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Unformatted text preview: Electronic copy of this paper is available at: Volatility and dividend risk in perpetual American options Miquel Montero Departament de F´ ısica Fonamental, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain. E-mail: [email protected] Abstract. American options are financial instruments that can be exercised at any time before expiration. In this paper we study the problem of pricing this kind of derivatives within a framework in which some of the properties —volatility and dividend policy— of the underlaying stock can change at a random instant of time, but in such a way that we can forecast their final values. Under this assumption we can model actual market conditions because some of the most relevant facts that may potentially affect a firm will entail sharp predictable effects. We will analyse the consequences of this potential risk on perpetual American derivatives, a topic connected with a wide class of recurrent problems in physics: holders of American options must look for the fair price and the optimal exercise strategy at once, a typical question of free absorbing boundaries. We present explicit solutions to the most common contract specifications and derive analytical expressions concerning the mean and higher moments of the exercise time. PACS numbers: 89.65.Gh, 02.50.Ey, 05.40.Jc 1. Introduction Pricing financial derivatives is a main subject in mathematical finance with clear implications in physics. In 1900, five years before Einstein’s classic paper, Bachelier [1] proposed the arithmetic Brownian motion for the dynamical evolution of stock prices with the aim of obtaining a formula for option valuation. Samuelson [2] noticed the structural failure of Bachelier’s market model: it allowed negative values for the stock price, what led to undesired consequences in option prices. For correcting these unwanted features he introduced the geometric Brownian motion. Within his log-normal model, Samuelson obtained the fair price for perpetual options, although he was unable to find a general solution for expiring contracts. The answer to this question must wait until the publication of the works of Black and Scholes [3], and Merton [4]. The celebrated Black-Scholes formula has been broadly used by practitioners since then, mainly due to its unambiguous interpretation and mathematical simplicity. It is well established, however, that this model fails to fit some features of actual derivatives. In particular, there is solid evidence pointing to the necessity of relaxing the Electronic copy of this paper is available at: Volatility and dividend risk in perpetual American options 2 assumption, present in the Black-Scholes model, that a constant volatility parameter drives the stock price. Many models have been developed with the purpose of avoiding this restrictive condition: in Merton [5] volatility was a deterministic function of time, in Cox and Ross [6] was stock-dependent, Hull and White [7] proposed a model where the...
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