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Quantitative Finance
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Efficient option pricing on stocks paying discrete or
path-dependent dividends with the stair tree
Tian-Shyr Dai a a
Department of Information and Finance Management and Institute of Finance , National Chiao-Tung University , 1001 Ta Hsueh Road, Hsinchu, Taiwan 300, ROC Published online: 12 Oct 2009.
To cite this article: Tian-Shyr Dai (2009) Efficient option pricing on stocks paying discrete or path-dependent dividends
with the stair tree, Quantitative Finance, 9:7, 827-838, DOI: 10.1080/14697680902814217
To link to this article: http://dx.doi.org/10.1080/14697680902814217
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Efficient option pricing on stocks paying discrete or
path-dependent dividends with the stair tree
TIAN-SHYR DAI*
Department of Information and Finance Management and Institute of Finance, National Chiao-Tung University, 1001 Ta Hsueh Road, Hsinchu, Taiwan 300, ROC
(Received 23 January 2007; in final form 6 February 2009)
Pricing options on a stock that pays discrete dividends has not been satisfactorily settled because of the conflicting demands of computational tractability and realistic modelling of the stock price process. Many papers assume that the stock price minus the present value of future dividends or the stock price plus the forward value of future dividends follows a lognormal diffusion process; however, these assumptions might produce unreasonable prices for some exotic options and American options. It is more realistic to assume that the stock price decreases by the amount of the dividend payout at the ex-dividend date and follows a lognormal diffusion process between adjacent ex-dividend dates, but analytical pricing formulas and efficient numerical methods are hard to develop. This paper introduces a new tree, the stair tree, that faithfully implements the aforementioned dividend model without approximations. The stair tree uses extra nodes only when it needs to simulate the price jumps due to dividend payouts and return to a more economical, simple structure at all other times. Thus it is simple to construct, easy to understand, and efficient. Numerous numerical calculations confirm the stair tree’s superior performance to existing methods in terms of accuracy, speed, and/or generality. Besides, the stair tree can be extended to more general cases when future dividends are completely determined by past stock prices and dividends, making the stair tree able to model sophisticated dividend processes.
Keywords: Option pricing; Stair tree; Dividends
By assuming that the stock price process follows a lognormal diffusion process, Black and Scholes (1973) arrive at their ground-breaking option pricing formula for non-dividend-paying stocks. Merton (1973) extends the model to the case where the underlying stock pays a non-stochastic continuous dividend yield. The resulting for-mula is often called the Black–Scholes–Merton forfor-mula. In reality, however, almost all stock dividends are paid at discrete time points rather than continuously. Pricing options on a stock that pays discrete dividends with known amounts seems to be investigated first in Black (1975). This dividend setting is called the discrete dividend for simplicity.
The discrete-dividend option pricing problem has drawn a lot of attention in the literature. According to Frishling (2002), the stock price with discrete dividends has been modelled by three following ways.
Model 1: This model, crystallized under the discussions of Roll (1977), Geske (1979), and Whaley (1981), assumes that the stock price is divided 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 (the net-of-dividend stock price) is assumed to follow a lognormal diffusion process, whereas the latter part is assumed to grow at the risk-free rate. Thus vanilla options can be computed by applying the Black–Scholes–Merton formula with the stock price replaced by the net-of-dividend stock price. Cox and Rubinstein (1985) call it ‘ad hoc adjustment.’
Model 2: Musiela and Rutkowski (1997), following Heath and Jarrow (1988), suggest that the cum-forward-dividend stock price, defined as the stock price plus the forward values of the dividends paid from the prevailing time up to maturity, follows a lognormal diffusion process. Thus vanilla options can be computed by applying the Black–Scholes–Merton formula by replacing
*Email: [email protected]
Quantitative Finance
ISSN 1469–7688 print/ISSN 1469–7696 onlineß 2009 Taylor & Francis http://www.informaworld.com
DOI: 10.1080/14697680902814217
the stock price with the cum-forward-dividend stock price and by adding the forward values of the dividends prior to maturity to the exercise price.
Model 3: The stock price decreases by the amount of the dividend paid at the ex-dividend date and follows a lognormal price process between adjacent ex-dividend dates.
Although the above three models attempt to solve the discrete-dividend option pricing problem, Frishling (2002) shows that they generate very different option prices. Roughly speaking, assume the volatility input to these three models is . Model 1 sets the volatility of the net-of-dividend stock price at , while Model 3 sets the volatility of the stock price at . The volatility of the stock price in Model 1 is lower than that in Model 3 because the volatility of the present value of future dividends, a component of the stock price, is assumed to be 0 in Model 1. Model 1 therefore produces lower option prices, and the difference becomes larger as becomes larger. Similarly, Model 2 produces higher option prices than Model 3 since Model 2 assigns the volatility of the forward values of the dividends, which is not a part of stock price, to be .
Although Model 1 and Model 2 are widely accepted in the literature (see Whaley 1982, Carr 1998, Chance et al. 2002) in solving the discrete-dividend problem, they suffer from many problems. For example, Frishling (2002) shows that Model 1 and Model 2 could incorrectly price barrier options. Bender and Vorst (2001) show that arbitrage opportunities exist in Model 1 if the volatility surface is continuously interpolated around ex-dividend dates. Bos and Vandermark (2002) show that both Model 1 and Model 2 violate a perfectly reasonable continuity requirement.
Although Model 3 is much closer to reality than the other two models, there is no exact pricing formula for European options. Hull (2000) recommends an approx-imate pricing formula by adjusting the volatility input to Model 1 using a simple formula. However, this paper shows that the performance of Hull’s volatility adjust-ment is mixed. Bos and Vandermark (2002) present an approach that is a mixture of the stock and exercise price adjustments (in other words, Model 1 and Model 2). Bos and Shepeleva (2002) claim that this approach results in some inaccuracies, especially for in- and out-of-the-money options. They suggest a different pricing formula by adjusting the volatility input to Model 1 using a complex formula. But their approach can not be easily extended for pricing American options. Besides, the numerical results in this paper also suggest that my approach provides more accurate option values than the aforementioned approaches for pricing European options.
Model 3 can be implemented by the tree or the related PDE method.y But a naive application of these methods results in combinatorial explosion. Take the well-known
CRR binomial tree proposed by Cox et al. (1979) as an example. Assume that the tree starts at time step 0 and ends at time step n. Let R stand for the gross risk-free return per time step. When the stock does not pay dividends, in one time step the price S becomes Su (the up move) with probability puand Sd (the down move) with
probability pd1 pu, where pu ðR dÞ=ðu dÞ.
The relation ud ¼ 1 is enforced by the CRR binomial tree. The black nodes at the first two time steps of the bushy tree in figure 1 forms a 2-time-step CRR binomial tree. The CRR binomial tree recombines; thus the size of the tree is only quadratic in n. Unfortunately, the recombination property disappears if the stock pays discrete dividends. Assume that a dividend D is paid at time step 2. The bushy tree splits into three trees after the ex-dividend date. Each such tree will be split further at each subsequent ex-dividend date. As a result, the tree size grows exponentially with the number of ex-dividend dates. The bushy tree implements Model 3 faithfully, but the exponential complexity renders it impractical.
In addition to Model 1 and Model 2, efficient numerical algorithms and simple formulas can also result by approximating the discrete dividend with either (1) a fixed dividend yield on each ex-dividend date or (2) a fixed continuous dividend yield. The first approach is
Ex-dividend date
S
2 (Sd −D) (S−D) 2 (Su −D)Sd
Su
0
1
2
3
4
S
2Su
2S d
up
dp
Figure 1. The bushy tree. The initial stock price is S. The upward and the downward multiplicative factors for the stock price are u and d, respectively. The upward and the downward branching probabilities are pu and pd, respectively. The black nodes in the first two time steps form a CRR tree. A dividend D is paid out at time step 2. The values in parenthesis at time step 2 denote the stock prices immediately after dividend payout. Three separate trees beginning at time step 2 are coloured in white, light grey, and dark grey, respectively.
yBasically, the trinomial tree is analogous to an explicit finite-difference model (see Lyuu 2002). Thus my method for handling known dividends or path-dependent dividends can be extended to an explicit finite-difference model.
followed by Geske and Shastri (1985). They replace the discrete dividends with fixed dividend yields. The resulting tree hence recombines and is efficient. My paper will show that this approach works well for American options but poorly for European options. Chiras and Manaster (1978), following Merton’s (1973) idea, adopt the second approach. They transform the discrete dividends into a fixed continuous dividend yield and then apply the Black–Scholes–Merton formula. As this approach is equivalent to the first approach in pricing European options, it shares the same faults.
The major contribution of this paper is a novel tree model, the stair tree, that faithfully implements Model 3 without combinatorial explosion. Numerical results in the paper will show that the prices calculated by the stair tree are extremely close to those generated by the Monte Carlo simulation for European options and those generated by the bushy tree for American options. The stair tree is furthermore efficient and general. In contrast, the Monte Carlo simulation cannot handle American options easily, and the bushy tree grows exponentially. Table 1 compares the sizes of the bushy tree and the stair tree. The size difference grows with the number of ex-dividend dates. Compared with the stair tree, existing schemes that implement Model 3 are less accurate, less efficient, and/ or less general.
The idea behind the stair tree is straightforward. The stair tree limits the stock prices at each time step t to be of the form Puk. Here P denotes the stock price of
some specific node at time step t, u denotes the upward multiplicative factor for the stock price in the CRR binomial tree, and k is some even integer. It therefore preserves the CRR tree structure at each time step. Consider a 4-time-step stair tree with a dividend payout D at time step 1 and time step 3 as illustrated in figure 2. The price drops due to the dividend payouts (at time step 1 and time step 3) resemble the riser. Note that the ex-dividend stock prices at nodes X and Y are Su X and Sd X, respectively. The time interval between time step 0 and time step 1 (an ex-dividend date), and the time interval between time step 2 and time step 3 resemble treads. This tree is therefore called the stair tree.
Assume S0denotes the largest stock price at time step 2.
Because the stock prices at time step 2 are restricted to be S0ukfor nonpositive even integers k, the stair tree remains
recombining at time step 3 and so on until the next ex-dividend date. In general, the stair tree follows the CRR tree structure between ex-dividend dates. This idea greatly reduces the number of tree nodes. For the nodes at the ex-dividend dates (like the grey nodes in figure 2), trinomial branching schemes are devised to connect the two adjacent CRR tree structures. The theoretical guar-antee that simple and efficient branching schemes exist constitutes a major contribution of the paper. The adaptive mesh model proposed by Figlewski and Gao (1999) and Gao et al. (1999) also adjusts the tree structure by adding trinomial branches at certain points in the tree. The adaptive mesh model focuses on suppressing the nonlinearity error which makes the pricing results oscillate, while the stair tree model focuses on implementing Model 3 faithfully without combinator-ial explosion.
Pricing options whose underlying stock pays stochastic dividends is discussed in Cox and Rubinstein (1985), Miltersen and Schwartz (1998), and Chance et al. (2002). This setting is important since the dividend payout is in practice not perfectly predictable, especially when the ex-dividend dates are far into the future. For example, even the so-called widow-and-orphan AT&T stock valued for its stable dividend payouts cut its quarterly dividend from 22 cents per share to 3.75 cents per share in the 4th quarter of 2000. Miltersen and Schwartz (1998) discuss pricing options on commodity futures with stochastic convenience yields. Chance et al. (2002) show that the Black–Scholes–Merton model is upheld by assuming that
Table 1. Sizes of the stair tree and the bushy tree. The stair tree (Stair) and the bushy tree (Bushy) are compared in terms of numbers of nodes. The stock price is 100, the volatility is 30%, the risk-free interest rate is 10%, and the time to maturity is 0.75 year. The number of time steps for both the stair and bushy trees is 300. The number of ex-dividend dates is in the first column. The exdividend dates divide the 0.75-year time span into equal-length time intervals. A 1-dollar dividend is paid at each ex-dividend date. For example, 2 ex-dividend dates means that a 1-dollar dividend is
paid at year 0.25 and year 0.5.
# Ex-dividend dates Stair Bushy
1 46,051 1,744,201 2 46,851 53,060,451 3 47,851 1,301,124,826 4 48,960 26,604,783,451 5 51,475 466,301,626,701 Y X D D S S'' (-2) (-4) (-6) Su Sd (-8) (0) (-10) (-12) S' 0 1 2 3 4
Figure 2. The structure of the stair tree. The initial stock price is S. The upward and the downward multiplicative factors for the stock price are u and d, respectively. The grey nodes are the nodes right after the dividend is paid. S0 and S00 denote the largest stock price at time step 2 and time step 4, respectively. The stock price for each node on the third tread is represented as S00uk¼S00ekpffiffiffiffit, where k is parenthesized.
the discretely stochastic dividends are uncorrelated with the stock price. Cox and Rubinstein (1985) argue that pricing options on dividend-paying stocks can be handled by the arbitrage-based pricing theory when the future dividends are known exogenously or completely deter-mined by past stock prices and dividends. I call their setting the path-dependent dividends as the dividends depend solely on the past history of the stock price path. In this setting, the dividend paid at time could be written as a function of stock prices and the dividends prior to time . This is more general and realistic than the discrete dividend setting in many ways. First, it can solve the negative stock price problem that occurred under the discrete-dividend setting. This problem happens as the stock price drop due to the discrete dividend payment is larger than the cum-dividend stock price at the ex-dividend date. The problem can be avoided by choosing a proper dividend-paying function so that the dividend payment is always less than the cum-dividend stock price. Second, the path-dependent dividends setting can fit the real world phenomenon by choosing a proper dividend function from empirical studies. Although it is well known that dividends can be explained by a variety of factors such as the net operating profits and long-run sustainable (or permanent) earnings, a dividend function that fits the path-dependent dividends setting can still be constructed if the stock prices and the dividends paid previously serve as good proxies of these factors. I will review one of such dividend models proposed by Marsh and Merton (1987). The stair tree can incorporate such dividend models by adding extra states to keep the information necessary for computing future dividends. A simple numerical example will be given to explain how that is done.
The paper is organized as follows. The mathematical model is briefly covered in section 1. The stair tree for the dividend-paying stock is discussed in section 2. A sample stair tree is given in section 3 to convey the main ideas. Experimental results given in section 4 verify the super-iority of the stair tree to other models. In section 5, I will first introduce the path-dependent dividends settings before going on to review Marsh and Merton’s dividend model and show how the stair tree incorporate their dividend model. Section 6 concludes the paper.
1. The models
In Model 3, the stock price under the risk-neutral probability is assumed to follow the lognormal diffusion process:
Sðt þ Þ ¼ SðtÞeðr0:52Þþ!, ð1Þ
where S(t) denotes the stock price at year t, r denotes the annual risk-free interest rate, denotes the volatility, and ! denotes the standard Brownian motion. In the
discrete-time tree model, it is assumed that there are n equal discrete-time steps between year 0 and year T. The length of each time step t is equal to T=n. Thus, time step i in the discrete-time model corresponds to year it in the
continuous-time model. The upward and downward multiplicative factors u and d for the stock price equal epffiffiffiffitand epffiffiffiffit, respectively, for the CRR and stair trees.
Si denotes the stock price at year it (or, equivalently,
time step i for a tree). The stock is assumed to pay m dividends Dt1, Dt2, . . . , Dtm, where Dti is paid out at time
step ti. I further assume t15 t25 5 tm for
conve-nience. Under the discrete dividend assumption, any arbitrary dividend Dtiis already known at time step 0. In
general, Dtican be determined by a function of stock prices
and/or the dividends paid up to time step tiunder the
path-dependent dividends assumption. The stock price simul-taneously falls by the amount Dti. For simplicity, is
assumed to be 1 throughout the paper, but a general poses no difficulties to the stair tree. When the ex-dividend stock price becomes negative, it is assumed to stay at zero from that point onward. Harvey and Whaley (1992), in contrast, assume that the dividend is not paid if its amount exceeds the prevailing stock price. The stair tree can easily incorporate their assumption, too.
The option is assumed to start at time step 0 and mature at time step n. The exercise price for this option is K. Define ðAÞþ to denote maxðA, 0Þ for simplicity. The payoff for a European option at maturity is
final payoff ¼ ðSnKÞ
þ
, for a call, ðK SnÞþ, for a put:
(
An American option gives the holder the right to exercise the option before maturity. The exercise value for an American option at a non-dividend-paying time step i is
exercise value ¼ SiK, for a call, K Si, for a put:
The exercise strategy for an American option at an ex-dividend date is only slightly more complicated. It is never optimal to exercise an American call immediately after the underlying stock pays a dividend because it is dominated by the strategy of exercising the call immedi-ately before. Similarly, it is never optimal to exercise a put before the stock pays a dividend. Consequently, the exercise value for an option at a dividend-paying time step i is exercise value ¼ S i; K, for a call, K Si, for a put, ð2Þ where S
i and Sidenote the cum-dividend stock price and
the net-of-dividend stock price at time step i, respectively. An option will be exercised early by the owner if the option’s continuation value (i.e. the value to hold the option) is smaller than its exercise value.
2. Construction of the stair tree
I illustrate the main ideas by the 4-time-step tree in figure 2. This 4-time-step stair tree contains two ex-dividend dates: one at time step 1 and the other at time step 3. For simplicity, the same D-dollar dividend is
paid at each ex-dividend date. The price drop due to the dividend payout is represented by a riser. Each tread covers a time interval between two adjacent ex-dividend dates except the first tread, which covers the time interval between time step 0 and the first ex-dividend date. The branches follow the CRR tree structure except those from the nodes at the ex-dividend dates. For example, the stock price at the root node is S. The stock prices for its two successor nodes are Su and Sd, where ud ¼ 1. Because of the CRR tree structure, the stock prices at the same time step are Puk, where P is the stock price of some
specific node at that time step and k is an even integer. For example, the stock price for each node at time step 4 can be represented as S00uk, where S00 denotes the largest
stock price at time step 4 and k is parenthesized. Technically, any node’s stock price can be picked for P because the stock prices at the same time step are part of the geometric sequence
. . . , Pu4, Pu2, P, Pu2, Pu4, . . .
Note that the first tread contains a single, complete CRR tree. The tree structure on each subsequent tread is composed of a CRR binomial tree with the initial section truncated.
I next construct the branches out of the grey nodes at an ex-dividend date to complete the stair tree. Figure 3 illustrates what happens at an ex-dividend date by zooming in the first three time steps of the stair tree in figure 2. Nodes X and Y are from the first ex-dividend date. The ex-dividend stock price at node X is SX¼Su D. The two branches from X follow the
CRR tree structure. S0, the stock price for the top node
at time step 2, therefore equals SXu. Define the
V-log-price of stock V-log-price V0 as lnðV0=VÞ; a V-log-price of z
implies a stock price of Vez. Since the stock price for each
node on the second tread can be expressed in terms of S0ukfor some even integer k, the S0-log-prices for nodes at
time step 2 in figure 3 are integral multiples of 2pffiffiffiffiffiffit. The branches from node Y are constructed as follows. Let the ex-dividend stock price for node Y be SY. At least
three branches are required for node Y so it has enough degrees of freedom to match the first two moments of the logarithmic stock price process and to satisfy the constraint that the sum of branching probabilities is 1. Three nodes at time step 2 follow node Y. By the log-normality of the stock price, the mean and the variance of the SY-log-prices of these nodes (under the risk-neutral
probability) equal
ðr 2=2Þ t, Var 2t,
which can be obtained by substituting t for into equation (1). Note that the distance between two adjacent nodes’ SY-log-prices at time step 2 is 2
ffiffiffiffiffiffi t p
. Thus there exists a unique node Z at time step 2 whose SY-log-price ^
lies in the interval
pffiffiffiffiffiffit, þ pffiffiffiffiffiffit
h
: ð3Þ
In other words, the SY-log-price of node Z, i.e. ^, is
closest to among the SY-log-prices of the nodes at time
step 2. I call ^ the mean tracker of node Y. The middle branch from node Y will be connected to node Z. Figure 3 illustrates the case where ^ ¼lnðS0=S
YÞ 4 ffiffiffiffiffiffi t p (or, two nodes below S0).
In general, the SY-log-prices of the two nodes
connected by the upper and lower branches from node Y can be expressed as ^ þ ‘u ffiffiffiffiffiffi t p and ^ ‘d ffiffiffiffiffiffi t p for some even positive integers ‘uand ‘d. It is clear that the
jump sizes ‘u and ‘d should be as small as possible to
minimize the size of the stair tree. And ‘uand ‘dshould
also be properly selected to make the branching probabilities of node Y valid. Let pu
Y, pmY, and pdY
denote the risk-neutral probabilities for the upper, middle, and lower branches from node Y, respectively. Define , , and as the SY-log-prices minus the mean
of the nodes connected by the middle, the upper, and the lower branches as follows:
,^ þ ‘u ffiffiffiffiffiffi t p , ‘d ffiffiffiffiffiffi t p :
Figure 3. Branching scheme at the ex-dividend date. Nodes X and Y are at the first ex-dividend date (time step 1). Both nodes are represented by dotted ellipses. The cum-dividend stock prices at X and Y are Su and Sd, respectively, whereas the net-of-dividend stock prices at X and Y are SXðSu DÞ and SYðSd DÞ, respectively. The stock price for the top node at time step 2 is S0 (¼ S
Xu). The integer k in parentheses for each node at time step 2 means the stock price equals S0ekpffiffiffiffit. The cross right above Z denotes the point with SY-log-price at time step 2. The three branches of Y are marked with thick solid lines. pu
Y; pmY; and pdY denote the probabilities for the upper, middle, and lower branches from node Y, respectively.
Note that the first equation implies that 2 ½pffiffiffiffiffiffit, pffiffiffiffiffiffitÞ. Note also that 4 4 . The prob-abilities can be derived by solving
puY þ pmY þ pdY ¼0, ð4Þ puY2þpmY2þpdY2¼Var, ð5Þ puYþpmYþpdY¼1: ð6Þ Equations (4) and (5) match the first two moments of the logarithmic stock price, and equation (6) ensures that pu
Y, pmY, pdY as probabilities sum to one. The three
equations do not automatically guarantee 0 pu Y,
pm
Y, pdY1. A proof to show that they actually do with
‘u¼‘d¼2 is given in Appendix A. The stair tree hence
does not lead to branches with huge jump sizes. This finding is essential to the efficiency of the algorithm. The same procedure can be repeated for nodes below Y.y To handle multiple dividends, just apply the procedure to each ex-dividend date.
Because the first and the second moments are matched via Equations (4)–(6), the stair tree converges to Model 3.
Unlike the bushy tree illustrated in figure 1, the stair tree faithfully implements Model 3 without combinational explosion.
3. A sample stair tree
Consider an American vanilla call with an exercise price of 70 that initiates at year 0 and matures at year 0.75. A 3-time-step stair tree is constructed in figure 4 to price this call. Thus the length of each time step t is 0.25 year. The initial stock price is 100, the risk-free interest rate is r ¼ 10%, and the volatility of the stock price is ¼30%. The multiplicative factors for the CRR binomial tree are u ¼ e0:3pffiffiffiffiffiffi0:251:162 and d ¼ e0:3pffiffiffiffiffiffi0:25 0:861.
The branching probabilities are pu¼ ðR dÞ=ðu dÞ
0:5466 and pd¼1 pu0:4534, where R ¼ e0:10:25
1:02532 denotes the gross risk-free return per time step. In the figure, the number at the upper cell of a node denotes the stock price at that node, whereas the number at the lower cell denotes the call option price.
Assume a five-dollar dividend per share is paid at year 0.25 (time step 1). Note that nodes X and Y (marked by
Figure 4. A 3-time-step tree for pricing an American vanilla call. The number at the upper cell of a node denotes the stock price at that node. The number at the lower cell denotes the call value. The grey cell denotes that the American call is exercised early. The two branches of X are marked with thick solid lines, whereas the three branches of Y are marked with thick dotted lines. The branching probabilities are listed in the lower-left table.
yThe aforementioned method can also be done by first adding CRR binomial branches to the bottom node (like node Y), and then inserting trinomial branches to other nodes without efficiency and accuracy penalties.
dotted ellipses) are at time step 1. The stock prices at X and Y before the dividend is paid are 100 u 116:183 and 100 d 86:071, respectively. The ex-dividend prices at X and Y are therefore 111.183 and 81.071, respectively. The stock price for the top node at time step 2 is then 111:183 u 129:177. The stock prices at time step 2 can be represented as 129:177 uk for
nonpositive even integers k.
Let us move on to the branching scheme of node Y. Node Z at time step 2 has a stock price of 70.894. Hence the SY-log-price of node Z equals lnð70:894=81:071Þ
0:13414. It is the mean tracker (i.e. ^) of Y because 0:13414 2 ½ pffiffiffiffiffiffit, þ pffiffiffiffiffiffitÞ,
where ¼ ðr 2=2Þt ¼ ð0:1 0:32=2Þ 0:25 ¼ 0:01375
and pffiffiffiffiffiffit¼0:3 pffiffiffiffiffiffiffiffiffi0:25¼0:15. Thus the SY-log-prices
of the nodes at time step 2 that will be connected to Y are ^ þ2pffiffiffiffiffiffit0:1659, ^ 0:1341, and ^ 2pffiffiffiffiffiffit 0:4341. To compute the branching probabilities from node Y, I substitute 0:1659 , 0:1341 , and 0:4341 into , , and , respectively, in Equations (4)–(6). The branching probabilities are illu-strated in the lower-left table of the figure. The value of the vanilla call is obtained by backward induction on the tree. For example, the continuation option value at node Y is
e0:10:25 ð0:49299 27:425 þ 0:50698 6:593 þ0:00002 0Þ 16:447,
and the continuation option value at node X is
e0:10:25 ð0:5466 60:905 þ 0:4534 27:425Þ 44:597: Note that an American call will be exercised early only at an ex-dividend date. Note also that it is more beneficial for an option holder to exercise a call immediately before the underlying stock pays a dividend than immediately after (see equation (2)). The call will be exercised early at node X since the exercise value 46.183 (¼ 116:183 70) is larger than the continuation value 44.597. The call value computed by the stair tree is 31.893.
4. Numerical evaluations
I first compare Geske and Shastri’s fixed dividend yield model, Hull’s volatility adjustment model, the stair tree
model, Model 1, Model 2, and Model 3 for pricing European options. Geske and Shastri (1985) use fixed dividend yields to approximate discrete dividends. The fixed dividend yield is defined as the discrete dividend amount divided by the initial stock price. For example, the dividend yield is 5% if the initial stock price is 100 and the discrete dividend is 5. I use FDY to denote their approach. Note that Chiras and Manaster (1978) approximate the discrete-dividend problem by transform-ing the discrete dividends into a fixed continuous dividend yield. This approach is equivalent to the FDY model in pricing a European option. Frishling (2002) argues that Model 1 generates lower option prices than Model 3. To remove this difference, Hull (2000) recommends that the volatility of the net-of-dividend stock price be adjusted by the volatility of the stock price multiplied by Sð0Þ= ðSð0Þ DÞ, where D denotes the present value of future dividends over the life of the option. I use Hull to denote Hull’s volatility adjustment approach. Besides, I use Model1 and Model2 to denote the option prices generated by Model 1 and Model 2, respectively. Stair denotes the prices generated by the stair tree model. Model 3 denotes the prices generated by Model 3 that based on the Monte Carlo simulation with 100, 000 trials. The numerical results for these models are listed in tables 2 and 3, where table 2 focuses on the single-discrete-dividend case and table 3 focuses on the two-discrete-dividend case. All the prices that deviate from Model3by 0.3 are marked by asterisks. Frishling (2002) claims that Model 1, Model 2, and Model 3 generate very different option prices. This can be verified in tables 2 and 3 that the option prices generated by Model 2 are higher than the prices generated by Model 3. On the other hand, Model 1 generates lower option prices than Model 3. The difference among these three models becomes larger as volatility increases. FDY does not approximate Model 3 well as it produces lower option prices than Model 1. The option prices generated by Hull’s volatility adjust-ment approach do not approximate the prices generated by Model 3 well. It can be observed that only the stair tree model produces options prices that are close to Model 3. Note that Model 3 seems to produce lower option price (generated by the Monte Carlo simulation) in each two-discrete-dividend case (except one case) in table 3 than that in the corresponding case in table 2. The stair tree model successfully captures this trend, but all other
Table 2. Pricing European call options with single discrete dividend. The initial stock price is 100, the risk-free rate is 3%, the time to maturity is 1 year, and a 5-dollar-dividend is paid at year 0.6. The volatilities of the stock price are shown in the first row. The exercise prices are listed in the first column. FDY denotes the fixed dividend yield approach of Geske and Shastri (1985). Model1and Model2 denote the option prices generated by Model 1 and Model 2, respectively. Hull denotes volatility adjustment approach of Hull (2000). Stair denotes the stair tree model in this paper. Model3 denotes the prices generated by Model 3 that based on Monte Carlo simulation with 100,000 trials. Option prices that deviate from Model3 by 0.3 are marked by asterisks.
0.4 0.5
X FDY Model1 Hull Model2 Stair Model3 FDY Model1 Hull Model2 Stair Model3
95 *16.263 *16.336 17.090 17.112 16.821 16.933 *19.890 *19.969 20.901 20.937 20.570 20.843 100 *14.214 *14.270 15.044 15.048 14.758 14.754 *17.964 *18.003 *18.959 *18.971 18.591 18.584 105 *12.400 *12.439 *13.222 *13.206 12.924 12.989 *16.194 *16.222 17.194 17.182 16.829 16.929
models fail. Note that both Model 1 and the Hull’s volatility adjustment approach produce similar option prices in the single-discrete-dividend case and the two-discrete-dividend case. This is because the net-of-dividend stock price in the single-discrete-dividend case (¼ 100 5e0:030:6) is almost equal to that in the
two-discrete-dividend case (¼ 100 2:5e0:030:42:5e0:030:8). Model 2 also produces similar option prices in both cases since the cum-forward-dividend stock prices for both cases are almost equal.
To derive approximation analytical formulas for Model 3, Bos and Vandermark (2002) present an approach (denoted as Mix) that is a mixture between the stock and the exercise price adjustment or, in other words, Model 1 and Model 2. Bos and Shepeleva (2002) suggest that the volatility of the net-of-dividend stock price can be adjusted by a complex formula. I use Vol to denote their approach. These two approaches and the stair tree approach are compared in tables 4 and 5. I use the Monte Carlo simulation that prices Model 3 (denoted as Model3) to serve as a benchmark to compute the root mean squared error and the maximum absolute error. Since both these two errors of the stair tree model are lower than the errors of Mix and Vol, I conclude that the stair tree provides more accurate values than these two approaches. Note that Model 3 seems to produce lower option price in each two-discrete-dividend case in table 5 than that in the corresponding case in table 4 as I mentioned before. Bos and Vandermark’s approach
successfully catches this trend, but Bos and Shepeleva’s approach fails.
For American calls with discrete dividends, I compare the stair tree with the popular analytical pricing formula of Roll (1977), Geske (1979), and Whaley (1981) (abbreviated as RGW), and the FDY model of Geske and Shastri (1985) in table 6. The parameters are from Cox et al. (1979). The benchmark option prices (B) are from Geske and Shastri (1985). Note that RGW is based on Model 1 and thus underprices the options. RGW focuses on single-dividend cases. Welch and Chen (1988) and Stephan and Whaley (1990) extend RGW for two-dividend cases. But it is hard to extend RGW for three or more dividends because this would have required RGW to evaluate a multivariate cumulative normal density function, whose deterministic computational cost is prohibitive. This phenomenon is known as the curse of dimensionality (see Lyuu 2002). Of course, even if the multivariate integral can be computed efficiently, there is no guarantee that the price is numerically accurate. Geske and Shastri (1985) claim that the FDY model performs well for pricing American calls. Numerical results in table 6 show that the stair tree outperforms the FDY model.
The delta of a call with respect to the stock price is illustrated in figure 5. I use a 140-time-step stair tree to evaluate a call option with 7 months to maturity, and the length of each time step is 0.004167(¼ (7/12)/140) year. 301 tree evaluations are performed by setting the initial stock price as 20 þ 0:1x, where 0 x 300. The resulting delta curve is very smooth. The stair tree’s quick convergence is verified in table 7, where the prices remain unchanged up to pennies when the number of
Table 3. Pricing European call options with two discrete dividends. The numerical settings are the same as those settings in table 2 except that a 2.5-dollar-dividend is paid at year 0.4 and year 0.8. Option prices that deviate from Model3 by 0.3 are marked
by asterisks.
0.4 0.5
X FDY Model1 Hull Model2 Stair Model3 FDY Model1 Hull Model2 Stair Model3
95 *16.303 *16.336 17.090 17.112 16.806 16.836 *19.931 *19.969 *20.901 *20.937 20.568 20.549 100 *14.250 *14.270 *15.044 *15.048 14.733 14.733 *18.001 *18.003 *18.959 *18.971 18.583 18.621 105 *12.433 *12.439 *13.222 *13.206 12.904 12.883 *16.228 *16.222 *17.194 *17.182 16.826 16.829
Table 4. Pricing European call options with single discrete dividend. The numerical settings are the same as those settings in table 2. Mix denotes the mixture approach of Bos and Vandermark (2002). Vol denotes the volatility adjustment approach of Bos and Shepeleva (2002). Model3 denotes the prices generated by Model 3 that based on Monte Carlo simulation with 100,000 trials. RMSE denotes the root mean
squared error. MAE denotes the maximum absolute error.
X Mix Vol Stair Model3
0.4 95 16.802 16.792 16.821 16.933 100 14.737 14.732 14.758 14.754 105 12.899 12.899 12.924 12.989 0.5 95 20.550 20.537 20.570 20.843 100 18.584 18.578 18.591 18.584 105 16.798 16.798 16.829 16.929 RMSE 0.147 0.152 0.130 MAE 0.293 0.306 0.272
Table 5. Pricing European call options with two discrete dividends. The numerical settings are the same as
those in table 3.
X Mix Vol Stair Model3
0.4 95 16.801 16.795 16.806 16.836 100 14.736 14.734 14.733 14.733 105 12.898 12.901 12.904 12.883 0.5 95 20.548 20.541 20.568 20.549 100 18.583 18.581 18.583 18.621 105 16.797 16.800 16.826 16.829 RMSE 0.026 0.027 0.023 MAE 0.038 0.041 0.038
time steps is at least 140. These experiments confirm the reliability of the stair tree.
The discrete dividend assumption is not so realistic since the dividend might not be perfectly predictable especially when the ex-dividend date is far into the future. A more realistic and generalized assumption, the path-dependent dividends assumption, is discussed in the next section. I will also show how the stair tree model can incorporate this assumption.
5. Path-dependent dividends
It is more general and realistic to assume that a stock pays a stochastic dividend rather than a dividend with known amounts at a future ex-dividend date. However, the option can only be hedged if the dividend is known exogenously or completely determined by the stock price process prior to the ex-dividend date as argued in Cox and Rubinstein (1985) unless one adds nonstandard derivatives such as the forward contracts on dividends in Chance et al. (2002). I call Cox and Rubinstein’s assumption the path-dependent dividends assumption since the future dividend, says Dti, completely depends on
the stock prices and the dividends prior to time step ti.
To be more precise, Dtican be expressed as
Dti fðS0, S1, S2, . . . , Sti, Dti1, Dti2, . . .Þ
for some function f. In reality, dividends can be explained by a variety of factors such as the net operating profits, long-run sustainable (or permanent) earnings, and so on. If the stock prices and the dividends paid previously serve as good proxies for these factors, a dividend function that fits the path-dependent dividends assumption can be constructed. Indeed, some empirical dividend models can fit path-dependent dividends assumptions with slight modifications. I will first review one of such dividend models proposed by Marsh and Merton (1987).
Table 6. Pricing American call options. The initial stock price is 40, the risk-free interest rate is 5%, and the volatility is 30%. The ex-dividend dates for the stock are 0.5, 3.5, and 6.5 months. The dividends to be paid at each ex-dividend date are shown in the first row. The exercise prices X are listed in the first column. The times to maturity T (in months) are in the second column. The values of American calls priced by the FDY model and the benchmark value are from Geske and Shastri (1985). RGW denotes the analytical pricing formula of Roll (1977), Geske (1979), and Whaley (1981) (for single-dividend cases) and the extended formula of Stephan and Whaley (1990) (for two-dividend cases). Stair denotes the stair tree model with 140 time steps. Option prices which
deviate from the benchmark values by 0.02 are marked by asterisks.
$1.0 $2.0 $3.0 $4.0
X T RGW FDY Stair B RGW FDY Stair B RGW FDY Stair B RGW FDY Stair B
35 1 *5.07 5.09 5.09 5.09 *5.05 5.09 5.09 5.08 *5.05 5.08 5.09 5.08 *5.05 5.08 5.09 5.08 4 *5.38 5.41 5.40 5.40 5.15 5.19 5.18 5.17 *5.08 5.12 5.12 5.11 *5.07 5.10 5.10 5.10 7 – *5.79 5.77 5.76 – *5.29 5.26 5.24 – *5.15 5.14 5.12 – 5.11 5.11 5.10 40 1 *1.14 1.17 1.17 1.17 *1.03 1.08 1.08 1.07 1.03 1.04 1.04 1.04 *0.93 1.02 1.03 1.02 4 *2.36 2.38 2.40 2.39 *1.89 1.91 1.93 1.92 1.57 1.60 1.60 1.58 *1.35 1.39 1.40 1.38 7 – 3.05 3.08 3.06 – 2.31 2.33 2.32 – 1.83 1.83 1.81 – *1.51 *1.51 1.48 45 1 0.08 0.09 0.09 0.09 0.04 0.05 0.06 0.05 0.04 0.04 0.04 0.04 0.02 0.03 0.03 0.03 4 0.87 0.87 0.88 0.88 0.62 0.62 0.64 0.64 *0.43 0.44 0.46 0.46 0.31 0.31 0.33 0.32 7 – *1.47 1.51 1.50 – *0.99 1.03 1.02 – *0.66 0.70 0.69 – *0.43 0.46 0.46
Table 7. Convergence of delta. The settings are identical to those in figure 5 except that the initial stock price is 40. The number of time steps n is selected to be a multiple of 14 so that each ex-dividend date coincides with a time step in the stair tree. Price and Deltadenote the option price and the delta computed by the stair tree, respectively. The numerical values remain unchanged (up
to pennies) for n 140. n Price Delta 56 5.13 0.97 70 5.13 0.97 84 5.13 0.96 98 5.12 0.97 112 5.12 0.97 126 5.12 0.97 140 5.11 0.97 154 5.11 0.97 0 0.2 0.4 0.6 0.8 1 20 25 30 35 40 45 Stock Price Delta
Figure 5. Delta: the x-axis denotes the initial stock price, and the y-axis denotes the delta of the vanilla call. The exercise price is 35, the risk-free interest rate is 5%, the volatility is 30%, and the time to maturity is 7 months. A 4-dollar dividend is paid at months 0.5, 3.5, and 6.5.
Then I will show how the stair tree can incorporate their dividend model.
Marsh and Merton (1987) derive a dividend model by following Linter’s (1962) stylized facts established by Linter in a classic set of interviews with managers about their dividend policies. Their dividend model can be expressed by a regression formula of the permanent earnings and the dividends paid previously. They argue that their formula can not be directly estimated because management assessments of changes in a firm’s perma-nent earnings are not observable. Thus they assume that the permanent earning to cum-dividend stock price ratio is a positive constant. Under this assumption, a future dividend in their dividend model can be expressed by a regression formula in terms of stock prices and dividends prior to the ex-dividend date. To illustrate how the stair tree incorporates the Marsh and Merton’s dividend model, I express their dividend formula by a discrete time model and assume that the length between two ex-dividend dates is two time steps:
log Dtþ2 Dt þ Dt St2 ¼a0þa1log StþDt St2 þa2log Dt St2 þuðt þ2Þ, ð7Þ where the dividends are paid at time step t and t þ 2, D denotes the dividend amounts paid at time step , S
denotes the net-of-dividend stock price at time step , and uðt þ2Þ denotes the disturbance term at time step t þ 2. By assuming that the disturbance term uðt þ 2Þ ¼ 0, equation (7) can be rewritten as
Dtþ2¼10 a0þa1log StþDtSt2 h i þa2log St2Dt h i St2Dtþlog Dt : ð8Þ Note that Dtþ2can be expressed as a function of St2, St,
and Dt. One of their empirical studies focuses on the
value-weighted NYSE index over the period 1926–81 and they estimate that a0¼ 0:101, a1¼0:437, and
a2¼ 0:042 by ordinary least squares method. A simple
numerical example is then given to demonstrate how the stair tree can incorporate the dividend model in equation (8) with aforementioned numerical settings.
A 4-time-step stair tree that prices a European vanilla call option with an exercise price of 50 is illustrated in figure 6. The underlying stock price at time step 0 is 100, the length of each time step of the stair tree is 0.25 year, the risk-free interest rate is 10%, and the volatility of the stock price is 30% . Note that the upward multiplication factor u ¼ e0:3pffiffiffiffiffiffi0:251:162 and the downward one
d ¼e0:3pffiffiffiffiffiffi0:250:861. I further assume that the
histor-ical net-of-dividend stock prices S1 and S3, and the
historical dividend D1to be 110, 80, and 5, respectively.
The underlying stock is assumed to pay two dividends (D1and D3) at time steps 1 and 3, respectively. The top
cell of each node denotes the stock price (at a non-dividend paying date) or the cum-non-dividend stock price (at an ex-dividend date) of that node. Each node contains at least one state (denoted by the cell following the top cell) to keep the option price. The nodes enclosed by
dotted ellipses contain two states to keep required information for computing D3 by equation (8) (to be
discussed later). Note that the net-of-dividend stock prices and the branching probabilities for the states at ex-dividend dates (time step 1 and time step 3) are illustrated in table 8.
Now I proceed to show how this 4-time-step stair tree is constructed. The cum-dividend stock prices at time step 1 are 100 u 116:183 and 100 d 86:071, respectively. The dividend D1is obtained by substituting D1(¼5), S1
(¼110), and S3(¼80) into equation (8) to get 4.518. Thus
the net-of-dividend stock prices for states A and B are 116:183 4:518 111:666 and 86:071 4:518 81:553, respectively. The stock price for the top node at time step 2 is then 111:666 u 129:737. Thus the stock prices at time step 2 can be represented as 129:737 uk for
nonpositive even integers k. The branches of state A follow the CRR tree structure. The mean tracker of state Bcan be found by equation (3) to be lnð71:201=86:071Þ ðÞ^ (expressed in SB-log-price). Thus the stock prices
of the nodes connected to state B are 96.112 (with SB-log-price ^ þ2
ffiffiffiffiffiffi t p
), 71.201 (with SB-log-price ^), Figure 6. A 4-time-step stair tree that incorporates the Marsh and Merton dividend model. The number at the top cell of each node denotes the stock price (at a non-dividend paying date) or the cum-dividend stock price (at a ex-dividend date) of that node. The number at the following cell(s) denotes the option price(s). Additional states are added to the nodes enclosed by dotted ellipses to keep the required information for computing D3. The net-of-dividend stock prices and the branching probabilities for all the states at ex-dividend dates are in table 8.
and 52.747 (with SB-log-price ^ 2
ffiffiffiffiffiffi t p
). The trinomial branching probabilities of state B can be computed by equations (4)–(6). The net-of-dividend stock prices and the branch probabilities for states A and B are illustrated in table 8.
To compute D3 by equation (8), S1, D1, and S1 are
required. While S1 and D1 are known to be 110 and
4.518, respectively, there are two possible S1 (111.666
and 81.553) in this stair tree. Additional states are added to the nodes enclosed by dotted ellipses to keep the information about S1. I colour all the cells and
corresponding branches from time step 1 to time step 3 in light-grey and dark-grey to denote the cases that S1¼111:666 and S1¼81:553, respectively. For example,
state F denotes the case that S1 ¼111:666 and the
cum-dividend stock price at time step 3 is 82.724, while state G denotes the case that S1¼81:553 and the cum-dividend
stock price at time step 3 is 82.724. Note that all the branches from the states at time step 2 follow the CRR tree structure.
Now I focus on time step 3. The dividend paid at state C is obtained by substituting D1¼4:518, S1¼111:666,
and S1¼110 into equation (8) to get 4.043. Thus the
net-of-dividend stock price for state C is 150:733 4:043 ¼ 146:690. Similarly, the net-of-dividend stock prices for states D and F are 111:666 4:043 ¼ 107:622 and 82:724 4:043 ¼ 78:681, respectively. The dividend paid at state E is obtained by substituting D1¼4:518,
S1¼81:553, and S1¼110 into equation (8) to get 3.547.
Thus the net-of-dividend stock price for state E is 111:666 3:547 ¼ 108:119. Similarly, the net-of-dividend stock prices for states G, H, and I are 82:724 3:547 ¼ 79:177, 61:283 3:547 ¼ 57:737, and 45:340 3:547 ¼ 41:853, respectively. The stock price for the top node at time step 4 is 146:690 u 170:249. All the stock prices at time step 4 can be represented as 170:249 uk for
nonpositive even integers k. The branches for state C follow the CRR tree structure. The trinomial branching schemes for states D, E, F, G, H, and I are constructed by following the method for constructing the branches for state B. The trinomial branching probabilities for these states are listed in table 8.
The value for the European vanilla call option can be obtained by backward induction. Note that some nodes have two different option prices due to different historical stock price paths. For example, the option price for state Fis
e0:10:25ð43:553 0:4699 þ 19:291 0:5296 þ1:332 0:0005Þ 29:917,
while the option price for state G is
e0:10:25ð43:553 0:4905 þ 19:291 0:5095 þ1:332 4:6 10530:410:
The call value computed by the stair tree is 46.804.
6. Conclusions
Pricing stock options with discrete dividend payouts has not been satisfactorily settled because of the conflicting demands of computational tractability and realistic modelling of the stock price process. It is realistic to assume that the stock price jumps down at an ex-divided date. However, pricing options under this stock price model can not be efficiently and/or accurately implemen-ted by analytical formulas and numerical methods. This paper suggests a recombining tree, the stair tree, that efficiently and faithfully implements this model. Numerical results confirm that the stair tree is both efficient and accurate. Moreover, the stair tree can be extended to more general cases when future dividends are completely determined by past stock prices and dividends. This extension, which is called ‘path-dependent dividends assumption’ in this paper, makes the stair tree model more realistic and flexible.
Acknowledgements
The author was supported in part by NSC Grant NSC 94-2213-E-033-024.
I thank two anonymous reviewers for comments that greatly improve the presentation and quality of the paper.
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Appendix A: Proof of valid risk-neutral probabilities Define
det ¼ ð Þð Þð Þ, detu ¼ ð þVarÞð Þ,
detm¼ ð þVarÞð Þ,
detd ¼ ð þVarÞð Þ:
Then Cramer’s rule applied to Equations (4)–(6) gives pu
Y¼detu=det, pmY¼detm=det, and pdY¼detd=det. Note
that det 5 0 because 4 4 . To ensure that the branching probabilities are valid, it suffices to show that
puY, pmY, pdY 0. As det 5 0, it is sufficient to show detu, detm, detd0 instead. Finally, as 4 4 , it
suffices to show that þ Var 0, þ Var 0, and þVar 0 under the premise 2 ½pffiffiffiffiffiffit, pffiffiffiffiffiffitÞ. Indeed,
þVar ¼ 22pffiffiffiffiffiffitþ2t ¼ ð pffiffiffiffiffiffitÞ2 0, þVar ¼ 242t þ 2t ¼ 232t 5 0, þVar ¼ 2þ2pffiffiffiffiffiffitþ2t ¼ ð þ pffiffiffiffiffiffitÞ2 0, as desired.
