yields an analytic lower bound for the American option price, which is as price-tight as the Barone-Adesi and Whaley (1987) analytic value proxy for short and medium maturities and exhibits good convergence to the Merton (1973) perpetual option price for large maturities. JEL-Classification: G12, G13. Keywords: American Options, Barrier Options. Alessandro Sbuelz is affiliated with the Department of Finance, Tilburg University, a.sbuelz@tilburguniversity.nl. The usual disclaimer applies.

3 Contents 1 Introduction 1 2 The American put 1 3 Conclusions 3

4 List of Figures 1 Critical-stock-price function: The put case Analytic American put price lower bound and percentage mispricing (T t = years, 1 years) Analytic American put price lower bound and percentage mispricing (T t = 2 years, 3 years) Analytic American put price lower bound and percentage mispricing (T t = 6 months, 12 months) Analytic American put price lower bound and percentage mispricing (T t = 1 month, 2 months)

5 1 Introduction The continuation value of an American put (call) price is equal to the price of the corresponding down-and-out put claim plus the price of a down-and-in rebate claim that pays off the difference between the difference between the exercise price and the critical stock price (the critical stock price and the exercise price). Both barrier claims are written on the same barrier. This barrier coincides with the critical stock price. However, to use this approach one needs to know the critical stock price, which may be defined in four equivalent ways. 1. It is the value of the stock price at which one is indifferent between exercising and not exercising the put (call). 2. It is the highest (lowest) value of the stock price for which the value of the put (call) is equal to the early exercise payoff (value matching condition, VMC). 3. It is the highest (lowest value) value of the stock price at which the put (call) value is smoothly pasted to the early exercise payoff (smooth pasting condition, SPC). 4. It is the highest (lowest) value of the stock price at which the put value does not depend on time to maturity (VM-SPC). The VMC is trivially satisfied by the aforementioned value breakdown. The VM-SPC comes from jointly considering VMC and SPC. Bunch and Johnson (2) use the VM- SPC in order to obtain an implicit equation for a reasonable proxy of the critical stock price. I use the SPC for the same purpose. This yields analytic American option price lower bounds via the value breakdown. The resulting critical stock price and American option values remain a proxy as, for analytic convenience, the critical stock price is in a first stage counter-factually assumed to be constant in time. The analytic American option price lower bound exhibits the same pricing error magnitude as the popular Barone-Adesi and Whaley (1987) pricing formula does. The lower bound has the advantage of consistent underpricing and of a faster convergence to the Merton (1973) perpetual price. 2 The American put I assume that the underlying stock price follows a Geometric Brownian Motion (GBM) with dividend yield q. The American put will be exercised as soon as the stock price hits the critical stock price S p. Thus, the put price can be expressed as the maximum between 1

6 its early exercise payoff and its continuation value. Since the put continuation value does not offer immediate payoffs, the fair American put value P is { P = max X S, e r dt E Q t (P + dp ) }, where S is the current stock price (at today s time t ), X is the exercise price, r is the risk-free rate, and E Q t is the risk-adjusted expectation operator conditional on the current stock price. Given a finite time to maturity T t (T t ) of the option contract, P must satisfy in the price region (S S p ) the Black-Scholes partial differential equation (BSE) with suitable terminal and boundary conditions: E Q t (dp ) = P r dt, (BSE) P (S + ) =, P (S T ) = max {X S T, }, P (S p ) = X S p, (VMC) P S (S S p) = 1, (SPC) where S T is the stock price at the maturity date. The solution P is given by P down out (S, X, S p (X, T t), T t) +RP down in (S, X, S p (X, T t), T t), S S p (X, T t) P (S, X, T t) = X S, S < S p (X, T t) and the critical stock price S p (X, T t) is given by the SPC equation. The fair values of the barrier contingent claims are given by P down out (S, X, S p, T t) = E Q t E Q t RP down in (S, X, S p, T t) = E Q t ( ) e r(t t) max {X S T, } ( )) (e r(tp t) 1 {Tp T }E Q e r(t Tp) max {X S Tp T, }, ( ) e r(tp t) 1 {Tp T } (X S p ), where T p denotes the first hitting time of the stock price at the lower barrier S p given a current stock price S S p. Under the GBM hypothesis and the first-stage assumption that S p is constant, such values and their Greeks do have closed-form solution. In particular, the SPC equation becomes exact because the deltas are known exactly. 2

7 Figure 1 plots the estimated critical stock price versus the time to maturity. Figure 2 shows the convergence of the analytic American price proxies to the Merton (1973) perpetual price as the maturity grows to infinity. Figures 3, 4, and document the pricing error magnitude for progressively shorter maturities. The benchmark American price is given by a projected successive overrelaxation (PSOR) algorithm price with a grid of 4 underlying price steps and 3 time steps. The PSOR algorithm numerically solves the BSE given the terminal and boundary conditions. 3 Conclusions I propose a barrier-option-based alternative to the analytic approximation to American option prices of Barone-Adesi and Whaley (1987). Such an alternative is a lower bound value that shows similar pricing error magnitude for short and medium maturities but greater accuracy for longer maturities. 3

8 4 38 critical price critical price 4 38 critical price critical price PUT CRITICAL STOCK PRICE PUT CRITICAL STOCK PRICE TIME TO MATURITY (YEARS) TIME TO MATURITY (YEARS) Figure 1: Critical-stock-price function: The put case This figure plots the critical stock price versus the time to maturity. The critical stock price proposed in this paper is labelled and the Barone-Adesi and Whaley (1987) critical stock price is labelled. The option parameters are r =.488, q =, σ =.3, and X = $4. These parameters are taken from Bunch and Johnson (2). 4

9 PUT PRICE ( T t = *12/12 ) PUT PRICE ( T t = 1*12/12 ) Figure 2: Analytic American put price lower bound and percentage mispricing (T t = years, 1 years) The graphs plot the analytic American put price proxy and its percentage mispricing against the Merton (1973) perpetual put price versus the current underlying stock price. The analytic price proxy proposed in this paper is labelled and the Barone-Adesi and Whaley (1987) proxy is labelled. The option parameters are r =.488, q =, σ =.3, and X = $4. These parameters are taken from Bunch and Johnson (2).

10 PUT PRICE ( T t = 2*12/12 ) PUT PRICE ( T t = 3*12/12 ) Figure 3: Analytic American put price lower bound and percentage mispricing (T t = 2 years, 3 years) The graphs plot the analytic American put price proxy and its percentage mispricing against the projected successive overrelaxation (PSOR) algorithm price versus the current underlying stock price. The analytic price proxy proposed in this paper is labelled and the Barone-Adesi and Whaley (1987) proxy is labelled. The PSOR price is implemented with a grid of 4 underlying price steps and 3 time steps. The option parameters are r =.488, q =, σ =.3, and X = $4. These parameters are taken from Bunch and Johnson (2). 6

11 PUT PRICE ( T t = 6/12 ) PUT PRICE ( T t = 1*12/12 ) Figure 4: Analytic American put price lower bound and percentage mispricing (T t = 6 months, 12 months) The graphs plot the analytic American put price proxy and its percentage mispricing against the projected successive overrelaxation (PSOR) algorithm price versus the current underlying stock price. The analytic price proxy proposed in this paper is labelled and the Barone-Adesi and Whaley (1987) proxy is labelled. The PSOR price is implemented with a grid of 4 underlying price steps and 3 time steps. The option parameters are r =.488, q =, σ =.3, and X = $4. These parameters are taken from Bunch and Johnson (2). 7

12 PUT PRICE ( T t = 1/12 ) PUT PRICE ( T t = 2/12 ) Figure : Analytic American put price lower bound and percentage mispricing (T t = 1 month, 2 months) The graphs plot the analytic American put price proxy and its percentage mispricing against the projected successive overrelaxation (PSOR) algorithm price versus the current underlying stock price. The analytic price proxy proposed in this paper is labelled and the Barone-Adesi and Whaley (1987) proxy is labelled. The PSOR price is implemented with a grid of 4 underlying price steps and 3 time steps. The option parameters are r =.488, q =, σ =.3, and X = $4. These parameters are taken from Bunch and Johnson (2). 8

13 References Barone-Adesi, Giovanni, and Robert E Whaley, 1987, Efficient analytic approximation of American option values, Journal of Finance 42, Bunch, David S., and Herb Johnson, 2, The American put option and its critical stock price, Journal of Finance, Merton, Robert C, 1973, The theory of rational option pricing, Bell Journal 4,