MEDDELANDEN FRÅN SVENSKA HANDELSHÖGSKOLAN SWEDISH SCHOOL OF ECONOMICS AND BUSINESS ADMINISTRATION WORKING PAPERS. Mikael Vikström

Transcription

1 MEDDELANDEN FRÅN SVENSKA HANDELSHÖGSKOLAN SWEDISH SCHOOL OF ECONOMICS AND BUSINESS ADMINISTRATION WORKING PAPERS 447 Mikael Vikström THE PRICING OF AMERICAN PUT OPTIONS ON STOCK WITH DIVIDENDS DECEMBER 2000

2 Key words: Option pricing, American put options, Approximation method, Simulation, Dividend Swedish School of Economics and Business Administration, Mikael Vikström Mikael Vikström Department of Finance and Statistics Swedish School of Economics and Business Administration P.O.Box Vaasa, Finland Distributor: Library Swedish School of Economics and Business Administration P.O.Box Helsinki Finland Phone: , Fax: SHS intressebyrå IB (Oy Casa Security Ab), Helsingfors 2000 ISBN ISSN

3 The Pricing of American Put Options on Stock with Dividends Abstract Pricing American put options on dividend-paying stocks has largely been ignored in the option pricing literature because the problem is mathematically complex and valuation usually resorts to computationally expensive and impractical pricing applications. This paper computed a simulation study, using two different approximation methods for the valuation of American put options on a stock with known discrete dividend payments. This to find out if there were pricing errors and to find out which could be the most usable method for practical users. The option pricing models used in the study was the dividend approximation by Blomeyer (1986) and the one by Barone-Adesi and Whaley (1988). The study showed that the approximation method by Blomeyer worked satisfactory for most situations, but some errors occur for longer times to the dividend payment, for smaller dividends and for in-the-money options. The approximation method by Barone-Adesi and Whaley worked well for inthe-money options and at-the-money options, but had serious pricing errors for out-ofthe-money options. The conclusion of the study is that a combination of the both methods might be preferable to any single model. Keywords: Option pricing, American put options, Approximation method, Simulation, Dividend. Comments by Kenneth Högholm, Johan Knif and Seppo Pynnönen are gratefully acknowledged. Kjell Blomqvist also provided a helping hand. This paper has also benefited from discussions with friends, colleagues and employees at Estlander & Rönnlund Financial Products Ltd.

4 2 1 Introduction The option pricing models has developed enormously since Black and Scholes (1973) and Merton (1973a) presented their valuation equation for European call and put options written on zero and constant proportional dividend yield stocks. Models now exist for pricing European and American options on a variety of underlying commodities ranging from financial assets such as common stocks and bonds to traditional agricultural futures contracts. A complex valuation situation is the one of American options written on commodities with discrete cash payments during the life of the option. Merton (1973a) has shown that an American call option on a non-dividendpaying stock should not be exercised prematurely. Rational early exercise can occur for call options on a dividend-paying stock if annualized dividend yield received over the remaining life of the option exceed the risk-free rate of interest. In this case, the opportunity cost of not receiving the dividend now outweighs the benefit of paying a lower present value for the stock later. Merton (1973b) has shown that American put options, on the other hand, can be rationally exercised prior expiration and therefore American put options are more difficult to value than European puts because at every instant for American puts there is a positive probability of premature exercise. The American put option may be exercised early because interest income can be earned on the exercisable proceeds of the option as soon as the option is exercised. Deferring exercise implicitly means interest is being forgone. The majority of listed put options are traded on stocks paying cash dividends. Geske and Shastri (1985) demonstrated that dividends significantly reduce the probability of early exercise for American puts. This diminishes the difference between American and European put option values, and consequently one might conclude that dividends simplify the valuation problem. Although this conclusion is correct in the sense that errors from using the European formula for American put options would be smaller, dividends complicate the exact valuation of American puts. When the stocks pays a dividend, the American put option holder is in a dilemma. If he continues to hold the put, he forgoes the interest, but if he exercises immediately, he will not profit from the discrete upward jump in the exercisable proceeds of the put when the stock goes ex-dividend.

5 3 While a usable valuation equation exists for the American call option written on a stock with known discrete dividends paid during the option s life (the Compound Option Model by Roll (1977), Geske (1979) and Whaley (1981)). The valuation of the American put option on a stock with discrete dividends remains largely unaddressed. Usually the valuation of such options involves implementing finite difference methods, like the one by Brennan and Schwartz (1978), or numerical methods like the Binomial option pricing model by Cox, Ross and Rubinstein (1979). Also a version of the Compound Option Model by Geske and Johnson (1984) the can be used. The problem with these models is that they are computationally expensive and impractical for real time pricing applications. Therefore it is usual to use approximation methods for the valuation of American put options on a stock with known discrete dividend payments. Blomeyer (1986) presented a fast algorithm that interpolates between known option values that surround the true unknown price. In his study he also simulated different cases and compared them with the Compound Option Model by Geske and Johnson (1984). The approximation gave values that where generally close to the Compound Option Model, with the largest difference for deep in-the-money options. The conclusion was that the American put option researcher, student, or trader, who is willing to sacrifice a small degree of put option valuation model accuracy for a high degree of computational efficiency, may view the approximation as an attractive alternative to the more complex analytic and numerical valuation models. Barone-Adesi and Whaley (1988) criticized Blomeyer s method and argued that his technique could lead to serious mispricing errors for typical parameter ranges. They therefore computed a study where they presented an alternative approximation method and a similar simulation study to the one made by Blomeyer. The result was that the put options slightly in-the-money had the greatest mispricing errors, but the magnitudes of the errors fall within transaction costs bands. Barone-Adesi and Whaley argued that the nonlinear approximation works remarkably well considering that it takes less than 1/1000 th of the time to compute the nonlinear interpolation than it does the finite difference value of these options. A common phenomenon with the pricing of options is that the models and much of the research is made according to American conditions, where dividends are paid quarterly and where the options maturity is perhaps a little shorter. This

6 4 phenomenon could lead to difficulties using the models according to European conditions where for example dividends are paid only once a year. The purpose of this paper is to compute a simulationstudy, using two different approximation methods for the valuation of American put options on a stock with known discrete dividend payments. This to find out if there are pricing errors and to find out which could be the most usable method for practical users. The option pricing models used in the study was the dividend approximation by Blomeyer (1986) and the one by Barone-Adesi and Whaley (1988). As a benchmark option price, the Binomial option-pricing model was used. The simulation parameters used in the study were based on normal market conditions. The study follows the setup by among others Geske and Johnson (1984), Cox and Rubinstein (1985), Blomeyer (1986), Barone-Adesi and Whaley (1987), Barone-Adesi and Whaley (1988), Blomeyer and Johnson (1988), Hull and White (1988), Ju (1998) and Little et al. (2000) were approximation model prices are compared against assumed correct model prices. The reason not to test against market prices here is because the purpose was to evaluate the accuracy and usability of the methods and this is rather difficult with market prices due to the problem of knowing whether your input estimates are wrong or your pricing method is wrong. Most of the input variables are easily accessible market-determined values, but in this study the problem parameters are the dividend and the volatility with the volatility as the most troublesome. The problem with the dividend comes among other things from that German dividends carry a tax credit, which makes the dividend worth more to a taxable German shareholder than to a tax-exempt or foreign shareholder. This raises the question of whether the net dividend, the gross dividend, or something between is relevant for pricing stock transactions. Also, as the focus on this paper is on evaluating pricing methods that can handle dividends, and the market naturally values only one dividend size at each time, the simulation setup were different dividend sizes are used gives a more flexible way of testing. To test the approximation methods against market prices can in a way be seen as a totally different setup that could be addressed in another paper. The simulation was computed on three dividend-paying stocks that were traded at the German market. The study showed that for the stock option parameter ranges considered, the pricing errors were around 1 2 percent, and there were not

7 5 any large differences between the presented approximation methods, although there were variations between the models. The outline of the paper is as follows. In section 2 the different pricing models are very briefly presented. First the assumptions and definitions used in the study are presented and then the approximation methods are presented. Section 3 presents the data and the simulation design and in section 4 the results of the simulation study are presented. The paper ends with a summary and concluding remarks. 2 Models In this section the option-pricing models are presented. The models are here presented very briefly, more deeply presentations are given in the original papers. The approximation methods relies on all of the standard Black and Scholes assumptions, except that, in place of the no dividend assumption, it is assumed that the underlying stock pays a single known dividend during the option s life. The approximation, however, employs the valuation equations for European and American put options on stocks with no dividends. From Black and Scholes, the European put option formula is, where and p (S,t) = Xe -rt N 1 (-d 2 ) - S N 1 (-d 1 ), (1) d 1 = d 2 = ln ln S X S X 2 σ + ( r + ) t 2 σ t 2 σ + ( r ) t 2 σ t (2) = d 1 σ t, (3) where p is the European put price, S is the current stock price, X is the exercise price of the option, r is the riskless rate of interest, σ is the standard deviation of the instantaneous rates of return on the underlying stock, and t is the time to expiration of the option. N 1 is the cumulative univariate normal density function.

8 6 From the Quadratic Approximation Method by MacMillan (1986), the American put-option valuation equation is, ={ p(s,t) + A(S/S*) q where S > S* P(S,t) X - S where S S* (4) where S * A = - (1 N1( d 1 ( S *))) (5) q q = 2 ( ( β 1 ) ( β 1 ) + 4 β / h) 2 (6) h(t) = 1 - e -rt (7) 2r β = (8) 2 σ and S* is the critical asset below above which the American put should be exercised immediately and is the solution to, X - S* = p(s*,t) - ( 1 1 N 1( d ( S *))) S *. (9) q In (9) the formula must be solved through iteration, but Barone-Adesi and Whaley (1987) presented an algorithm that normally finds a good approximation for S* after three or even less iterations (see Appendix 1). 2.1 Blomeyer`s dividend approximation The Blomeyer (1986) approach in this case recognizes that there are upper and lower bounds on the American put values when the underlying stock pays a cash dividend. These bounds are valued directly using the Black and Scholes (1), model and the

9 7 Quadratic Approximation Method (4), by MacMillan. Linear interpolation is used to obtain the American put value. An American put that goes exdividend at expiration will not be exercised prior to expiration if the cash dividend D*, is large enough to satisfy, rt D * = X ( e 1 ). (10) This American put option is equivalent to a European put option, that is, P(S,t;D*,t) = p(s,t;d*,t). (11) An American put option that has just gone exdividend with dividend D* will be at least as valuable as an American put option that goes exdividend at maturity with dividend D*, and we can state, P(S,t;D*,0) P(S,t;D*,t) = p(s,t;d*,t). (12) An American put that goes exdividend prior to expiration, at time t D, will be at least as valuable as an American put that goes exdividend at expiration. Also, an American put that has just gone exdividend will be at least as valuable as an American put that goes exdividend prior to expiration, and we can write, P(S,t;D*,0) P(S,t;D*,t D ) p(s,t;d*,t). (13) The European put value can be calculated from the Black and Scholes model by escrowing the discounted dividend. In a similar manner, the American put value P(S,t;D*,0), can be calculated from the Quadratic Approximation Method. The remaining American put value can be found by linear interpolation as, t t D P( S, t ; D *, t D ) = p ( S, t ; D *, t ) + ( )( P( S, t ; D *, 0 ) p ( S, t ; D *, t )) t (14)

10 8 The American put option value is an increasing function of the dividend size. If the dividend D* required to satisfy (10), is larger than the actual cash dividend, D, we can write, P(S,t;D* t D ) > P(S,t;D,t D ) > P(S,t). (15) Where P(S,t) is the American put value without cash dividends. By interpolation we obtain P(S,t;D,t D ), from, P( S, t ; D, t ) = P ( S, t ) + ( D / D *)( P ( S, t ; D *, t ) P ( S, t )). (16) D D If the cash dividend is large enough such that D D*, (14) can be used instead of (16), resulting in only one interpolation. 2.2 Barone-Adesi and Whaley`s dividend approximation Barone-Adesi and Whaley (1988) addressed that the tradeoff between the interest income and the dividend can be expressed D > X(e rt D - 1). This condition can be used to gather further insight about the put option pricing problem. To do so, define t N as a point in time before which it may be optimal to exercise the put early prior to exdividend, but after which it will not be optimal to exercise until after the dividend has been paid. This point in time is defined by the solution to, D = X(e r (t D - t N ) - 1) (17) and ln( 1 + D / X ) t N = t D -. (18) r The approximation solution to the put option pricing problem described has two partsfirst where early exercise before the dividend is not possible and second where early exercise before the dividend is possible. In both cases, the approach to finding the solution is the same. Known option prices to which the true put option value

11 9 converges are identified and then weighted by the probabilities of their occurrence. The weighted average approximates the true option value. The first case to examine is t N 0. An upper bound for the put option P(S,t;D,t D ) is the value of an American put written on an non-dividend-paying stock (4), where the stock price net of the present value of the escrowed dividend, S # = S - rtd D e, replaces the stock price parameter P(S #,t). A lower price bound for P(S,t;D,t D ) is found by subtracting the early exercise premium from the upper bound, that is, P(S #,t) - ε p (S #,t D ) (19) where the early exercise premium is defined by, ε p (S #,t D ) = P(S #,t D ) - p(s #,t D ). (20) The upper and lower bounds of the true price are now weighted to provide an approximation for the true value. The approximation formula is, # # # P(S,t;D,t D ) = w P ( S, t ) w [ P ( S, t ) ε ( S, t ) ] +. (21) 1 2 p D Here w 1 and w 2 sum to one. To compute this probability, (4) is first used to evaluate the price of an American put with time to expiration (t - t D ). A by-product of the valuation is the computation of the critical stock price S*(t - t D ) below which the American put option holder will exercise at t D. With the critical stock price in hand, the probability of early exercise at t D is the computed as, w 2 = N 1 (-b) (22) where b = ln # [ S / S * ( t t ) ] σ D t 2 + ( r 0, 5 σ ) t D D (23) and w 1 = N 1 (b). (24)

12 10 The weight w 1 is the complement to w 2 or the probability that the put will not be exercised at time at t D. The case where t N is positive is slightly more complex. Here we also use the pricing of an American put option on a non-dividend-paying stock. The price of such an option can be approximated using (4), that is, P(S,t N ). When t N > 0, the approximation for the American put is, # # # P(S,t;D,t D ) = w P ( S, t ) w [ P ( S, t ) ε ( S, t ) ] + + w P ( S, t ) (25) 1 2 p D 3 N where w 1 + w 2 + w 3 = 1. The weight w 3 is the probability that the put will be exercised in the interval 0 to t N, that is, w 3 = N 1 (-a) (26) and a = ln [ S / S * ( t t ) ] N σ 2 + ( r 0, 5 σ ) t t N N. (27) The weights w 1 and w 2 remain as above except that they are now joint probabilities. In order for the put life s to be extended beyond t N, the weights must reflect the probability that the put is not exercised at t N, that is, N 1 (a). The weights are now, w 1 = N 2 (a, b; t / t ) (28) N D and w 2 = N 2 (a, - b; - t / t ) (29) N D where N 2 (a, b; ρ) is the cumulative bivariate normal density function with upper integral limits a and b and correlation coefficient ρ. Note that the weights w 1 + w 2 sum to the probability of no exercise at t N, N 1 (a).

13 11 3 Data and simulation design The simulation parameters used in the study were based on normal market conditions. The study did not compute any statistical based test against market prices because it was difficult to get a clear picture of this kind of complex option pricing problems from market observations. The simulation was computed on three dividend-paying stocks that were traded at the German Xetra and Eurex 1. These stocks were Daimler, Deutsche Bank and Volkswagen. The examined option situations were based on put options that had their expiration day the 20 of June Data comes from Estlander & Rönnlund Ltd and are based on market values of the 22 of October In the study 9 different stock prices were simulated (movements of + / - 20 percent) and for 10 different time periods, in other words the trading day moved closer and closer to the dividend payment. The study also used 5 different dividend sizes. Option pricing has many dimensions and the pricing dynamics was here limited to using simulations for only one strike price for each option 2 ; also the interest rate and volatility were kept constant according to the values of the 22 of October Effects of different volatilities would, to some extent, still be made because the three different stocks had different volatilites. Another feature of option pricing is also that different maturities have the same effects on option prices as different volatilities. The study came up to 135 different situation for each stock or 405 different situations in total. Table 1 shows the used parameters in the study. Table 1 Parameters used in the study Stock Daimler Deutsche Bank Volkswagen Stock price Strike price Dividend Time to dividend days days days Maturity days days days Interest rate Volatility Parameters are based on market values from the 22 of October Eurex was formerly Deutsche Terminebörse and Xetra was introduced 1997 and replaced the IBIS. 2 When the stock price instead was changed with + / - 20 percent there would still be out-of-the-money, at-the-money and in-the-money options.

14 12 The option pricing models used in the study were Blomeyer`s dividend approximation and Barone-Adesi and Whaley`s () dividend approximation. As a benchmark correct option-price, the Binomial option pricing model with 150 steps was used, according to Geske and Johnson (1984) and Cox and Rubinstein (1985) 150 steps has to be used in order for the model to be accurate. In similar studies where different models are compared with each other either the Binomial option pricing model or finite difference methods are used as the benchmark option price. These studies are among others those of Geske and Johnson (1984), Cox and Rubinstein (1985), Barone-Adesi and Whaley (1987), Barone-Adesi and Whaley (1988), Blomeyer and Johnson (1988), Hull and White (1988), Ju (1998) and Little et al. (2000). A comparison of the Binomial option pricing model and the finite difference method was computed by Geske and Shastri (1985), both methods gave similar results, but Geske and Shastri argued that researcher prefer the Binomial option pricing model, because it was pedagogically superior. To be able to compare the differences from the benchmark option price, percentage deviations (PD) were calculated as PD M = P M P P C C *100 (30) where P M was the price for the model and P C was the benchmark price from the Binomial option pricing model. To facilitate the graphical presentation the absolute percentage deviations (APD) were also computed as APD M = PD M. (31) The presentation of the results also uses the mean and the standard deviation of the absolute percentage deviations and the percentage deviations. In the presentation the options were categorazied according to the stock price in relation to the strike price S/X, and at-the-money options were defined as 0.95 S/X 1.05, in-the-money options were defined as S/X < 0.95 and finally out-of-the-money options were defined as S/X > In the next section the result of the simulation study is presented.

15 13 4 Results To illustrate the dynamics of the pricing errors of the different methods, the results of the study are first presented graphically for all the stocks put together. A more detailed graphical presentation of each stock is presented in Appendix 2. After the graphical results, statically measures of deviations for each stock are presented separately. Deviations for the approximation methods, when changing the time and the stock price are presented in Figure 1 3. Blomeyer APD LONGER TIME SHORTER IN-THE-MONEY AT-THE-MONEY OUT-OF-THE -MONEY APD LONGER TIME SHORTER IN-THE-MONEY AT-THE-MONEY OUT-OF-THE-MONEY Figure 1 Deviations for the models, changing the time and price The maturity was between 210 and 80 days, the time to dividend between 182 to 64 days and the stock price was + / - 20 percent from the strike price. Daimler: strike price 90, dividend 1.00, interest rate 3.15% and volatility 14.50%. Deutsche Bank: strike price 70, dividend 2.15, interest rate 3.15% and volatility 15.00%. Volkswagen: strike price 600, dividend 6.00, interest rate 3.15% and volatility 17.50%. The absolute percentage deviations were calculated as, APD M = [(P M P C )/ P C ]*100, where P M was the price for the model and P C was the benchmark price from the Binomial option pricing model. 3 In Figure A1 the deviations are presented separately for each stock.

16 14 For Blomeyer it could be seen that the model had larger deviations for in-the-money options and also for out-of-the options for longer time periods. For shorter time periods the deviations were largest for at-the-money options. Overall the Blomeyer approximation worked satisfactory. Examining the same simulation for the method, it could be seen that the method worked well for in-the-money options, but for out-of-the-money options the errors were obvious. Blomeyer APD SMALL DIVIDEND LARGE IN-THE-MONEY AT-THE-MONEY OUT-OF-THE-MONEY APD SMALL DIVIDEND LARGE IN-THE-MONEY AT-THE-MONEY OUT-OF-THE-MONEY Figure 2 Deviations for the models, changing the dividend and price The dividend in relation to the strike price varied between 0.55 percent to 4.00 percent and the stock price was + / - 20 percent from the strike price. Daimler: strike price 90, time to dividend 122 days, maturity 151 days, interest rate 3.15% and volatility 14.50%. Deutsche Bank: strike price 70, time to dividend 122 days, maturity 145 days, interest rate 3.15% and volatility 15.00%. Volkswagen: strike price 600, time to dividend 122 days, maturity 137 days, interest rate 3.15% and volatility The absolute percentage deviations were calculated as, APD M = [(P M P C )/ P C ]*100, where P M was the price for the model and P C was the benchmark price from the Binomial option pricing model. Deviations for the methods, when changing the dividend size and the stock price are presented in Figure 2 4. Here it could be seen that the Blomeyer model had larger deviations for out-of-the-money options with smaller dividends and also for at-the- 4 In Figure A2 the deviations are presented seprarately for each stock.

17 15 money options with larger dividends. Examining the deviations for the method it could again be seen that the deviations were larger for out-of-the-money options. The largest absolute percentage deviations for the Blomeyer model was 4.62 percent and for the model it was 10.1 percent, both for the stock Daimler. Overall the percentage errors for were much larger than for Blomeyer. The graphical presentation revealed that Blomeyer works quite well for most of the situations. Some errors occur for longer times to the dividend payment, for smaller dividends and for in-the-money options. tended to have problems with the pricing of out-of-the-money options. With the graphical presentation it was quite difficult to compare the methods and therefore additional statically measures were computed. In Table 4 it could be seen that the mean and the standard deviation was less for Blomeyer. In the column All options it could be seen that Blomeyer tended to undervalue the options and that tended to overvalue the options. Table 2 Statically measures of deviations for the approximation methods Daimler Deutsche Bank Volkswagen All options Blomeyer Blomeyer Blomeyer Blomeyer APD Mean Std.dev PD Mean Std.dev The percentage deviations were calculated as, PD M = [(P M P C )/ P C ]*100, where P M was the price for the model and P C was the benchmark price from the Binomial option pricing model. The absolute percentage deviations were calculated as, APD M = PA M. The deviations were based on all the situations simulated in the study. From Table 2, Blomeyer seemed to be the most usable method. A reason for this could be that had quite large errors for out-of-the-money options and therefore it could be interesting to examine mean deviations for different option types. The results were interesting and in Table 3 it could be seen that Blomeyer had smaller errors for out-of-the-money options. For at-the-money options there seemed to be no different and for in-the-money options the result differs. For in-the-money options Blomeyer had smaller errors for Deutsche Bank and had smaller errors for Daimler and Volkswagen.

18 16 Table 3 Mean absolute percentage deviations for different types of options Daimler Deutsche Bank Volkswagen All options Blomeyer Blomeyer Blomeyer Blomeyer In At Out The absolute percentage deviations were calculated as, APD M = [(P M P C )/ P C ]*100, where P M was the price for the model and P C was the benchmark price from the Binomial option pricing model. The deviations were based on all the situations simulated in the study and were here divided into different option types, in-the-money options, at-the-money options and out-of-the-money options. To test for significant differences between the methods a non-parametric method used by among others Sterk (1982) and Sundkvist (2000) was computed next. The Wilcoxon matched-pairs signed-rank test is a preferable method to us to compare the performance of one approximation method relative to another. To implement this test, the difference between the absolute percentage deviations (31) were computed for each observation as follows D = APD Blomeyer, i APD, i i = 1, 2,..., 405. (32) The difference D tests which of the method that performed better. If the Blomeyer method were closer to observed market prices, then D should be predominantly negative. The Wilcoxon test ranks the differences and computes the sum of the negative and positive ranks, it therefore also accounts for the magnitude of the differences. The sum of the negative ranks should exceed the sum of the positive ranks if the Blomeyer method performs better relative to the. Table 4 gives the results of the tests. Table 4 Z-scores for Wilcox signed-rank tests comparing the approximation methods Daimler Deutsche Bank Volkswagen All options Blomeyer 0.14 Blomeyer 6.21* Blomeyer 1.38 Blomeyer 4.38* The method specified by the label to the left provided prices closer to the reference model prices than the method specified by the label at the top of the column. The deviations were based on all the situations simulated in the study. * indicates significance at level. The Wilcoxon test confirmed the results indicated by the mean absolute percentage deviations, that was, the Blomeyer method gave prices that were significantly closer to the assumed true price than those of the method, although there existed

19 17 contradictions. For Daimler the Wilcoxon test showed that performed better than Blomeyer although the Z-scores were insignificant. The results for different option types are presented in Table 5. Table 5 Z-scores for Wilcox signed-rank tests for different types of options In Daimler Deutsche Bank Volkswagen All options Blomeyer Blomeyer 3.22* Blomeyer 5.79* 3.33* Blomeyer 1.52 At Blomeyer 0.05 Blomeyer 0.15 Blomeyer 2.77* Blomeyer 2.11* Out Blomeyer 5.75* Blomeyer 5.09* Blomeyer 5.28* Blomeyer 8.87* The method specified by the label to the left provided prices closer to the reference model prices than the method specified by the label at the top of the column. The deviations were based on all the situations simulated in the study and were here divided into different option types, in-the-money options, at-the-money options and out-of-the-money options. * indicates significance at level. Again the Wilcoxon test confirmed the earlier results, but for Volkswagen and at-themoney options the Blomeyer method performed significantly better than the method. Also, these results indicate that for all the stock and at-the-money options the Blomeyer method seemed to be superior. An explanation for the varying results of in-the-money options and for the better performance of Blomeyer, for Deutsche Bank, could be that the dividend was quite large. The size of the dividend in relation to the strike price for Deutsche Bank varied between 2.14 percent and 4.00 percent. For Daimler it was between 0.55 percent and 1.67 percent and for Volkswagen it was between 0.67 percent and 1.33 percent. When the dividend is large the proceeds of the put when the stock goes exdividend is more valuable to the put holder than the forgone of the interest. The equations (17) and (18) gave the point in time before which it may be optimal to exercise the put early prior to ex-dividend, but after which it will not be optimal to exercise until after the dividend has been paid. This point in time might be used to explain the behavior of the errors for the different models for in-the-money options. Before this point in time the MAPD for in-the-money options were 0.90 percent for Blomeyer and 0.27 percent for. After this point in time the MAPD for in-themoney options were 0.19 percent for Blomeyer and 0.33 percent for.

20 18 Looking at the presented results it should be noted that they are percentage errors and therefore and small error for an in-the-money option could be quite large in absolute terms. The other way around also holds, a large percentage error for an outof-the money option might be due to a quite small absolute error. One conclusion of the study might be to use the method by Blomeyer if one model for all situations is used. Although is should again be remembered that the pricing errors of in-the-money options might be quite large in absolute terms and therefore a combination of the both methods might be preferable. Such a combination could be to use Blomeyer for out-of-the-money options, for at-the-money options and depending on the point in time after which it will not be optimal to exercise until after the dividend has been paid, either Blomeyer or for in-themoney options. could be used before this point in time and Blomeyer after this point in time for in-the-money options. A combination of the methods could reduce the MAPD to 0.63 percent for all the situations, compared to a MAPD of 0.70 percent for Blomeyer and a MAPD of 1.32 percent for. 5 Summary and conclusions Pricing American put options on dividend paying stocks has largely been ignored in the literature because the problem is mathematically complex and valuation usually resorts to computationally expensive and impractical pricing applications. Therefore it is usual to use approximation methods for the valuation of American put options on a stock with known discrete dividend payments. This paper provided a simulation study of two commonly used approximation methods, the one by Blomeyer (1986) and the one by Barone-Adesi and Whaley (1988). This to find out if there were pricing errors and to find out which could be the most usable method for practical users. The study showed that the approximation method by Blomeyer worked satisfactory for most situations, but some errors occur for longer times to the dividend payment, for smaller dividends and for in-the-money options. The approximation method by Barone-Adesi and Whaley () worked well for in-the-money options and at-the-money options, but had serious pricing errors for out-of-the-money options.

21 19 The conclusion of the study is that a combination of the both methods might be preferable to any single model. This comes from the fact that although the method by Blomeyer worked satisfactory for most situations, the pricing errors of in-themoney options might be quite large in absolute terms. A simple method could be to use Blomeyer for out-of-the-money options, for at-the-money options and depending on the point in time after which it will not be optimal to exercise until after the dividend has been paid, either Blomeyer or for in-the-money options. could be used before this point in time and Blomeyer after this point in time for inthe-money options. A combination of the methods could reduce the MAPD to 0.63 percent for all the situations, compared to a MAPD of 0.70 percent for Blomeyer and a MAPD of 1.32 percent for. This kind of strategy is easily programmed and does not change the calculation speed by any noteworthy amount. When examining these results it has to be remembered that the result stands for the situations simulated in this paper. Because the option prices are dependent on many different factors and the combination possibilities are enormous the simulations has to be limited in someway. Possible further studies could address the pricing problem by for example using different volatility and more deeply analyze different dividend sizes. As mentioned earlier to test the approximation methods against market prices could also be a proposal for further research. Finally it has to be remembered that the discussed methods are only approximations and that the absolute pricing errors with these methods might be quite large for some situations. This is a cost the user might have to sacrifice for a high degree of computational efficiency. But the computer business is changing rapidly and it could be possible to use more accurate models such as the Binomial option pricing model or finite difference methods also for traders in a near future.

22 20 References Barone-Adesi, G. and Whaley, R. E. (1987), Efficient analytic approximation of American option values Journal of Finance, Vol. 42, No. 2, June, Barone-Adesi, G.and Whaley, R. E. (1988), On the valuation of American put options on dividend-paying stocks Advances in Futures and Options Research, Vol. 3, Black, F. and Scholes, M. (1973), The pricing of options and corporate liabilities Journal of Political Economy, Vol. 81, May - June, Blomeyer, E. C. (1986), An analytic approximation for the American put price for options with dividends Journal of Financial and Quantitative Analysis, Vol. 21, No. 2, June, Blomeyer, E. C. and Johnson, H. (1988), An empirican examination of the pricing of American put options Journal of Financial and Quantitative Analysis, Vol. 23, No. 2, March, Brennan, M. J. and Schwartz, E. S. (1978), Finite difference methods and jump process arising in the pricing of contingent claims: A synthesis Journal of Financial and Quantitative Analysis, Vol. 13, September, Cox, J. C. and Ross, S. A. and Rubinstein, M. (1979), Option pricing: A simple approach Journal of Financial Economics, Vol. 7, September, Cox, J. C. and Rubinstein, M. (1985), Options market, New Jersey: Prentice Hall Inc. Geske, J. (1979), The valuation of compound options Journal of Financial Economics, Vol. 7, March, Geske, J. and Johnson, H. E. (1984), The American put option valued analytically Journal of Finance, Vol. 39, No. 5, December,

23 21 Geske, J. and Shastri, K. (1985), Valuation by Approximation: A comparison of alternative option valuation techniques Journal of Financial and Quantitative Analysis, Vol. 20, No. 1, March, Hull, J. and White, A. (1988), The use of control variate technique on option pricing Journal of Financial and Quantitative Analysis, Vol. 23, No. 3, September, Ju, N. (1998), Pricing an American Option by Approximating Its Early Exercise Boundary as a Multipiece Exponential Function, Review of Financial Studies, Vol. 11, No. 3, Little, T. and Pant, V. and Hou, C. (2000), A new integral representation of early exercise boundary for American put options, Journal of Computational Finance, Vol. 3, No. 3, MacMillan, L. W. (1986), Analytic approximation for the American put option Advances In Futures and Option Research, Vol. 1, Merton, R. C. (1973a), Theory of rational option pricing Bell Journal of Economics and Management Science, Vol. 4, Spring, Merton, R. C. (1973b), The relationship between put and call option prices: Comment Journal of Finance, March, Roll, R. (1977), An analytical formula for unprotected American call options on stoch with known dividends Journal of Financial Economics, Vol. 5, November, Sterk, W. (1982): Comparative Performance of the Black-Scholes and Roll-Geske- Whaley Option Pricing Models, Journal of Financial and Quantitative Analysis, Vol. 18,

24 22 Sundkvist, K. (2000): Evaluating Option Pricing Models Different Ways of Modelling Time, Working Paper, Swedish School of Economics and Business Administration. Whaley, R. E. (1981), On the valuation of American call options on stocks with known dividends Journal of Financial Economics, Vol. 9, June,

25 23 Appendix 1 To find the critical commodity price S*, it is necessary to solve (9). Since this cannot be done directly, an iterative procedure must be developed. Barone-Adesi and Whaley (1987) presented an algorithm that normally finds a good approximation for S* after three or even less iterations. To begin, evaluate both sides of (9) at some seed value S 1, that is, and LHS(S i ) = X - S i RHS(S i ) = p(s i ) - ( 1 N1( d 1 ( S i ))) S i q (A1) (A2) where i = 1. Naturally, it is unlikely that LHS(S i ) = RHS(S i ) on the initial guess of S 1, and a second guess must be made. To developed the next guess S i+1, first find the slope b i, of the RHS at S i, that is, 1 n1 ( d 1 ( S i )) b i = 1 ( 1 N1 ( d 1 ( S i ))) * ( 1 ) +. (A3) q σq t Next find where the line tangent to the curve RHS at S i, intersects the exercisable proceeds of the American put, X - S, that is, RHS(S i ) + b i (S - S i ) = X S (A4) and then isolate S to find S i+1, ( X RHS( S i ) b i S i ) Si+ 1 =. (A5) (1 b ) i The formula (A5) will provide the second and subsequent guesses of S, with new values of (A1), (A2), (A3) and (A5) computed with each new iteration. The iterative

26 24 procedure should continue until the relative absolute errors falls within an acceptable tolerance level; for example, LHS( S i ) RHS( S i ) / X < 0, (A6) The speed, with which the algorithm finds the critical commodity price, can be improved by using a starting value S 1, closer to the solution. The critical commodity price must satisfy (9). At expiration the critical commodity price will be the exercise price of the option, X. At the other extreme, if the time remaining to expiration is infinite, the critical commodity price may be solved exactly by substituting t = + in (9), that is, S*( ) = βx. (A7) ( 1 + β ) It is worthwhile to point out that (A7) is exactly the same as Merton s (1973a). In (9), the critical commodity price is a decreasing function of time to expiration. With this and other information from the put option-pricing problem in hand, it is possible to derive an approximate analytic solution to find the critical commodity price 5, that is, where S* = S*( ) + (X - S*( ))e k (A8) X k = (rt - 2σ t )*( ). (A9) ( X S * ( )) The formula (A8) provide the seed value for the iterative procedures that determine the critical commodity price in the American put option algorithm. It is a straightforward procedure, which usually ensures convergence in three iterations or less. 5 For more information see Barone-Adesi and Whaley (1987).

27 25 Appendix 2 Blomeyer DAIMLER Blomeyer DEUTSCHE BANK Blomeyer VOLKSWAGEN , , , ,5 702 Figure A1 Deviations for the models, changing the time and price Daimler: strike price 90, dividend 1.00, interest rate 3.15% and volatility 14.50%. Deutsche Bank: strike price 70, dividend 2.15, interest rate 3.15% and volatility 15.00%. Volkswagen: strike price 600, dividend 6.00, interest rate 3.15% and volatility 17.50%. The absolute percentage deviations were calculated as, APD Mi = [(P Mi P Ci )/ P Ci ]*100, where P Mi was the price for the model and P Ci was the benchmark price from the Binomial option pricing model.

28 26 Blomeyer DAIMLER 0,5 0,5 1,25 1, Blomeyer DEUTSCHE BANK 1,5 1,5 2,475 2, Blomeyer VOLKSWAGEN , , , ,5 702 Figure A2 Deviations for the models, changing the dividend and price Daimler: strike price 90, time to dividend 122 days, maturity 151 days, interest rate 3.15% and volatility 14.50%. Deutsche Bank: strike price 70, time to dividend 122 days, maturity 145 days, interest rate 3.15% and volatility 15.00%. Volkswagen: strike price 600, time to dividend 122 days, maturity 137 days, interest rate 3.15% and volatility The absolute percentage deviations were calculated as, APD Mi = [(P Mi P Ci )/ P Ci ]*100, where P Mi was the price for the model and P Ci was the benchmark price from the Binomial option pricing model.