The American straddle close to expiry

Abstract

We address the pricing of American straddle options. We use a technique due to Kim (1990) to derive an expression involving integrals for the price of such an option close to expiry. We then evaluate this expression on the dual optimal exercise boundaries to obtain a set of integral equations for the location of these exercise boundaries, and solve these equations close to expiry.

Integral Equation Formulation for Shout options

“…Although a detailed study of multiple-shout options is beyond the scope of this study, we will touch on the free boundary for a two-shout option. As we noted earlier, for a two-shout option the payoff at the free boundary S (2) f (τ) is the present value of the difference between the stock price and the strike price together with an at-themoney one-shout option. Using this payoff (2.12) and (2.13) in the formulae (2.4) and (2.5), respectively, it is possible to arrive at a set of integral equations somewhat similar to those presented above for the one-shout option.…”

“…Using this payoff (2.12) and (2.13) in the formulae (2.4) and (2.5), respectively, it is possible to arrive at a set of integral equations somewhat similar to those presented above for the one-shout option. However, they are rather more complicated with these equations involving S (1) f (τ), which, in principle, is known as the solution to (3.1) and (3.2) for the call and (3.3) and (3.4) for the put as well as S (2) f (τ). As with the one-shout option, we solve these equations close to expiry to find expressions for the location of the free boundary x (2) f (τ) = ln(S (2) f (τ)/E (2) ) in the limit τ → 0.…”

“…However, they are rather more complicated with these equations involving S (1) f (τ), which, in principle, is known as the solution to (3.1) and (3.2) for the call and (3.3) and (3.4) for the put as well as S (2) f (τ). As with the one-shout option, we solve these equations close to expiry to find expressions for the location of the free boundary x (2) f (τ) = ln(S (2) f (τ)/E (2) ) in the limit τ → 0. In doing so, we will use the series we found for x (1) f (τ) earlier and this again is an example of how the pricing of shout options is a recursive problem: to find the free boundary S (n) f (τ) for an n-shout option, we first need to know S (1) f (τ), S (2) f (τ), .…”

“…Returning to our analysis, we assume that x (1) f (τ) has the form (4.1) with the coefficients found earlier and that x (2) f (τ) has the form…”

Pricing American Options under Stochastic Volatility and Stochastic Interest Rates

“…(2002)). Although a high-order short-maturity asymptotic approximation of S(τ ) is available in an analytical form (see Alobaidi and Mallier (2004)), accuracy remains an issue. Chen and Chadam (2007) use short-maturity asymptotics to derive an implicit approximation that appears to be accurate for time-to-maturity less than several months.…”