A hybrid approach to valuing American barrier and Parisian options

A hybrid approach to valuing American barrier and Parisian options M. Gustafson & G. Jetley Analysis Group, USA Abstract Simulation is a powerful tool for pricing path-dependent options. However, the possibility of early exercise has limited the use of such methods for valuing American options. Recently, a few methodologies have been presented for pricing American options, such as the least-squares-monte-carlo approach proposed by Longstaff and Schwartz [16]. In this paper, we propose a hybrid methodology combining simulation and the Cox-Ross-Rubinstein [10] binomial method to value American barrier and Parisian options. Our methodology uses the insight that once the barrier condition is met, the exercise strategy of an up-and-in or down-and-in barrier option is identical to that of an option without the restriction. This insight allows us to use the binomial method to determine the optimal exercise strategy. The simulation then incorporates both the optimal exercise strategy and the barrier condition to value the option. 1 Introduction Over the last decade a number of simulation-based methodologies have been proposed to value American options. In fact, most, if not all, of these approaches deal with Bermudian options, i.e., options that permit early exercise at prespecified discrete points in time prior to maturity. Three types of methodologies have been suggested to determine the optimal exercise boundary, i.e., the value of the underlying asset at which early exercise is optimal at each permitted early exercise date. The first set of methodologies combines forward moving simulation with backward-moving dynamic programming techniques. Examples of such

110 Computational Finance and its Applications methodologies include, Tilley [21], Grant, Vora and Weeks [13], Broadie and Glasserman [6] and Broadie, Glasserman and Jain [7]. The second set of methodologies combines regression and simulation techniques. In these methodologies, the early exercise decision for all early exercise dates is based on a comparison of the intrinsic value of the option to the expected value that is likely to be realized from holding the option. Regression is used to estimate the conditional expectation of continuing to hold the option for each early exercise date. Algorithms suggested by Carriere [9], Tsitsikilis and Roy [22] and Longstaff and Schwartz [16] are examples of valuation techniques that combine regression and simulation. The third set of methodologies treats the early exercise option as an additional parameter of the derivative security, and then employs simulation to maximize payoff to the option holder. Examples include the methodologies proposed by Fu and Hu [12] and Fu, Laprise, Madan, Su and Wu [11] use the parameterization technique. In this section we provide brief descriptions of the methodologies suggested by Tilley [21], Broadie and Glasserman [6] and Longstaff and Schwartz [16] (we focus on the first two types of algorithms, as our methodology is broadly similar to the dynamic-programming based methods and the regression-based methods to value options). For more detailed summaries of other simulation-based valuation methods see Boyle, Broadie and Glasserman [4], Fu, Laprise, Madan, Su and Wu [11] and Tavella [20]. Tilley [21] was one of the first to propose a simulation-based algorithm to value American options. Tilley proposed a multi-stage algorithm that involves, for each date on which exercise is permitted: sorting the simulated asset values in descending order, grouping the ordered values into bundles, and then determining the tentative optimal stopping rule for each path in each bundle by comparing the immediate exercise value to the bundle-specific average asset value for the subsequent time period. The bundle-specific average asset value is equal to the simple average of asset values in the time period immediately following the exercise date across all paths in the bundle. Then, based on a review of the tentative stopping rule, an optimal or final stopping rule is determined. Barraquand and Martineau [1] modified Tilley s methodology to make it applicable to higher-dimension securities. Grant, Vora and Weeks [13] proposed a methodology that also used simulation along with a sequential dynamic programming algorithm to value path-dependent American options. The key steps in their algorithm are: identifying a range of possible critical prices (i.e., the asset price below (above) which early exercise is optimal for a put (call)) for a given early exercise date; using simulation to compute the expected value of the option at the date subsequent to the early exercise date for all prices within the critical range; identifying the critical value that maximizes the expected value of the option at the next date (for a given early exercise date, no methodology is suggested for identifying the critical range of values and the difference between two successive values within the range). Grant, Vora and Weeks [13] recommend that for practical purposes the critical value be estimated for subset of all possible early

Computational Finance and its Applications 111 exercise dates, and then linear interpolation be used to estimate the critical prices for other early exercise dates (the authors recommend that the number of early exercise dates for which the critical price is computed using simulation should be the smallest number that ensures the desired level of accuracy). Broadie and Glasserman [6] suggested a non-recombining simulated trees approach to value American options. Unlike the standard Monte Carlo approach for pricing European options in which, sample paths are generated using simulation, the Broadie and Glasserman method uses random trees. Starting from the spot price, a random tree is constructed by using simulation to generate a pre-specified number of branches from each node prior to expiration. After constructing the random tree, the approach suggests using two separate dynamic programming algorithms to compute a high and a low estimate of the option value. The high estimate is based on the assumption that the early exercise decision is made after taking into account all prices of the underlying security in the subsequent time period that originate from that node. For purposes of computing the low estimate, the early exercise decision is based on a subset of all security prices in the subsequent time period that originate from that node. The high and low estimates bracket the true value of the option. Broadie, Glasserman and Jain [7] refined the Broadie and Glasserman [6] algorithm by suggesting ways to improve the efficiency of the high and low estimates by using variance reduction techniques, such as antithetic branching, and by reducing the number of nodes at which branching is required. The Longstaff and Schwartz [16] algorithm begins by using simulation to generate possible future price paths of the underlying asset. Then, starting from the maturity date and moving backward, for each possible early exercise date the algorithm compares the intrinsic value of the option for each path for which early exercise is possible (i.e., the option is in the money) to the path-specific continuation value. The path-specific continuation value is the predicted value based on the result of the regression i of the cross section of discounted value of holding the option to maturity on the intrinsic value of the option related to the paths which are in the money at the time when early exercise is possible. Note that, the predicted value is interpreted as the conditional expectation of continuation cash flows. Every time the intrinsic value of the option is greater than the continuation value, an exercise decision is made (Longstaff and Schwartz [16] suggest the use of the first three Laguerre polynomials as the basis function for valuing an American put). The simulation-based methods that have been proposed to date suffer from at least one or both of the following weakness: (1) the methods cannot deal with continuous exercise and thus are suitable for valuing options which limit early exercise to certain pre-specified dates or periods, and (2) the methods may not be suitable for path-dependent options such as Parisian options. For example the transformations required by Tilley [21] and the regression required by Longstaff and Schwartz [16] on each possible date of early exercise render these methodologies impractical for valuing options with unrestricted early exercise. On the other hand the Broadie, Glasserman and Jain [7] methodology cannot effectively deal with path-dependency.

112 Computational Finance and its Applications In this paper, we propose a hybrid methodology combining simulation and the binomial method to value American barrier and Parisian options. The proposed methodology is simple to understand and easy to implement. The remainder of the paper is organized as follows. Section 2 describes our approach. Section 3 describes the numerical algorithm in detail. Section 4 discusses the results of using the algorithm to value: an American barrier option, a European Parisian option, an American Parisian option and an American cumulative Parisian option. A brief discussion of possible refinements to the algorithm is included in Section 5. 2 Overview of the methodology Given any barrier, the sum of the knock-in and knock-out barrier option is equal to the value of plain vanilla option (this holds true for European and American options as well as for knock-out options). V = V + V (1) c ki cu ko cu ki Vc is the value of a call option, V cu is the value of n knock-in-up barrier option ko and V cu is the value of n knock-out-up option. Equation (1) holds for options with any type of a barrier, e.g. fixed, floating, etc. Furthermore, (1) also holds for Parisian and cumulative Parisian options (Haber, Schonbucher and Wilmott [14] refer to these securities as Parasian options). Note that Parisian options can only be exercised after the value of the underlying asset has met a pre-specified barrier condition for a pre-specified continuous amount of time, while a cumulative Parisian option can be exercised after the barrier condition has been met for a pre-specified cumulative length of time. Early exercise and path-dependency are the two key issues that need to be addressed when valuing securities such as American Parisian options. Dealing with early exercise involves identifying the early exercise boundary ( EB ). For a call (put) option, the EB identifies, at each point in time prior to maturity, the lowest (highest) value of the underlying asset at which early exercise is optimal. The EB for plain vanilla options can be easily computed using numerous wellaccepted numerical techniques, for example, the binomial method proposed by Cox, Ross and Rubinstein [10]. Our methodology uses the insight that the EB of a knock-in barrier or Parisian option is identical to that of an option without the barrier/parisian restriction. Thus, for any point in time, the lowest (highest) asset value at which early exercise of a given American call (put) is optimal is the same as for that American option with an knock-in Parisian restriction. The only difference is that the Parisian option holder is precluded from exercising until the restriction is met. As the path dependency of a Parisian knock-in option can be ignored while estimating its EB; standard American option valuation techniques can be used to estimate its EB. Once the EB for the knock-in Parisian option has been

Computational Finance and its Applications 113 estimated it can be very simply incorporated into a standard Monte Carlo simulation to determine the value of the option. The different steps of the algorithm can be easily understood with the help of a simple example that involves valuing a Parisian down-and-in American put option. The characteristics of the option are: Spot price = S o ; Strike price = K; Restriction = the option can only be exercised after the value of the stock has remained below H for two days continuously; and Maturity = 3 days Figure 1: The bigger nodes are the ones at which exercise is optimal. S e 1 and e S 2 represent the maximum price at which early exercise is optimal at t 1 and t 2. Figure 1 represents a very simple binomial model. The exhibit shows that if the e stock price is less than or equal to S 1 at the end of day 1 the put option should be exercised. Similarly, early exercise is optimal if the stock price is less than or e equal to S 2 at the end of day 2. After the EB has been estimated it can be treated as an exogenous barrier and thus can be easily incorporated into a simulation to compute the value of the option. Exhibit 2 illustrates a very simple simulation. m 1 through m 4 are the simulated paths of the stock price from t 0 through t 3. The dashed horizontal line represents the value below which the stock price has to remain for at least two days continuously before the option can be exercised. The dashed grey line that slopes upwards is the EB. The small grey squares represent the highest price for a given day at which the option should be exercised. The large marker on m 4 indicates early exercise. Note that even though, for m 3, the stock price at the end of day 2 is below the EB, the option cannot be exercised because the restriction has not been met.

114 Computational Finance and its Applications Figure 2: Simulation of the stock price path. The methodology described above cannot be used to value a knock-out Parisian option (In instances where the EB lies completely above (below) the barrier restriction, our methodology can be used to price Parisian American knock-out put (call) options). The knock-out restriction effectively reduces the expected time to maturity of the option and thus impacts its EB. All else being equal, a knock-out call (put) option will be exercised at a stock price that is lower (higher) than the price at which a plain vanilla American call (put) will be exercised. However, equation (1) can be used to value the knock-out as the difference between the value of the American option and the same option with a knock-in restriction. In addition to being simpler than the explicit finite difference solution of the partial differential equation formulation of an American Parisian and an American cumulative Parisian option proposed by Haber, Schonbucher and Wilmott [14], our methodology does not impose restrictions on the delta of the option. Restrictions imposed include that the delta of the option is continuous across all values of the underlying stock (this restriction is acknowledged by Haber, Schonbucher and Wilmott [14]), and that the delta at any two adjacent points along the time dimension is the same (this is an implicit assumption underlying the explicit finite difference method. For more details see Hull [15]). Note that the delta of an American Parisian option is discontinuous when the price of the underlying asset is close to meeting the restriction (this holds true for any type of barrier option). 3 Numerical algorithm di In this section we describe the algorithm in detail. Assume that V is the value of an American down-and-in Parisian put option ( ADIPP ). The strike price is

Computational Finance and its Applications 115 K, time to maturity is T, the price barrier is H, the spot price of the stock is S 0, and the option will be knock-in after it has remained below H for length of time τ. The binomial method (for ease of exposition we have not assumed any dividends, however, incorporating dividends, especially continuous dividends, is very easy) is used to compute the value of the plain vanilla put and, more importantly, S, the highest stock price at each time-step t such that * t V (S, t) = K - S t V (S, t) is the value of the plain vanilla put at time-step t. Given that the stock price has to remain below H for τ, implies that the S has to remain below H for α consecutive time-steps. α = τ / t t is the time elapsed between any two consecutive time-steps. Thus the number of time-steps used in the simulation is n (n = T / t ). Note that the same t and n should be used for running the simulation and constructing the binomial lattice. The value of the ADIPP at t=0 along each simulated path m is equal to di rnλt V 0 m = MAX[0, e (K-S nm )] if S tm S * t t = 1 to n and c Tm =α r( n t) Λt = e (K-S tm )] if S tm < S and c tm =α = 0 otherwise * t After the first time the stock price satisfies the barrier restriction at time-step n, c tm is set to α t n. Note that c tm is the value of the counter at time t. V di paths. is then equal to 1/p p di V0m m= 1 where p is equal to the number of simulated 4 Results All results are for a down-and-in put with a strike price of $100, a barrier of $80, and a one-year maturity. The risk-free rate is assumed to be 10%. Table 1

116 Computational Finance and its Applications presents the calculated values for American barrier, European Parisian, American Parisian and American cumulative Parisian options. For each type of option, values were calculated with volatilities of 15% and 30%, the length of time required under barrier varying from 0.05 years to 0.15 years and the spot price varying from $98 to $102. Table 1: Sensitivity of option values to spot price, time under barrier and volatility. American Barrier European Parisian American Parisian American Cumulative Parisian Spot Price Fraction of year under barrier Fraction of year under barrier Fraction of year under barrier Volatility = 15% 0.05 0.10 0.15 0.05 0.10 0.15 0.05 0.10 0.15 98 1.251 0.666 0.500 0.385 0.772 0.558 0.420 0.896 0.687 0.557 99 1.037 0.570 0.427 0.331 0.656 0.474 0.347 0.725 0.570 0.460 100 0.897 0.463 0.332 0.258 0.531 0.387 0.292 0.607 0.457 0.385 101 0.738 0.395 0.279 0.208 0.458 0.311 0.235 0.523 0.387 0.325 102 0.604 0.336 0.225 0.167 0.375 0.256 0.189 0.414 0.323 0.247 Volatility = 30% 0.05 0.10 0.15 0.05 0.10 0.15 0.05 0.10 0.15 98 8.331 6.183 5.504 4.913 7.159 6.235 5.501 7.546 6.886 6.357 99 7.920 5.924 5.143 4.533 6.713 5.875 5.104 7.155 6.540 5.995 100 7.498 5.516 4.842 4.234 6.328 5.489 4.786 6.695 6.163 5.618 101 7.090 5.190 4.566 4.026 6.049 5.176 4.504 6.357 5.749 5.249 102 8.773 4.869 4.305 3.784 5.598 4.918 4.199 6.033 5.428 4.980 Number of time-steps is 300 and 95,000 runs are used in the simulation. As expected, the values of all the options decline as the spot price increases. The results also show the impact of the restriction that the stock price be below the barrier for some length of time. For example, given a spot price of $100, the value of an American barrier option is $0.897. As the fraction of year to be spent under the barrier increases to 0.05 the value drops by over 40% to $0.531, and it continues to fall to $0.292 as the length of time under the barrier increases to 0.15 years. Volatility affects knock-in barrier and Parisian options in two ways. Not only does an increase in volatility increase the expected payoff of the vanilla option, and therefore the value of the Parisian option (recall that the value of a knock-in option is equal to the value of the vanilla option minus the knock-out option. (see equation 1)), but it also increases the likelihood that the barrier restrictions will be satisfied. Table 1 shows that the value of all options increase substantially. For example the value of the American barrier option (spot $100) increases from $0.897 to $7.498 as the volatility increases from 15% to 30%. In

Computational Finance and its Applications 117 relative terms, the increase in the value of the other options is more pronounced. For example the value of the Parisian American ($100 spot and τ =0.05) increases by over 10 times from $0.531 to $5.489 as the volatility increases form 15% to 30% (the impact of volatility on the value of the options is also driven by the difference between the spot price (S 0 ) and the barrier price (H). The sensitivity of the value of knock-in barrier / Parisian options is positively related to the difference between S and H). The difference between the American Parisian and the American cumulative Parisian option becomes evident as the length of the barrier increases. When the time to be spent under the barrier is very small, the prices of the two options are about the same. However, as the length of time to be spent under the barrier increases the prices begin to diverge. This trend is illustrated in Table 2. Note that as the length of time under the barrier increases, the difference as a percentage of the American Parisian option value increases. Table 2: Relative value of Parisian and cumulative Parisian options. Fraction of year under barrier Percentage Difference Difference 0.01 0.022 2.60% 0.05 0.077 14.41% 0.10 0.069 17.87% 0.15 0.093 31.84% The difference is based on the results in Table 1 (volatility 15% and spot price $100). Number of time-steps is 300 and 95,000 runs are used in the simulation. 5 Refinements The accuracy of the results is dependent on the EB. Note that the actual EB is likely to be a smooth curve (EB for a put is convex (i.e., the price at which exercise is optimal increases with time), while the EB for a call is likely to be concave). Given the discrete structure of the binomial model, the estimated EB based on such a binomial model is not likely to be smooth and will lie below (for a call option the estimated EB will lie above the actual EB) the actual EB (at any given time-step the actual EB is likely to lie between two nodes. However, in the binomial model the early exercise decision will be made only for the lower of the two nodes). This in turn indicates that prices based on our methodology will have a downward bias. The smoothness and accuracy of the estimated EB can be increased by either increasing the number of time-steps that are used to model the stock price over any given length of time or increasing the number of branches at each node (one way to do this would be to use a trinomial tree).

118 Computational Finance and its Applications Furthermore, other numerical techniques such as one of the finite difference methods may be used instead of the binomial model to estimate the EB. The finite difference methods are more flexible and incorporate time-varying volatility and drift. References [1] Barraquand, J. & Martineau, D., Numerical Valuation of High Dimensional Multivariate American Securities. Journal of Financial and Quantitative Analysis, 30, pp. 383-405, 1995. [2] Brandimarte, Paolo, Option Valuation by Monte Carlo Simulation (Chapter 7). Numerical Methods in Finance, John Wiley: New York, pp. 315-346, 2002. [3] Boyle, P.P., Options: A Monte Carlo Approach. Journal of Financial Economics, 4, pp. 323-338, 1977. [4] Boyle, P.P., Broadie, M. & Glasserman, P., Monte Carlo methods for security pricing. Journal of Economics Dynamics and Control, 21, pp. 1267-1321, 1997. [5] Boyle P.P. & Lau, H.S., Bumping Up Against the Barrier with the Binomial Method. Journal of Derivatives, 1, pp. 6-14, 1994. [6] Broadie, M. & Glasserman, P., Estimating Security Price Derivatives Using Simulation. Journal of Economic Dynamics and Control, Vol. 21, pp. 1323-1352, 1997. [7] Broadie, M., Glasserman, P. & Jain, G., Enhanced Monte Carlo Estimates for American Option Prices. Journal of Derivatives, 5, pp. 25-44, 1997. [8] Cheuk, T.H.F. & Vorst, T.C.F., Complex Barrier Options. Journal of Derivatives, Fall, pp. 8-22, 1996. [9] Carriere, J.F., Valuation of the early-exercise price for options using simulations and nonparameteric regressions. Insurance: Mathematics and Economics, 19, pp. 19-30, 1996. [10] Cox J., Ross, S. & Rubinstein, M., Option Pricing: A Simplified Approach. Journal of Financial Economics, Vol. 7, No. 3, pp. 229-263, 1979. [11] Fu, M.C., Laprise, S.B., Madan, D.B., Su, Y. & Wu, R., Pricing American Options: A Comparison of Monte Carlo Simulation Approaches, Working Paper, 2000. [12] Fu, M.C. & Hu, J.Q., Sensitivity analysis for Monte Carlo simulation of option pricing. Probability in the Engineering and Information Sciences, 9, pp. 417-446, 1995. [13] Grant, D., Vora, G. & Weeks, D., Path-Dependent Options: Extending the Monte Carlo Simulations Approach. Management Science, Vol. 43, 11, pp. 1589-1602, 1997. [14] Haber, R.J., Schonbucher P.J. & Wilmott, P., Pricing Parisian Options. Journal of Derivatives, Spring, pp. 71-79, 1999.

Computational Finance and its Applications 119 [15] Hull, J.C., Numerical Procedures (Chapter 16). Options, Futures, and Other Derivative Securities, Prentice Hall: Upper Saddle River, pp. 388-434, 2000. [16] Longstaff, F.A. & Schwartz, E.S., Valuing American Options by Simulation: A Simple Least-Squares Approach. The Review of Financial Studies, Vol. 14, No. 1, pp. 113-147, 2001. [17] Moraux, F., On cumulative Parisian Options. Working paper, 2002. [18] Randall, C. & Tavella, D.A., Numerical Examples (Chapter 6). Pricing Financial Instruments: The Finite Difference Method, John Wiley & Sons: New York, pp. 183-230, 2000. [19] Ritchken, P., On Pricing Barrier Options. Journal of Derivatives, 5, pp. 19-28, 1995. [20] Tavella, D.A., Simulation for Early Exercise (Chapter 6). Quantitative Methods in Derivates Pricing, John Wiley: New York, pp. 177-206, 2002. [21] Tilley J.A., Valuing American Options In A Path Simulation Model. Transactions of Society of Actuaries, 45, pp. 499-520, 1993. [22] Tsitsikilis, J. & Roy, V., Regression Methods for Pricing Complex America-Style Options. Working paper, Stanford University, 2000.