### Asymptotic Approximations for Asian, European, and American Options with Discrete Averaging or Discrete Dividend/Coupon Payments

Sam Howison
2012 SIAM Journal on Financial Mathematics
We develop approximations to the pricing of options on an asset which makes discrete dividend payments, focusing on the case of frequent payments. The principal mathematical tool is the method of multiple time scales, allied to matched asymptotic expansions. We first analyse European style options, deriving the continuouslypaid dividend equation from the relevant discrete problem, and we analyse the range accrual note to compute the relevant 'continuity correction'. We also carry out the same
more » ... arry out the same analysis for Asian options with discrete averaging. We then give a detailed description of the intricate exercise policies that arise for American put (and, to a lesser extent, call) options when dividends are paid discretely, for the cases of proportionate and fixed-amount dividends. * Mathematical Institute, Oxford University, 24-29 St Giles, Oxford OX1 3LB, UK & Oxford-Man Institute of Quantitative Finance. I am grateful for helpful conversations with Gunter Meyer and Christoph Reisinger. The Black-Scholes model for an American put option has been widely studied as a canonical early-exercise problem. The majority of these studies have assumed a continuously paid constant dividend yield (which is taken equal to zero in some cases). Apart from existence/uniqueness studies, attention has been paid to the properties of the early exercise boundary such as its convexity [4] , and in particular its behaviour at times close to expiry, where approaches divide between those that use the Green's function to transform the problem into an integral equation (see, for example, [2, 9, 10] ) and those that exploit matched asymptotic expansion methods directly on the underlying partial differential equation (for example, [8]). Less attention has been paid to cases in which the contract has 'discrete' features. The relationship between a Bermudan option with frequent exercise opportunities and its continuously-exercisable equivalent is discussed in [6]; in this paper, we consider the behaviour of a put option on an asset which pays a large number of discrete dividends, which we refer to as the discrete version of the equivalent option with continuous dividend payments (we return to the precise definition of these terms below). Rather than deriving a continuity correction, as in [1, 5] for discretely and continuously activated barrier options and [6] for Bermudan and continuously-exercisable American ones, we focus on the structure of the exercise region between dividend dates, which is quite intricate. This has been recognised for some time: the current study was motivated by Figure 5 -37 of the early text [3] , and by the numerical calculations of [11] which are illustrated with similar figures. A sketch of the exercise boundary for a put option is given in Figure 1 . The exercise region for the continuous option is to the left of the dashed line, which is the optimal exercise boundary for that contract. However, the exercise region for the discrete option on an asset paying proportionate dividends consists of many disjoint portions, one for each interval between dividend payments (plus final and initial ones), and its exercise boundary, shown solid, has correspondingly many components; each of these, save only the final one, extends to S = 0 as time approaches a dividend date from below, and then the next component starts from a finite value of S. Furthermore, the exercise boundary has very nearly constant speed for an appreciable fraction of the interval between payment dates. This is the structure that we seek to explain; and having done so, we repeat the analysis for call options, and for both puts and calls on an asset paying a fixed dividend.