Options with Extreme Strikes

Abstract

In this short paper, we study the asymptotics for the price of call options for very large strikes and put options for very small strikes. The stock price is assumed to follow the Black-Scholes models. We analyze European, Asian, American, Parisian and perpetual options and conclude that the tail asymptotics for these option types fall into four scenarios.

Discrete sums of geometric Brownian motions, annuities and Asian options

“…A similar result is obtained for the left tail asymptotics of the time integral of the GBM (Proposition 1(ii) in [42]). The corresponding asymptotics for the left tail of the sum of GBM is however different, as seen from Proposition 24.…”

“…Remark 26. The right tail asymptotics of the discrete sum of GBM ( 133) is similar to the right tail asymptotics of the time integral of the GBM which was studied in [42] in relation to the large strike asymptotics of the out of money Asian call options in the Black-Scholes model. From Proposition 1(i) in [42] one finds…”

“…The case m = 0 is particularly simple, as we have α = 1. For this case the expectation ( 41) is 1, and we get (42) c = 2 σ 2 τ .…”

Subleading Correction to the Asian Options Volatility in the Black–scholes Model

“…Restricting to analytical approaches we mention here the Geman-Yor approach [9,5], the Laguerre polynomial expansion method [3], the PDE expansion method [22,7], and the spectral method [12]. We mention also the large-and small-strike asymptotics of Asian options in the Black-Scholes model obtained in [11] and [24]. The short maturity asymptotics of Asian option prices has been studied using probabilistic methods from Large Deviations theory [2,16,19], assuming that S t follows a one-dimensional diffusion (3) dS t = σ(S t )S t dW t + (r − q)S t dt , under suitable technical conditions on the volatility function σ(•).…”

Option Pricing, Zero Lower Bound, and COVID-19

“…The optimal exercise boundary and the corresponding valuation formula for contracts with nonconvex payoffs, such as American call options with constant and growing caps, are studied in Broadie and Detemple (1995), whereas the analytic valuation formula for American options written on assets that pay continuous dividends is introduced in Allegretto et al (1995), where the pricing is obtained as the continuous limit of the valuation formula for American calls with early exercise at a finite number of points in time. More recently, Zhu (2015) derived asymptotic prices of call options with very large strikes and put options with very small strikes. Gapeev and Al Motairi (2018) obtained closed-form formulas for the pricing of perpetual American random dividend-paying put and call options in a set-up that extends Black-Merton-Scholes model considering full and partial information.…”