Exotic Option Pricing: Analysis and Presentation Using Excel
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JOURNAL OF ECONOMICS AND FINANCE EDUCATION ∙ Volume 12 ∙ Number 1 ∙ Summer 2013 Exotic Option Pricing: Analysis and Presentation Using Excel Steve Johnson1 and Robert Stretcher2 ABSTRACT Analysis and presentation of derivatives analysis presents challenges to educators. The nature of financial derivatives is complex and varied, requiring structured modeling techniques. As sophisticated as these techniques may be, ineffective communication of the pricing process can impair the application of the models to investment scenarios. This paper presents a possible framework for analyzing and presenting exotic option pricing, and demonstrates the use of the model through three example analyses: Asian options, down-and-out options, and down-and-in options (barrier options). Introduction Teaching techniques vary greatly in the presentation of option pricing. The different types of options are numerous, and their individual pricing nuances must be taken into account when forming an analytical basis for pricing. A readily available software package in both educational and professional settings is Excel. It is possible in Excel to create a pricing framework that can handle a variety of different option types. It is the purpose of this paper to present one such framework, a binomial model that determines the contribution to the value of the exotic option of each possible price path. Exotic option values have path dependency, unlike the typical American or European options, where the value of the underlying asset at maturity determines the option value regardless of the pricing path during the term of the option. Motivation A clear presentation of exotic options is important in both the business and academic settings. Exotic options are integral to finance topic areas. Both Asian and barrier options have practical uses for investors. Zhang (1995) notes that Asian options are popular in the commodities and currency markets. Many consumers of commodities are concerned about hedging across multiple periods. For these types of consumers, a single spike in the price would have less effect than an upward trend in the price. With an Asian option, the consumer can hedge the average price over a period of time. Also, Asian options may be cheaper, since at inception, the volatility of the average price over the life of the option will be less than the volatility of the price at expiration. Derman and Kani (1996) note that barrier options have a number of uses for investors. First, buying a barrier option allows the investor to eliminate scenarios that seem unlikely. An investor that believes that stock prices will go up could buy a “down-and-out” call option that becomes worthless if the price drops below the barrier price. Since this option is less likely to expire in the money (it could be “knocked out”), it will cost less than a “plain vanilla” call option. Second, there is an application of barrier options to capital structure theory. Brockman and Turtle (2003) argue that barrier options represent a more accurate model of capital structure than plain vanilla options because the firm can be "knocked out" anytime that interest is not paid. Reisz and Perlich (2007) model equity as a down-and-out barrier option as part of a bankruptcy prediction model. ___________________________________ 1Assistant Professor of Finance, Sam Houston State University, Huntsville, TX 77341, [email protected] 2Professor of Finance, Sam Houston State University, Huntsville, TX 77341, [email protected] 7 JOURNAL OF ECONOMICS AND FINANCE EDUCATION ∙ Volume 12 ∙ Number 1 ∙ Summer 2013 A good understanding of the nature of these options can undergird students' understanding of not only options, but also various other similar structures in finance. A clear exposition method of exotic option pricing therefore has value for students of finance. Spreadsheet Construction Since, with exotic options, the price of the underlying asset at different points during the option's term affects the value of the option, it is necessary to model the possible paths and to establish probabilities associated with each path. In order to be effective for a classroom presentation, a limited number of possible paths should be presented; enough to where the variety of paths and end nodes is sufficient to illustrate a range of option values, but few enough that the lesson is not lost in the detail. We begin by constructing a four-period binomial option pricing model for a European call option. In this model, the underlying asset price can change four times. The resulting "tree" of possible outcomes is depicted in Exhibit 1.At each node of the binomial tree, the stock price is displayed above the call price. The spreadsheet is structured according to a binomial valuation model.From basic option theory (Hull 2000), the value of a plain vanilla European call option at any final node is equal to max (Stock price – X, 0) where “Stock price” refers to the price of the stock at that node “X” is the strike price of the option. The stock price follows a binomial pricing model where S_0 = stock price at time zero Rf = risk-free rate, continuously compounded Sigma = the standard deviation of returns, continuously compounded DeltaT = 0.25 u = 1 + percent increase (continuously compounded) in price if the stock price moves up d = 1 + percent increase (continuously compounded) in price if the stock price moves down a = exp(Rf * deltaT), a variable used to calculate the probability of an upward price movement p = the probability of an upward movement in the price The stock price at any given node equals (# of upward movements) (# of downward movements) S_0 * u * d The call option price at t-1, Ct-1, equals the discounted weighted average of the call option prices: (p*Ct| stock price increase +(1-p)* Ct| stock price decrease)*e-Rf*t where: p is the probability of a stock price increase (1-p) is the probability of a stock price decrease. The outcomes can be expressed in terms of the possible combinations of turning points during the option term, as in Exhibit 2. Referring to the specific cells in the exhibit, the inputs (risk free rate, sigma, strike price, and deltat) are placed in cells B8:B12, and the path nodes in cells E1:I27. Columns K-N contain 0,1 representations of the direction the underlying asset value took at each critical point (1 for up, 0 for down), and the stock prices at each point in columns O-S. Applied to a European option, where the value of the underlying stock at expiration determines whether the option is in the money, we see that the path is immaterial; only the value of the stock at expiration matters. With exotic options discussed in this paper, the path does matter, since the value of the option depends on the stock price movement from initiation to expiration. 8 JOURNAL OF ECONOMICS AND FINANCE EDUCATION ∙ Volume 12 ∙ Number 1 ∙ Summer 2013 Exhibit 1.Possible outcomes from binomial model. Column T contains the call value at expiration for each possible path. The total of the 0,1 indicators will have the same total value (for example , 1110 would result in the same ending node as 1101, 1011, and 0111) and therefore, the same call value at expiration. The risk free probabilities associated with each end node (Column U) represent the probability specified for each upward move raised to the number of upward moves in the end node power. For example, if the end node represents three upward moves and one downward move, the risk free probability associated with that path is .537813times (1-.53781)1. Note that all paths leading to the same end node will thus have the same risk free probability. Multiplying the call value at expiration times the cumulative probability times the discount factor (using the risk free rate as the discount rate), the result in column W represents the weighted contribution for each value at expiration. Summed, the result is the option value, in this case $4.99. 9 JOURNAL OF ECONOMICS AND FINANCE EDUCATION ∙ Volume 12 ∙ Number 1 ∙ Summer 2013 Exhibit 2.Initial Spreadsheet. Asian Options Applied to Asian options, the value is determined not by the value of the underlying asset at maturity, but rather by the average price of the underlying asset during some specified period, specified here as the lifetime of the option (although terms within the lifetime of the option are sometimes specified, as well). Thus, the value of an Asian option is path dependent. The average of the stock prices for each of the five nodes in the path is calculated, therefore, and the result placed in column T. Note that although the ending price (column S) is the same for all paths leading to the same ending node, the average price is not the same, since different paths were taken. As a result, the value at expiration (Column U) will have different values within one end node. The discount factors and the probabilities are the same as in exhibit 2, since the same probabilities accompany the same paths. Again, the sum of the contribution column is the intrinsic value of the option (Exhibit 3). The value of the Asian option, based on averages over the term, is less than the value of the call option from exhibit 2 (using the exact same inputs), since the average value does not vary as much as the possible price at expiration. 10 JOURNAL OF ECONOMICS AND FINANCE EDUCATION ∙ Volume 12 ∙ Number 1 ∙ Summer 2013 Exhibit 3.Asian Option Pricing. Down and Out Call Option The down and out call option is a type of barrier option. If the price falls to (or below) the value of the barrier value "H" (cell B13, $46 in this scenario), then the option is "out," and the value of the option is zero, regardless of price movement after it reaches or falls below the barrier.