An Accurate Approximate Analytical Formula for Stock Options with Known Dividends

Transcription

1 An Accurate Approximate Analytical Formula for Stock Options with Known Dividends Tian-Shyr Dai Yuh-Dauh Lyuu Abstract Pricing options on a stock that pays known dividends has not been satisfactorily settled in the literature. Many assume that the stock price minus the present value of future dividends follows the lognormal diffusion. Unfortunately, this approximation undervalues the option because the volatility of the underlying stock s price is underestimated. This problem can be addressed by adjusting the volatility parameter, but the approach remains a heuristic. The known dividend is also suggested to be approximated by a fixed discrete dividend yield or a fixed continuous dividend yield. But both approximation schemes are shown to be numerically deficient. In our paper, a known dividend is approximated by an equivalent continuous dividend yield that can be represented as a function of the known dividend and the stock return. Thus the stock price process follows a lognormal diffusion process and the option pricing formula can be easily derived. Numerical results show that our approximation pricing formula is more accurate than other existing approaches. 1 Introduction Solving option pricing for dividend-paying stocks is a long-standing question. By assuming that the stock price follows the lognormal diffusion, Black and Scholes (1973) arrive at their ground-breaking option pricing model for non-dividend-paying stocks. Merton (1973) Corresponding author. Department of Information and Finance Management, National Chiao-Tung University, 1001 Ta Hsueh Road, Hsinchu, Taiwan 300, ROC. d88006@csie.ntu.edu.tw. Tel: # Fax: The author was supported in part by NSC grant E Department of Finance and Department of Computer Science & Information Engineering, National Taiwan University, No 1, Sec 4, Roosevelt Rd, Taipei, Taiwan lyuu@csie.ntu.edu.tw. The author was supported in part by NSC grant E

2 extends the model to the case where the underlying stock pays a non-stochastic continuous dividend yield. He defines the cost of carrying of a stock as the risk-free interest rate less the dividend yield, and the stock is assumed to grow at the cost of carrying rate. This continuous dividend yield assumption is widely adopted for pricing options as in Barone- Adesi and Whaley (1987), Broadie and Detemple (1995,1996), and so on. However, almost all stock dividends are paid discretely rather than continuously. We call this dividend setting the known dividend if the amounts of future dividends are known today. Pricing options on a stock that pays known dividends seems to be investigated first in Black (1975). The known-dividend option pricing problem has drawn a lot of attention in the literature. Roll (1977) suggests that the stock price minus the present value of future dividends over the life of the option follows the lognormal diffusion. The Black-Scholes formula is then applied with the cum-dividend stock price replaced by the stock price net of the present value of future dividends. We follow Cox and Rubinstein (1985) in calling the above approximation methodology the ad hoc adjustment. The ad hoc adjustment is widely accepted in academic literature. Geske (1979) and Whaley (1981) develop pricing formulas for American calls with the ad hoc adjustment. Carr (1998) derives a semi-explicit formula for American puts with the ad hoc adjustment. Chance et al. (00) show that the Black-Scholes-Merton model is upheld when the underlying stock pays discrete stochastic dividends under the assumptions that the dividends are uncorrelated with the stock price process and forward contracts on dividends exist. It can be said that most analytical approaches to the known-dividend option pricing problem rely on the ad hoc adjustment. The ad hoc adjustment is problematic when it comes to pricing accuracy. It divides the stock price into two parts: the stock price minus the present value of future dividends over the life of the option and the present value of future dividends. The former part (netof-dividend stock price) is assumed to follow a lognormal diffusion process, whereas the dividend is assumed to grow at the risk-free rate. The volatility of the net-of-dividend stock price is set as the volatility of the cum-dividend stock price. Thus the ad hoc adjustment underestimates the volatility of the stock price because the volatility of the present value of future dividends is not taken into account. Options are hence undervalued, and the biases become larger as the volatility of the stock price or the dividend amount becomes larger. To remove this systematic bias, Hull (000) recommends that the volatility of the net-ofdividend stock price be adjusted by a simple formula. The paper shows that the performance of Hull s volatility adjustment is mixed. Besides the analytical method, options can also be priced by the tree or the related PDE method. But a naive application of these methods results in combinatorial explosion. Consider the 4-time-step binomial tree with an ex-dividend date at time step in Fig. 1. This tree splits into 3 trees after the ex-dividend date. Each such tree will be split further

3 Figure 1: A Tree Model for Pricing Stock Options with Known Dividends. Ex-dividend date A dividend D is paid out at time step. Three separate trees beginning at time step are colored in white, light gray, and dark gray, respectively. into more separate trees for each additional ex-dividend date. As a result, the tree size grows exponentially. Although its pricing results converge to the theoretical option value, the exponential complexity renders it impractical. Other than the ad hoc adjustment, efficient numerical algorithms and simple formulas can also result by approximating the known dividend with either (1) a fixed dividend yield on each dividend-paying date or () a fixed continuous dividend yield. Geske and Shastri (1985) construct a recombing tree by following the first approach. Although their tree model is efficient, our paper will show that it produces inaccurate pricing results for European options. The second approach is followed by Chiras and Manaster (1978). They transform the discrete dividends into a fixed continuous dividend yield and then apply the Merton formula. As this approach can be shown to be equivalent to the first approach in pricing European options, it shares the same faults. This paper provides a new formula for pricing stock options with known dividends. Unlike the naive, ad hoc approximations mentioned above, the known dividend in this paper is replaced by an equivalent continuous dividend yield which can be represented as a function of the known dividend and the stock return. Thus the known dividends can be reinterpreted as the shift of the drift and the volatility of the stock price. Under this setting, the stock price follows the lognormal diffusion price process and the pricing formulas for vanilla options can be easily derived. Our formula can provide more accurate pricing results than other approximation methods. Moreover, it can be further extended to price an option with multiple known-dividend payouts. This property is useful as a stock can pay up to 4 dividends per annum in the U.S., for example. Numerical results verify the superiority of our pricing 3

4 formula to other models. The paper is organized as follows. The mathematical model is briefly covered in section. The approximation formulas are derived in section 3. In this section, we first derive a pricing formula for stock options with one known dividend. Then we extend our formula for the multiple-known-dividend case. Experimental results given in section 4 verify the superiority of our pricing formula to other pricing methods. Section 5 concludes the paper. The Models The stock price is assumed to follow the lognormal diffusion process: ds(t) = rdt+ σd B(t), S(t) where S(t) denotes the stock price at time t, r denotes the annual risk-free interest rate, σ denotes the volatility, and B(t) denotes the standard Brownian motion. Then the stock price S(t) can be represented as S(t) =S(s)e (r 0.5σ )(t s)+σ(b(t) B(s)), if no dividend is paid between time s and time t. Assume that a fixed dividend D is paid at ex-dividend date τ, then the stock price simultaneously falls by the amount αd at time τ. For simplicity, α is assumed to be 1 in our pricing formulas. α can be less than 1 when considering the effect of tax on dividend income. A general α poses no difficulties to our model. Assume that a stock option initiates at time 0 and matures at time T. Then the payoff at maturity date is (S(T ) X) + for a vanilla call option and (X S(T )) + for a vanilla put option, where X denotes the strike price and (A) + denotes max(a, 0). The underlying stock is assumed to pay n known dividends between time 0 and time T,wherenis a positive integer. The i-th dividend (c i )ispaidattime i j=1 t j,wheret j > 0. 3 The Analytical Formula We will first derive a pricing formula for pricing stock options with one known dividend. Then we extend our pricing formula to the multiple-known-dividends case. Although our discussions focus on call options, extension to put options is straightforward. 3.1 A Stock Option with Single Known Dividend First, consider a stock that pays one known dividend c 1 at time t 1. Then the stock price at time t 1 is S(t 1 )=S(0)e µt 1+σ(B(t 1 ) B(0)) c 1,whereµ r 0.5σ. Note that the distribution 4

5 of S(T ) is no longer lognormal and a simple and accurate option pricing formula is hard to derive. To solve this problem, we make the stock price process S(t) 1 follow a lognormal diffusion process by replacing the known dividend c 1 by an equivalent continuous dividend yield q 1 as follows: S(t 1 )=S(0)e µt 1+σ(B(t 1 ) B(0)) c 1 S(0)e (µ q 1)t 1 +σ(b(t 1 ) B(0)) (1) S(0)e µt 1+σ(B(t 1 ) B(0)) (1 e q 1t 1 )=c 1 1 e q 1t 1 = c 1e µt 1 S(0) e σ(b(t 1) B(0)) The left-hand side of the above equation can be approximated by the first order Taylor series expansion as 1 (1 q 1 t 1 ), and the right-hand side is approximated to be k 1 (1 σ(b(t 1 ) B(0))), where k 1 = c 1e µt 1.Thuswehaveq S(0) 1 k 1(1 σ(b(t 1 ) B(0))) t 1. The stock price at maturity date T canbeexpressedasfollows: S(T ) = [ S(0)e µt 1+σ(B(t 1 ) B(0)) c 1 ] e µ(t t 1 )+σ(b(t ) B(t 1 )) = S(0)e (µ q 1)t 1 +σ(b(t 1 ) B(0))+µ(T t 1 )+σ(b(t ) B(t 1 )) S(0)e (µ k 1/T )T +k 1 σ(b(t 1 ) B(0))+σ(B(T ) B(0)), which follows the lognomral distribution. Let Var(X) denote the variance of the random variable X. Defineσ 1 by Var [k 1 σ(b(t 1 ) B(0)) + σ(b(t ) B(0))] = (1 + k 1 ) σ t 1 + σ (T t 1 ) σ 1T Then the known dividend c 1 paid at time t 1 approximately takes a from that can be interpreted as the shift of the drift of the stock price from µ to µ k 1 /T and the volatility from σ to σ 1. The value for a vanilla call option can be calculated by the risk-neutral variation method as follows: e rt E(S(T ) X) + = e rt E [ S(0)e (µ k 1/T )T +k 1 σ(b(t 1 ) B(0))+σ(B(T ) B(0)) X ] + = S(0)e a N(d 1 ) Xe rt N(d ), () where a = σ 1 σ S(0) ln X T k 1, d 1 = k 1+(µ+σ1 )T σ 1 T,andd = d 1 σ 1 T. The above formula is simple but not accurate enough. This is because the random variable B(t 1 ) B(0) might not be small in magnitude and thus the approximation k 1 e σ(b(t1) B(0)) k 1 (1 σ(b(t 1 ) B(0))) is not accurate enough. To improve accuracy, k 1 e σ(b(t1) B(0)) can 1 You commented that This is confusing. in section, S(t) is the cumdividend price, I presume. But here, is it still so? Probably not. Yes, S(t) in section is the cum-dividend price. I substitute the known dividend c 1 with a continuous equivalent dividend yield q 1. So the dividend is removed from S(t) (0<t<t 1 )ata rate q 1 under this setting. q 1 is selected so S(t 1 )=S(0)e µt1+σ(b(t1) B(0)) c 1 = S(0)e (µ q1)t1+σ(b(t1) B(0)). 5

6 ( be further expanded as k 1 1 σ [B(t 1 ) B(0)] + σ [B(t 1 ) B(0)] q 1 is solved to be k 1 [ ] 1 σ(b(t 1 ) B(0))+σ (B(t 1 ) B(0)) t 1 ). Thus the dividend yield, and the stock price at time T is S(T )=S(0)e (µ k 1/T )T +σ(b(t ) B(0))+k 1 σ(b(t 1 ) B(0)) k 1 σ [B(t 1 ) B(0)]. (3) Note that it is hard to derive an analytical formula in this case since the distribution of the exponent term in Eq. (3) is unknown. We approximate the chi-square random variable k 1 σ [B(t 1 ) B(0)] with its mean E( k 1σ [B(t 1 ) B(0)] )= k 1σ t 1 δ 1. A more accurate formula for a call option is obtained by replacing a and d 1 in Eq. () with σ 1 σ T k 1 δ 1,and ln S(0) X k 1+(µ+σ 1 )T δ 1 σ 1 T, respectively. 3. Multiple Known Dividends Our formula can be easily extended to price a stock option with multiple known dividends. We will first focus on the two-known-dividend case. Then we will list the generalized pricing formula for the multiple-known-dividend case. The first dividend c 1 is paid at time t 1, and the second dividend c is paid at time t 1 + t. Thus the exdividend stock price at time t 1 + t can be approximated by replacing the known dividend c with an equivalent continuous dividend yield q as follows: S(t 1 )e µt +σ(b(t 1 +t ) B(t 1 )) c S(t 1 )e (µ q )t +σ(b(t 1 +t ) B(t 1 )) 1 e q t = c e µt S(t 1 ) e σ(b(t 1+t ) B(t 1 )). (4) By substituting Eq. (1) into Eq. (4) where q 1 k 1[1 σ(b(t 1 ) B(0))]+δ 1 t 1, q canbesolvedto be k [1 (1 + k 1 )σ(b(t 1 ) B(0)) σ(b(t ) B(t 1 ))] + δ, t where k = c e µ(t 1 +t )+k 1 +δ 1 and δ S(0) = k [(1+k 1 ) σ t 1 +σ t ]. Thus the stock price at time T can be represented as follows: S(T ) = S(0)e (µ q 1)t 1 +σ(b(t 1 ) B(0)) e (µ q )t +σ(b(t 1 +t ) B(t 1 )) e µ(t t 1 t )+σ(b(t ) B(t 1 +t )) = S(0)e (µ k 1 +k +δ 1 +δ T )T +(1+k 1 +k +k 1 k )σ(b(t 1 ) B(0))+(1+k )σ(b(t 1 +t ) B(t 1 ))+σ(b(t ) B(t 1 +t )), which follows the lognormal distribution. Define σ by Var [(1 + k 1 + k + k 1 k )σ(b(t 1 ) B(0)) + (1 + k )σ(b(t 1 + t ) B(t 1 )) + σ(b(t ) B(t 1 + t ))] =(1+k 1 + k + k 1 k ) σ t 1 +(1+k ) σ t + σ (T t 1 t ) σt. Again, the known dividends c 1 and c approximately take a form that can be interpreted as the shift in the drift of the stock price from µ to µ k 1+k +δ 1 +δ and the volatility from T 6

7 σ to σ. The value for a vanilla call option can be calculated by the risk-neutral variation method as follows: e rt E(S(T ) X) + = S(0)e σ σ T k 1 k δ 1 δ N(d 1) Xe rt N(d ), where d S(0) ln X 1 = k 1 k +(µ+σ )T δ 1 δ σ T and d = d 1 σ T. The pricing formula for stock options with more known dividends can be derived by iteratively repeating the method mentioned above. A pricing formula for n known dividends are illustrated with the aid of recursive formulas listed as follows: 0, if i>j, a i,j = σ if i = j, j 1 h=1 a i,hk h + σ if i<j, δ i = k i i j=1 a j,it j, k i = c ie µ σ i = i i j=1 t i+ i 1 j=1 (k j+δ j ) S(0) The pricing formula of a vanilla call option is then j=1 a j,i+1 t j + a i+1,i+1 (T i h=1 t h). T S(0)e σ n σ T n i=1 (k i+δ i ) N(d 1) Xe rt N(d ),, where d 1 = S(0) ln X +(µ+σ n)t n i=1 (k i+δ i ) σ n T and d = d 1 σ n T. 4 Numerical Results Since the ad hoc adjustment underestimates the volatility of the stock price, the options are undervalued and the implied volatilities overestimated. This observation is confirmed in Table 1, where the Monte Carlo simulation (MC) serves as the benchmark. Indeed the Black- Scholes formula with the ad hoc adjustment (denoted as BS) underestimates the true option price in every scenario, and the biases become larger as the volatility or the dividend amount becomes larger. The pricing errors between BS and MC listedintheerror%(bs) row demonstrate that the maximum pricing error can be as high as 15.9%. The Implied Vol.(BS) row tabulates the implied volatilities calculated by the Black-Scholes formula with the ad hoc adjustment (i.e., BS) giventhemc option prices as inputs. These implied volatilities are larger than the volatilities that generate the MC option prices. Hull (000) recommends that the volatility that will be substituted into Black-Scholes formula be adjusted by the volatility 7

8 of the cum-dividend stock price multiplied by S/(S D), where S and D denote the stock price and the present value of future dividends, respectively. Hull s volatility adjustment (see BS in Table 1) does not seem to work very well. The Error%(BS ) row denotes the relative pricing errors of the Black-Scholes formula with the ad hoc adjustment and Hull s volatility adjustment. The pricing error can be as high as 1.4%. The Implied Vol.(BS ) row denotes the implied volatilities calculated by BS. They consistently underestimate the volatilities that generate the MC option prices. Ours denotes the pricing formulas suggested in this paper. The Error%(Ours) and the Implied Vol.(Ours) rows denote, respectively, the relative pricing errors and the implied volatilities calculated by the formulas suggested in this paper. Prices generated by Ours are very close to the benchmark option prices, and the implied volatilities are also much more accurate than the methods based on the ad hoc adjustment. We conclude that the ad hoc adjustment as an approximation method even with additional heuristics does not work as well as our pricing formula. Geske and Shastri (1985) use fixed dividend yields to approximate the known dividends. The fixed dividend yield is defined as the known dividend amount divided by the initial stock price. For example, the dividend yield is 5% if the initial stock price is 100 and the dividend is 5. We use FDY to denote their approach. We compare FDY, BS, andours in Table and 3 with the Monte Carlo simulation (MC) as the benchmark. Table focuses on the single-known-dividend case and Table 3 focuses on the two-known-dividend case. It can be observed that FDY undervalues the options as BS does. The pricing errors of both FDY and BS are larger than that of Ours. It is interesting to note that prices produced by BS in both Table and 3 are almost equal. This is because the net-of-dividend stock price in Table (=100 5e ) is almost equal to that in Table 3 (=100.5e e ). However, MC shows that the option value in each case in Table 3 tends to be lower than that in the corresponding case in Table. Ours successfully captures this property while both FDY and BS fail. 5 Conclusions Traditional approaches for pricing options on known-dividend-paying stocks either produce biased results or are inefficient. Our paper suggests efficient and accurate pricing formulas by approximating known dividends with equivalent continuous dividend yields that can be represented as functions of known dividends and stock returns. Numerical results show that our formulas are more accurate than other existing approaches. 8

9 Table 1: Biases Caused by the Ad Hoc Adjustment and Hull s Volatility Adjustment. Volatility 30% 30% 70% 70% Ex-dividend date (years) Dividend (dollars) MC BS Error%(BS) 1.58% 15.9% 5.70% 7.68% Implied Vol.(BS) 30.09% 30.9% 71.73% 7.38% BS Error%(BS ) 3.6% 1.4% 6.19% 3.76% Implied Vol.(BS ) 9.80% 9.39% 68.3% 68.90% Ours Error%(Ours) 1.13% 1.9% 0.95%.64% Implied Vol.(Ours) 9.93% 30.07% 69.7% 69.4% The initial stock price is 100, the strike price is 150, the risk-free interest rate is 10%, the time to maturity is 0.5 year, and the volatilities of the cum-dividend stock prices are listed in the first row. The stock pays a dividend at the ex-dividend date. MC denotes the Monte Carlo simulation based on 100, 000 paths. BS denotes the Black-Scholes formula with the ad hoc adjustment, and BS denotes the Black-Scholes formula with the ad hoc adjustment and Hull s volatility adjustment. Ours denotes our pricing formula. Error%(A) denotes the relative pricing error of method A relative to MC: (MC A)/MC, wherea can be BS, BS,or Ours. Implied Vol.(A) denotes the implied volatility calculated based on method A. Acknowledgement We thank Wang Ren-Her for his useful suggestions. References [1] Barone-Adesi., and R.E. Whaley. Efficient Analytic Approximation of American Options Values. Journal of Finance, 4 (1987), pp [] Black, F. Fact and Fantasy in the Use of Options. Financial Analysts Journal, 31 (1975), pp ,

10 Table : Pricing a Call Option with One Known Dividend X FDY BS Ours MC FDY BS Ours MC FDY BS Ours MC * * * * * * * * The initial stock price is 100, the risk-free rate is 3%, a 5-dollar-dividend is paid at 0.6 year. The volatilities of the stock price are shown in the first row of the table. The strike prices are listed in the first column. FDY denotes the fixed dividend yield settings proposed in Geske and Shastri (1985). BS denotes the Black-Scholes formula with the ad hoc adjustment, Ours denotes our pricing formula, and MC denotes the Monte Carlo simulation based on 100, 000 paths. We use MC as the benchmark. Option prices which deviate from the benchmark values by 0.5 are marked by asterisks. Table 3: Pricing a Call Option with Two Known Dividends X FDY BS Ours MC FDY BS Ours MC FDY BS Ours MC * * * * * *16.8 * The numerical settings are the same as that in Fig. except that a.5-dollar-dividend is paid at 0.4 year and 0.8 year. Option prices which deviate from the benchmark values by 0.5 are marked by asterisks. [3] Black, F., and M. Scholes. The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81 (1973), pp [4] Broadie, M., and J.B. Detemple. American Capped Call Options on Dividend Paying Assets. Review of Financial Studies, 8 (1995), pp [5] Broadie, M., and J.B. Detemple. American Options Valuation: New Bounds. Approximations and a Comparison of Existing Methods. Review of Financial Studies, 9 (1996), pp [6] Carr, P. Randomization and the American Put. The Review of Financial Studies, 11 (1998), pp [7] Chance, D.M., R. Kumar, and D. Rich. European Option Pricing with Discrete Stochastic Dividends. Journal of Derivatives, 9 (00), pp

11 [8] Chiras, D.P., and S. Manaster. The Informational Content of Option Prices and a Test of Market Efficiency. Journal of Financial Economics, 6 (1978), pp [9] Cox, J.C., and M. Rubinstein. Options Markets. Englewood Cliffs, NJ: Prentice- Hall, [10] Dai, T.-S., and Y.-D. Lyuu. An Exact Subexponential-Time Lattice Algorithm for Asian Options. In Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms, Philadelphia: Society for Industrial and Applied Mathematics, January 004, pp [11] Geske, R. A Note on an Analytical Valuation Formula for Unprotected American Call Options on Stocks with Known Dividends. Journal of Financial Economics, 7 (1979), pp [1] Geske, R., and K. Shastri. Valuation by Approximation: a Comparison of Alternative Option Valuation Techniques. Journal of Financial and Quantitative Analysis, 0 (1985), pp [13] Harvey, C.R., and R.E. Whaley. Dividends and S&P 100 Index Option Valuation. Journal of Futures Markets, 1 (199), pp [14] Hull, J. Options, Futures, and Other Derivatives. 4th ed. Englewood Cliffs, NJ: Prentice-Hall, 000. [15] Lyuu, Y.-D. Financial Engineering & Computation: Principles, Mathematics, Algorithms. Cambridge, U.K.: Cambridge University Press, 00. [16] Merton, R.C. Theory of Rational Option Pricing. Bell Journal of Economics and Management Science, 4 (1973), pp [17] Roll, R. An Analytic Valuation Formula for Unprotected American Call Options on Stocks with Known Dividends. Journal of Financial Economics, 5 (1977), pp [18] Whaley, R.E. On the Valuation of American Call Options on Stocks with Known Dividends. Journal of Financial Economics, 9 (1981), pp [19] Whaley, R.E. Valuation of American Call Options on Dividend-Paying Stocks: Empirical Tests. Journal of Financial Economics, 10 (198), pp [0] Wu, C.-N. Option Pricing under GARCH Volatility Process. MBA Thesis. Department of Finance, National Taiwan University, Taiwan,