STATIC REPLICATION

Static Replication

Static Replication

Introduction

Pricing and hedging stock saving accounts is an important yet difficult problem in the finance research. In the past three decades, many researchers have developed methods to overcome this problem (e.g. see Bunch and Johnson, 2000 for a good review). Many successful methods, including numerical methods and analytical approximation formulae, are able to Price stock saving accounts efficiently. For example, the recent advanced numerical methods include the binomial methods of (Broadie and Detemple, 1996), (Figlewski and Gao, 1999), (Heston and Zhou, 2000) and (Chung and Shih, 2007), the Monte Carlo simulation methods of (Broadie and Glasserman, 1997) and (Longstaff and Schwartz, 2001), and the quadrature integral methods of Sullivan (2000).

In the literature, the static replication method is formulated in two different ways. The first approach, proposed by (Bowie and Carr, 1994) and (Carr et al., 1998), etc., is to construct static positions in a continuum of standard European options of all strikes, with the maturity date T matching that of the exotic option (e.g. a barrier option). The second approach, developed by Derman et al. (1995), uses a standard European option to match the boundary at maturity of the exotic option and a continuum of standard European options of maturities from time 0 to time T to match boundary before maturity of the exotic option, with the strike equaling the boundary before maturity (e.g. the barrier level of a knock-out option). Our static replication methods of stock saving accounts take advantage of both approaches by using standard European options with multiple strikes and multiple maturities. The reason for using standard options with multiple strikes and multiple maturities is because the early exercise boundary of the stock saving account is time variant.

2. Static hedging and pricing of American put options under the Black-Scholes model

As an illustration, we discuss the static hedging and pricing for American put options in the rest of the paper. The extensions to American call options are straightforward and thus are omitted.

Under the assumptions of Black-Scholes model, it is well known that the Price F of any American or European option written on a stock satisfies the following partial differential equation (PDE):(1)

where S is the stock Price.

At time tn-1 when the stock Price equals the critical exercise price Bn-1, value-matching and smooth-pasting conditions imply that(2)(3)

where PE(S, X, s, r, q, t) and ?E(S, X, s, r, q, t) are the Black-Scholes formulae of the price and Delta, respectively, for the European put option and are given by(4)PE(S,X,s,r,q,t)=Xe-rtN(-d2)-Se-qtN(-d1),(5)?E(S,X,s,r,q,t)=-e-qtN(-d1),

where t is the time to maturity, N(·) is the cumulative distribution function of the standard normal distribution and

Substituting the Black-Scholes formula of Delta (i.e., Eq. (5)) into Eq. (3) yields:(6)

Substituting Eq. (6) into Eq. (2) leads to a nonlinear equation of Bn-1 which can be solved numerically using the Newton-Raphson method. Then we have wn-1 by substituting Bn-1 into Eq. (6). Using similar procedures, we work backward to determine the number of units of the standard European option, wi, and its strike price, Bi, at ...