Path Dependent Options: “Buy at the Low, Sell at the High”

Abstract

IT IS WELL KNOWN that the valuation of European puts and calls with fixed exercise prices is solely dependent on the distribution of the terminal price of the underlying stock. This paper examines the properties of European options with exercise prices that are functions of the realized sample path of the stock. In particular, the commonplace shareholder desire to "buy at the low" and sell at the high" can be satisfied with a combination of a call on the stock with an exercise price equal to mino.T S(r) and a put with exercise price equal to maxo,TS(T) where S is the stock price and T is the term of the option.'Strictly speaking, the creation of these new options in a frictionless context would not expand the investor's opportunity set. However, in a realistic market setting such new options might very well acquire substantial popularity. The appeal would be threefold: (1) the options would guarantee the investor's fantasy of buying at the low and selling at the high, (2) the options would, in some loose intuitive sense, minimize regret, and (3) the options would allow investors with special information on the range (but possibly without special information on the terminal stock price) to directly take advantage of such information.In this paper we analyze the hedging, pricing, and economic properties of these options. Wherever possible we compare and contrast these options with their traditional counterparts. In section two we establish that these options can be hedged and that closed-form valuation equations exist. Particular emphasis here centers on the hedgeability of these options when the stock is at an extremum (i.e., equal to its current maximum or minimum). In section three, by analysis and simulation, we establish the properties of these options and contrast them with those of their traditional counterparts. In particular we examine: (1) the functional dependence of these options with respect to two state variables-stock price and time to expiration, and (2) the pricing of these options relative to the stock and traditional options at the time of inception. Due to the ungainliness of the general pricing relations developed in section two, we found it convenient throughout section three to provide detailed analyses and explicit derivations of the properties of these options for the particularly intuitive case where the logarithm of the adjusted geometric mean return of the stock is zero. We then illustrate by simulation that, qualitatively, the results of our specific example carry over to the general case. We conclude in section four with a general discussion of path-dependent options.' Another example of a path-dependent option that has been examined in the pricing of an American put. See Parkinson [7] for details. 1112The Journal of Finance Throughout the paper we will adhere to the following notation Timing conventions: 0:All options are assumed to be written at time zero, T:The expiration date of all options, Ir:The current date, t: T -T, the time to expiration. Remaining notation: S(,r):Stock price...

Risk in Fixed-Income Hedge Fund Styles

“…The look-back put option gives the owner the right (but not the obligation) to sell an asset at the highest price during the life of the option. Goldman, Sosin, and Gatto [1979] describe the theoretical pricing of a European-style look-back option. The look-back straddle consists of both the look-back call and look-back put.…”

“…At the beginning of each month, we purchase a look-back straddle on the spread. The price of the straddle is based on the theoretical pricing formula in Goldman, Sosin, and Gatto [1979], using the historical volatility of the spread for the previous 21 trading days. At the end of the month, the payoff of the look-back straddle is the difference between the maximum and minimum spreads during the month.…”

Double Lookbacks

“…A standard one-asset lookback call (or put) gives its holder the right to buy (or sell) the underlying stock at its historical minimum (or maximum) price over a certain period. Analytical solutions for standard lookback options have been found by Goldman, Sosin, and Gatto (1979) and Goldman, Sosin, and Shepp (1979). Lookback options are appealing because they offer investors the opportunity (at a price, of course) of buying a stock at its lowest price and selling a stock at its highest price.…”

Early Exercise Regions for Exotic Options

“…Goldman, Sosin, and Gatto [1979] derive valuation formulas for European look-back options. Ritchken, Sankarasubramanian, and Vijh [1989], Hull and White [1993], and Neave and Tumbull [1993] have attempted to design efficient computational algorithms for valuing Asian options.…”