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日期：2024-09-02 07:44

Applied Mathematical Finance

Closed Formula for Options with Discrete Dividends and Its Derivatives

CARLOS VEIGA & UWE WYSTUP

Frankfurt School of Finance & Management, Centre for Practical Quantitative Finance, Sonnemannstraße 9-11, 60314 Frankfurt am Main, Germany

(Received 23 May 2008; in revised form 4 March 2009)

ABSTRACT        We present  a closed pricing formula for European options under the Black–Scholes model as well as formulas for its partial derivatives. The formulas are developed making use of Taylor series  expansions  and a proposition  that  relates  expectations  of partial  derivatives  with partial derivatives themselves. The closed formulas are attained assuming the dividends are paid in any state of the world. The results are readily extensible to time-dependent volatility models. For completeness, wereproducethenumericalresultsin VellekoopandNieuwenhuis,coveringcallsandputs, togetherwith results on their partial derivatives. The closedformulas presented here allow a fast calculation of prices or implied volatilities when compared with other valuation procedures that rely on numerical methods.

KEY WORDS:  Equity option, discrete dividend, hedging, analytic formula, closed formula

1.    Introduction

1.1    Motivation

The motivation to return to this issue is the fact that whenever a new product, model or valuation procedure is developed, the problem that arises with discrete dividends is dismissed or overlooked by applying the usual approximation that transforms the discrete dividend into a continuous stream of dividend payments proportional to the stock price. After all that has been said about the way to handle discrete dividends, there are still strong reasons to justify such an approach.

We here recall the reasons that underlie the use of this method by the majority of market participants and pricing tools currently available. We choose the word method instead of modelalthough one could look for what model would justify such calculations and find the Escrowed Model, as it is known in the literature. We do not follow this reasoning because we consider that such a model would be unacceptable since it admits arbitrage. The reason being that such a model would imply two different diffusion price processes for the same underlying stock under the same measure if two options were considered with different maturities and spanning over a different number of dividend payments. We thus refuse the model interpretation and consider the procedure that replaces the discrete dividend into a continuous stream of dividend payments as an approximation to the price of an option, under a model that remains arbitrage free when several options coexist. The models in Section 1.2 belong to that class.

The drivers behind the huge popularity of this method are mostly due to the (i) tractability of the valuation formulas, (ii) applicability to any given model for the underlying stock, and (iii) the preserved continuity of the option price when crossing each dividend date.

However, the method has some significant drawbacks. First and foremost, no proof has  ever  been  present  that  this  method  would  yield  the  correct  result  under  an acceptable  model  in  the  sense  above.  In  fact,  for  the  natural  extensions  to  the Black–Scholes  (BS) model described in  Section  1.2, the error grows larger as the dividend date is farther away from the valuation date. This is exactly the opposite behaviour of what one would expect from an approximation – a larger period of time between the valuation date and the dividend date implies that the option valuation functions are smoother and thus easier to approximate. The other side of the inaccu- rate pricing coin is the fact that this method does not provide a hedging strategy that will  guarantee  the  replication  of the  option  payoff at  maturity.  To  sum  up,  no numerical procedure based on this method returns (or converges to) the true value of the option, in any of the acceptable models we are aware of. It still seems like the advantages outweigh the drawbacks since it is the most widely used method.

An example may help to demonstrate this. Consider a stochastic volatility model with jumps. Now consider the valuation problem of an American style option under this model. The complexity of this task is such that a rigorous treatment of discrete dividends, i.e. a modification of the underlying’s diffusion to account for that fact, would render the model intractable.

1.2    Description of the Problem

In the presence of discrete dividend payments, diffusion models like the BS model are no longer an acceptable description of the stock price dynamics. The risks that occur in this  context  are  mainly  the  potential  losses  arising  from  incorrect  valuation  and ineffective hedging strategy. We address both of these issues in this article.

The most natural extension to a diffusion model to allow for the existence of discrete dividends is to consider the same diffusion, for example

dSt  = St (rdt + σdWt ),                                           (1)

and add a negative jump with the same size as the dividend, on the dividend-payment date as StD      = StD-   - D. This gives rise to the new model diffusion

dSt  = St (rdt + σdWt ) - DⅡ{t≥tD } ,                                  (2)

where S is the stock price, r is the constant interest rate, σ is the volatility and  Wis a standard Brownian motion. StD-   refers to the time immediately before the dividend- payment moment, tD, and StD   to the moment immediately after.

There are though some common objections to this formulation. A first caveat may be the assumption that the stock price will fall by the amount of the dividend size. This objection is mainly driven by the effects taxes have on the behaviour of financial agents and thus market prices. We will not consider this objection in this article and thus assume Model (2) to be valid. A second objection may be that the dividend-payment date and amount are not precisely known until a few months before their payment. We also believe  this  to be  the case, but  a more  realistic model in this respect would significantly grow in complexity. Our goal is rather to devise a simple variation that can be applied to a wide class of models that does not worsen the tractability of the model and produces accurate results.

Finally, the model admits negative prices for the stock price S. This is in fact true and can easily be seen if one takes the stock price StD-  to be smaller thanD at time tD- . A simple solution to this problem is to add an extra condition in Equation (2) where the dividend is paid only if StD-   > D, i.e.

dSt  = St (rdt + σdWt ) - DⅡ{t≥tD }Ⅱ{StD- >D} .                           (3)

However, in most practical applications, this is of no great importance as the vast majority of the companies that pay dividends have dividend amounts that equal a small fraction of the stock price, i.e. less than  10% of it, rendering the probability assigned to negative prices very small. For this reason we may drop this condition whenever it would add significant complexity.

In the remainder of this section we review the existing literature on the subject and the reasons that underlie the use of the method most popular among practitioners. We then turn to develop the formulas in Section 2, and in Section 3 we reproduce the numerical results in Vellekoop and Nieuwenhuis (2006) together with put prices and partial derivatives. Section 4 concludes.

1.3    Literature Review

Here we shortly review the literature on modifications of stock price models coping  with discrete dividend payments. Merton (1973) analysed the effect of discrete divi-  dends  in American  calls  and  states  that  the  only  reason  for  early  exercise  is  the  existence of unprotected dividends. Roll (1977), Geske (1979) and Barone-Adesi and  Whaley  (1986)  worked  on  the  problem  of  finding  analytic  approximations  for  American options. John Hull (1989) in the first edition of his book establishes what  was to be the most used method to cope with discrete dividends. The method works by  subtracting from the current asset price the net present value of all dividends occurring  during the life of the option. On the other end of the spectrum, Musiela and Rutkowski  (1997) propose a model that adds the future value at maturity of all dividends paid  during the lifetime of the option to the strike price. To balance these two last methods,  Bos and Vandermark (2002) devise a method that divides the dividends in ‘near’ and  ‘far’  and  subtracts  the  ‘near’  dividends  from  the  stock  price  and  adds  the  ‘far’ dividends  to  the  strike  price.  A  method  that  considers  a  continuous  geometric  Brownian motion with jumps at the dividend-payment dates is analysed in detail by  Wilmott (1998) by means of numerical methods. Berger and Klein (1998) propose a non-recombining binomial tree method to evaluate options under the jump model. Bos et  al.  (2003)  devise  a method  that  adjusts  the volatility  parameter  to correct  the subtraction method stated above. Haug et al. (2003) review existing methods’ perfor- mance and pay special attention to the problem of negative prices that arise within the context of the jump model and propose a numerical quadrature scheme. Bjrk (1998) has one of the clearest descriptions of the discrete dividends problem for European options and provides a formula for proportional dividends. Shreve (2004) also states the result for proportional dividends. Vellekoop and Nieuwenhuis (2006) described a modification to the binomial tree method to account for discrete dividends preserving the crucial recombining property.

2.    Closed Formula

The derivation of the closed formula assumes a BS model as in Equation (1) with constant interest rate rand constant volatility σ . However, the following can be easily modified to allow for time-dependent volatility. Furthermore, our arguments consider and are only valid for European-style options.

We assume aproblem with n dividends Di, with i = 1, . . ., n, having theirrespective payment dates on ti ordered in this manner t0 < t1 < ... < tn < T and having t0 and T as the valuation date and the option’s maturity date, respectively.

We take a vanilla call option as our working example. We start by focusing on the time point just after the last dividend payment, which we will refer to astn. We choose this point in time because it is the earliest moment on which we can make a conjecture with respect to the price of the option, i.e. from this point on, we know how to price and hedge a claim, for there are no dividends left until the option matures. The price of our call would thus be a function C(Stn, tn )1 of the stock price and time, the celebrated BS formula that we state here for completeness sake:

As usual,K and T are the strike price and the maturity date, respectively. As it is going to be used extensively throughout this article, we take here the opportunity to also present the general formula for the ith derivative with respect to the first variable, St, of Formula (4) developed by Carr (2001)

where  N/ (x)  denotes the probability density function of the standard normal distribution,  S1 (i;j) the Stirling number of the first kind and Hi (d) are Hermite polynomials.

The problem we face now is how to move one step back in time to t < tn. For that we take Assumption (2) in Section 1.2. This assumption yields C(Stn; tn ) = C(Stn-   - Dn; tn ), and its right-hand side already refers to the stock price at a time point just before tn. We now wish to move further back in time but still without crossing any other dividend date, that is, to tn- 1. This task is a straightforward application of option pricing theory yielding

e-r(tn -tn-1)Etn(Q)- 1  [ C (Stn-   - Dn; tn )];                                      (6)

which is the discounted expectation of the random variable C(Stn-   - Dn; tn ) under the risk-neutral measure Q with respect to the σ-algebra Ftn- 1 .

Unfortunately, Expression (6) is not directly solvable into a closed formula for it includes the random variable log {Stn-   - Dn },  which has no known or explicit distributiona.

At this point our hope is to replace the formula C(Stn-   - Dn; tn ) by an equivalent representation that would not involve  log {Stn-   - Dn }. The natural candidate is the Taylor series expansion of C taken at the point Stn-   and with  a shift of size -Dn. Unfortunately, we know from the works of Estrella (1995) that such replacement is not valid for all values of Stn- , and thus

The reason for this is the fact that the Taylor series expansion of the BS formula for calls is convergent only for shifts of a size smaller than Stn-   and diverges otherwise. In our  case, where  Stn-   < Dn   the  Taylor  series  does  not  produce  the  same values  as C (Stn-   - Dn; tn ), and in turn, the expectation in (6) will also be affected. We acknowl- edge, though,that the risk-neutral probability of Stn-   < Dn  is very small, and thus the effect in (6) of the divergence of the Taylor series for the values Stn-   < Dn will not affect our approximation too much.

Confronted with this result, we tried to carry the derivation forward on rigorous grounds, rewriting Expression (6) by introducing an indicator function for the set A = {W : Stn-   > Dn },

e-r(tn -tn-1)Etn(Q)- 1  [ C (Stn- ; tn ) + ( C (Stn-   - Dn; tn )- C (Stn- ; tn )) . ⅡA].            (8)

This approach did lead to a closed formula for the case of problems with only one dividend payment (see Veiga and Wystup, 2007). However,the cost of this rigour was a highly complex formula that cannot be generalized to fit multiple dividend problems. For this reason we here take a different route.

with ηn  high enough to approximate C(Stn-   - Dn; tn ) reasonably well, for all Stn- . We thus trade the error of this approximation for the tractability that it enables. We do so because we believe that in almost all realistic scenarios the error is not significant. In fact, our results in Section 3 based on this assumption do provide very good results with scenarios even more demanding than realistic market conditions.

Hence, we rewrite Expression (6) as

Since we have a finite series as integrand function, we can safely interchange the integral with the summation, yielding

Finally, to turn Expression (11) above into an explicit formula we use the following:

Proposition 2.1.    Let C(St; t) and all its derivatives be continuous functions in its first variable, then

Proof.    We prove the proposition by mathematical induction. For i = 0 we get

C(St; t) = e-r(tk -t)Et(Q) [C(Stk; tk )];                                    (13)

which is true, for it states that the discounted BS price is a martingale under the risk- neutral measure Q.

Now we need to check that the proposition for i implies the same proposition for i + 1. Changing variables by Pt  = Ste-iσ2 (tk -t) we get

Et(Q) [∂1(i) C(Stk; tk )] = e-(

where now Stk   = Pt exp {(r - 2 ) (tk  - t)+ σ(Wtk   - Wt ) }.

We differentiate the left-hand side with respect toPt, and since ∂1(n)C is continuous for

all n ∈ N, we apply Leibniz integral rule, getting

R

Taking z = y + σ√-k(--------)-t(--) leaves us with

which equals

with Stk   = Pt exp , a Brownian motion.

Taking the derivative of the right-hand side of (14) with respect to Pt, we obtain

Equating the derivatives of both sides of (14) with respect to Pt, i.e. (17) and (18), rearranging and using the fact Pt  = eσ2 (tk -t)Ste-(i+1)σ2 (tk -t), we get

which is exactly the claim for i + 1 with a positive factor multiplying the first argument of C on both sides of the equation.

We can now write Equation (19) considering a new initial stock price t(I) = eσ2 (tk -t)St

and rely on the geometric nature of the diffusion St to have also St(I)k   = eσ  (tk -t)Stk , then

yielding

Hence, to get a closed formula for a call option maturing at T with one discrete dividend payment at time tn of amount Dn, we explicitly rewrite Expression (11) and we denominate as Cn (Stn- 1; tn- 1 ),

In what follows, we will require a more condensed notation, so we introduce the abbreviations below and suppress the time variable from all C functions.

gjtn   = exp {-jσ2 (tn  - tn- 1 )}                                            (23)

Now, Formula (21) becomes

We  can  now  resume  our  movement  backwards  in  the  time  axis  using  the  same programme that led us here, namely, apply (2) to move over the dividend date tn- 1; apply Approximation (9) now for Cn yielding2 

take the discounted expectation under the measure Q with respect to the σ-algebra Ftn-2   and apply Proposition 2.1 to solve the expectation and get 

Running this programme for all n dividends returns the formula for an arbitrary number of dividend payments

with Il  = Σm(n)=l im  and GI  = Πh(n)= 1 gt(I)h(h) .

Before we conclude this section, we remark that even though we developed our analysis focused on a European call, it remains valid for other types of options. In fact, the above analysis is valid for all options that satisfy all conditions it involved, namely, European-style options that are priced by only taking expectations under the risk-

neutral measure, Approximation (9); ∂1(n)C is continuous for all n ∈ N to apply Leibniz

integral rule. The existence of a closed formula for an arbitrary derivative of the option price,  e.g.  Formula  (5),  greatly  accelerates  the  calculation  process.  However,  the analysis remains valid if the derivatives are replaced by numerical approximations. This alternative may be useful for problems solved by finite difference methods that return a vector of option prices for different stock prices, thus enabling the calculation of numerical derivatives for all the necessary stock price levels.

Therefore, a European put is an example of another option type covered in this analysis and for which a closed formula for an arbitrary derivative is also available in Carr (2001). In Section 3 we also consider European puts and observe that their prices are coherent with the respective call prices.

2.1    The Greeks

A  closed  formula  for  the  derivative3   of the  option  price  of arbitrary  order  is  a straightforward application of the chain rule. Thus, for the dth derivative of the call price with one discrete dividend payment we have 

The derivatives with respect to other variables, namely σ and r, require similar derivations that we skip here since they constitute simple calculus exercises. There is one exception worth mentioning though: the theta, i.e. the derivative with respect to valuation time t. The theta can be calculated by making use of the BS partial differ- ential equation, yielding