A TAX-ADJUSTED ALGORITHM FOR PRICING C)ERIVATIVE SECURITIES USING T H E SYMBOLIC COMPUTATIONAL L,4NGUAGE MAPLE

M. A. Milevsky and E. Z. Prisman*

... taxes are largely a source of embarrass- ment t o financial economists.. .partial equilib- rium models lead very quickly t o numerical problems ...( Dybvig and Ross, Journal of Fi- nance, 1986)

I ABSTRACT: This paper develops a symbolic computational pricing al- gorithm tha t locates the No-Arbitrage price of a contin- gent claim (derivative security) in a world with realistic income taxes. By maintaining the price of the contingent claim as a symbolic function, we can use the concept of a fixed-point theorem t o locate the price of the option t,hat will make it impossible to create after-tax arbitrage opportunities.

2 INTRODUCTION: The effect of taxes on derivative security pricing mod- e:ls has been mostly ignored in the academic literature. We suggest tha t this oversight is mainly attributed to ab theorem by Scholes (1976). Scholes stated that when taxes are paid and refunded in continuous time at iden- tical capital gains and ordinary income rates, the deriv- a-tive security pricing formula is independent of the tax predicament of the holder.

In the real world, taxes are paid at distinct points in time and at different rates. Thus, we start from the premise tha t taxes are non-symmetric, non-linear and non-additive and hence taxes should matter for pricing derivatives.

A methodological dilemma we confront is the issue of iterative simultaneity. The theoretical Ne-Arbitrage

'Dr. Milevsky is Assistant Professor of Finance and can be reached at Tel: (416) 736-2100 ext: 66014, Fax: (416) 225-5034, Email: milevsky@yorku.ca. Dr. Prisman is Nigel Martin Pro- f a so r of Finance and can be reached a t Tel: (416) 736-2100 ext: 77948, Fax: (905) 882-9414, Email: eprismanQyorku.ca, http://www.yorku.ca/faculty/academic/prisman. Both authors are a t the Schulich School of Business, York University, North York, Ontario, M35 1P3, Canada. The authors would like to ac- knowledge financial assistance from the York University Research Authority and helpful comments from Narat Charupat, Eliakim Katz and Tom Salisbury.

price of any derivative security is a function of its after- tax cash flows. However, in most tax systems, t he after- tax cash flows from a financial security depend on the cost basis of that particular security. This is because tax liability depends on profit, and profit is a function of purchasing price. In fact, t o make matters worse, in some cases the after-tax cash flows depend on the actual mar- ket price-path of the derivative itself. We are inevitably confronted with apparent circular reasoning. How can one determine the theoretical tax-adjusted price of a derivative security, when the payoff from the security depends on its initial price, and its path, which is the unknown parameter (function) m$ are looking for in the first place? W e are confronted, with a path-dependent o p - tzon, where the path dependency in questaon is the path of the actual denvatave. The price of t he option satis- fies a certain functional relationship which depends, in an iterative fashion, on the function itself. The power of a symbolic computational language lies in its ability to maintain and manipulate functions without assign- ing them particular values. By maintaining the price of the derivatiive security as a symbolic function, we call

use the concept of a fixed-point theorem t o locate t h e price of the option that will make i t impossible t o create after-tax arbitrage opportunities.

The remainder of t,his paper is organized as follows Section 2 provides a motivational example of the taxa- tion issues that arise when hedging and pricing options. Section 3 is a brief review of the literature as it per- tains to the subject of taxes and derivatives. Section 4 describes our computer algorithm in great technical de- tail, coded in Maple V, with some examples. Section 5 summarizes the paper with some final thoughts.

The interested reader is referred to the companion pa- per by Milevsky and Prisman (1997) for an in-depth analysis of pricing, existence and uniqueness issues with realistic income taxes in continuous time.

3 MOTIVATION: The multitu'de of tax jurisdictions and financial products make i t almost impossible to create a universal t ax ad- justed derivative pricing model. Rather, each situation

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must 1s analyzed separately within the framework that we propose.

This section explores a representative numerical exam- ple in which taxes dzstort denvatzve payoffs. The main features of our approach are thereby illustrated and con- trasted with traditional methodologies.

The most liquid exchange traded equity index option in the world is SPl00 OEX traded on the C.B.O.E. A simple two period binomial model illustrates tha t the after-tax, based on the U.S. Internal Revenue Code, sec- tion 1256, pay-off from writing a delta hedged SPl00 In- dex “vanilla” long-term call option, is path dependent. In other words, the payoff is contingent on the interme- diate values of the underlying security during the life of the option.

Assume tha t the SPlOO Cash Index is currently trad- ing at s = 100 and that the one period interest rate is lo%, where, for convenience a period is taken to be one year. We construct a binomial model for the evolution of the SP100. Let the binomial parameters be U = 1.25 and d = 0.8 respectively. This corresponds t o a volatil- ity of 22.3% per annum, which is not unreasonable for the SP100. Now, consider a long-term call option with an exercise price of X = 75 and a two period maturity horizon. T h e example is constructed so that the op- tion’s lifetime transcends tax year bou$daries and hence induces a tax liability t o the holder at the end of the first time period. Using the standard, non tax-adjusted, binomial techniques for valuing an option, we obtain the market prices of the call at every node. It is important t o note tha t pre-tax payoff from the option at node [2,1] does not depend on the path taken by the underlying security.

For illustrative purposes, we will now assume an in- vestor in the 50% marginal tax bracket. Now, due to the fact tha t the option is being delta hedged, the U.S. tax authorities view the entire position as a strad- dle and hence there is a suspension of the holding pe- riod on the underlying security as well as a deferral on losses to t h e extent of unrecognized gains. Therefore, at the end of period number one, if the stock moves to a value of d S = 80, the call option will be tax- marked-to-market and the investor will profit on the option (lose on the stock) and incur a tax liability of:

0.5($39.03-$15.15) = $11.94, which is due immediately. However, if on the other hand, at the end of period num- ber one, the stock n iove~ to a value of us = 125, even though the position is tax-marked-to-market, the loss of ($56.82 - $39.03) = $17.79 on the option can not be used t o offset other gains, because the option is part of a straddle position. (Note the crucial asymmetry.)

Consequently, at the end of period two, if the path of the underlying stock was: 100 4 80 + 100, there will be an after-tax profit of $5.82 on the short call option.

- f A \

(39.03 - 25.00) - (39.03 - 15.15)(0.5)(1.1)

F.V. of 1 s t per iod tax pre- tax p r o f i t

t a x rebate on loss

+ ‘(25.m - 15.15)(0.5j = $5.82

On the other hand, if the path of the stock was: 100 +

125 + 100, there is no tax liability until the end of the second period, because the position moved against the short (and the loss can not be used). This m&ns that there will be an after-tax profit of $7.01 as a direct result of the cash flow timing of tax liabilities.

pre-tax p r o f i t t a x l i a b i l i t y - P h 7

(39.03 - 25.00) - (39.03 - 25.00>(0.5)

= $7.01

The difference in profit, between the two possible paths, amounts to ) = 20.5% which is quite sub- stantial.

We have illustrated that when the maturity horizon of a standard exchange traded index option transcends tax years, the after-tax pay-off from transacting in the secu- rity becomes path dependent. The writer of the generic call option “hopes” that the underlying stock price-path will be 100 4 80 4 100 as opposed t o 100 +125 ---f 100. The traditional, ‘Lcontinuous taxes” approach, summa- rized in the next section, ignores this asymmetry and non-linearity. Likewise, in a discrete time framework, the pricing operator results of Prisman (1986) and Ross (1987) no longer apply due t o the “portfolio” nature of the taxation treatment.

4 LITERATURE Myron Scholes (1976) used reasoning similar t o tha t used by Black and Scholes (1973) t o derive the price of a call and put option in a world with taxes. The resulting par- tial differential equation (P.D.E.) for t he price of the call (or put) with taxes is surprisingly similar t o the P.D.E. derived by Robert Merton (1973) for the price of a call (or put) option on a constant dividend yielding stock.

The argument was formulated as follows. Let S de- note the price of the underlying security, which obeys

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the Geometric Brownian Motion. Likewise, let f repre- sent the price of a derivative security which is a n explicit function of S .

Scholes (1976) argued that if the portfolio is con- [strutted by modifying the amount of stock held for each unit of derivative sold; one can in theory create a riskless investment. In this case, let:

where ' rd is the marginal tax rate on the derivative se- curity and T~ is the marginal tax rate on the underlying security. Thus, if we purchase units of stock for every unit of derivative sold. We obtain that in a short instant of time:

af d S

AW = -A f (1 - T , ) + -As (1 - 7,)

Now, by substituting values for A S and Af (using Ito's lemma), t he above expression can be represented as:

AW = -(I - 7 d ) X

af \ +- [pS At + US AB] (1 +- 7 d ) d S

which is instantaneously riskless because the variables multiplying the Brownian motion term AB cancel each other out.

For the sake of model completeness, if we allow the un- derlying security S to contain a constant dividend yield denoted by q , then the after-tax rate of return from the dividend yield is q ( l - T ~ ) . Hence, by substituting and simplifying in the classical manner, we obtain:

where:

Solving the P.D.E. with the standard boundary condi- tions, results in the prosaic.

where

I~[S/X] + (i; - q + a 2 / 2 ) ( ~ - t ) a J F 2 dl =

d2 = dl - a m

The Tax-a.djustec1 Black-Scholes (TaBS) formula shares the exact functional form with the classical Black-Scholes formula. In fact, when all marginal tax brackets are set equal to each other, equations (2) and (3) imply that the continuous-tax assumption will self-dissipate and the Tax-adjusted Black-Scholes formula will collapse t o its classical progenitor. Perhaps this similarity in qualita- tive substance has motivated the conviction that taxes do not alter the Black-Scholes formalism.

However, there are some problems which we believe render the Scholes approach ineffective..

(A) Crucial to Schole (1976) entire argument, is the gradual incremental tax liability tha t is paid on an on- going basis. Continuous-time cash Aows can arguably be justified with dividend payments on a large stock index, however using the analogy for taxes pushes the limit of reasonability. Besides ignoring the time value of money, the above mentioned approach does not distinguish b e tween the important t iming of tax liabilities.

(B) In rnost cases, the writer (seller) does not have to pay tax until the derivative position is liquidated, an event that can take place after many tax years. The writer (seller) of a n O.T.C. option can operate without paying tax for a long period of time and the effective tax rate r d on the derivative can be coerced to zero.

( C ) In many cases the position in t h e underlying secu- rity (or the derivat,ive) may be tax-marked-to-market a t the end of the calendar year creating a possible tax lia- bility without an offsetting tax gain. This mismatch can be quite large in the case of long term options (LEAPS).

(D) This approach does not take into account t he dif- ference between hedging price and replicating price dis- cussed above. An economic agent tha t is contemplating purchasing .a derivative security will not be willing t o pay any more than its replicating cost. T h e hedging cost is the cost of .a portfolio of primitive underlying securities that replicates the exact opposite of t he after-tax payoff from the derivative security when they are all held in one portfolio.

5 SYMBOLIC MODEL Given the above mentioned (and some other) difficul- ties in using a continuous time model for tax-adjusted derivative pricing, we advocate using a path-dependent binom'ial laiiice model with the help of a symbolic com- putatio nul language. We will illustrate this technique by demonstrating how t o compute the tax-adjusted hedging price and hedging parameters for a European call option on a non-dividend paying stock. T h e use of a symbolic computational language t o price derivative securities was independently implemented by Benninga, Steinmetz and

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Stroughair (1993) for Asian path-dependent options.

general form, map the paths of the stock portfolio, bond <zs <zb (8)

long position in stock is: We introduce three tax functions which, in their most

ST + b$ - “[ST - K,O] - - e(n) - *(n)

portfolio and call option into their respective tax liabili- ties and/or refunds. Let C+[O, T] denote the set of pm- itive real functions defined on the time scale [0, TI. The symbol S( , ) E C+[O, T ] denotes the entire sample path of the stock between time zero and maturity. The symbol A;,, E C+[O,T] denotes the entire sample path of delta (the amount of stock held to hedge the short call op- tion), strongly dependent on the realization of the stock process. The symbol b;,, E C- [0, T] denotes the entire sample path of bond holdings (which is always negative, by definition of t he hedge) between time zero and ma- turity. Finally, the symbol c* E C+[O,T] denotes the entire sample path of the option between time zero and one. Define the following function:

I.)

%E : c+ [0, TI x C+[O, TI -4 R 92) : c- [O,T] --t R- (6) e ( n ) . Rn+l -4 s z c .

Where a negative number denotes a tax liability and a positive number denotes a tax refund. The function maps the paths S(,) and At,) into a tax liabilitylrefund that is due at time T. The superscript‘(n) denotes the number of symmetric tax due-dates during the life [0, T ] of the hedge. For simplicity we assume that time T (ma- turity) is a tax due-date and that all other tax due-dates are evenly dispersed over the time interval [0, TI. Thus, for example, there may be three (symmetric) tax events during the life of the hedge (i.e. tax due-dates), but the actual payment of tax will take place at maturity. (On every tax due-date, you borrow the amount that is due and pay the loan back at maturity.) The function 92) implicitly takes into account the realized net gain and adjusted cost basis calculation. Similarly, the function 32) maps the path bi., into a tax refund at T (maturity).

Finally, the function Sk:”,’ maps discrete points on the into a tax liability/refund that is due at ma-

::::y:;+ his t ax function does not depend on the entire sample path but rather on the value of the call option at particular points in time. Specifically, the functional form is:

n

The first two expressions in equation (8) correspond to the payoff from the delta hedge, the third expression cor- responds to the payoff from the option. The last three expressions in equation (8) correspond t o the tax liabili- ties/refunds that a re due from the total position, taking into account the portfolio nature of t he transaction. One must remember that (despite t he simplified notation) the actual tax liabilities will depend on the entire pricepath history.

For there to be No Arbitrage in the market, the after- tax payoff in equation (8) should equal t o zero in all states of nature. The trading strategy tha t hedges the option on an after-tax basis will be very different from the one used on a pre-tax basis. The main reason for this is the restriction on ‘wash sales’ tha t would apply if the position was being used to reduce taxes by realizing losses at the end of every tax year. In general, modify- ing the (pre-tax) delta hedge at a finite number of points (with measure zero) will not change the density function of the outcome. The stochastic ifiegral will remain the same. However, in reality, once taxes a re taken into ac- count, modifying the hedge on a finite number of points, will have a severe affect on the after-tax payoff. Realiz- ing all losses at the end of the tax year will be optimal, regardlm of the value of t he derivative being hedged.

Let us now examine a very simplified tax structure, namely, when there is only one tax liability date on the call option and there is no t ax due on profits/losses from the underlying security. (This may seem somewhat ar- tificial, but Scholes (1976) analyzed such a case, albeit with continuous taxes on t h e call option, which corre- sponds t,o no taxes on capital gains.) In this event, we employ the fundamental theorem of Harrison and Pliska (1981) which states tha t when there is No Arbitrage in the market, there exists a n equivalent martingale mea- sure under which the underlying stock price process is a discounted martingale. Under this measure all deriva- tive securities can be priced by discounting expected cash fiows. In our case, from equation (7) 3:’ = 0,%2) = o and 3:) = (c; - c f ) ~ and we obtain the following:

Note: All t ax liabilities/refunds are paid/received at ma- turity, but are computed based on path behavior during the life of the hedge.

Using the above stated notation, the final (time T ) after-tax payoff from selling a (properly priced tax- adjusted) call option and (properly) hedging i t with a

(4 -

where Et(.] denotes expectations at time t under the risk neutral probability measure and 7 is the (constant) mar- ginal tax rate on call option profits/losses. The second part of equation (9) is simply the after-tax payoff from the call option. Thus, t he only difference between equa- tion (9) and the standard option pricing expression, is

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t h a t after-tax cash flows have been substituted for pre- tax cash flows. Simplifying equation (9) we obtain that:

ct' = Et e - r ( ~ - t ) (max[ST - K , ( B I ( ~ - 7) + CE.)] [ or, by substituting the integral for the expectation,

c i = ezr(T-t) x

iw max[sty - K , ol(1- T)q(g)dg + e-+-t)cFT.

where 9(y) is the appropriate stock price density func- tion. Notice the fixed-point aspect of the problem in which the tax-adjusted value of t he call option is equal t o a function of itself, namely ct = f(c;). The expression for the call can be further simplified t o yield:

.t' (1 - 7 e - r ( T - t ) = e-r(T-t) X ) lw "[St'ty - K , Ol(1 - T)q(Y)dY.

However, by extracting constants we obtain

- < p ( T - t ) max[Sty - K , O]q(y)dy = ct

Indeed, the tax-adjusted price of the call option will be less than the no-tax value, analogous t o the result of Scholes (1976), but differing in magnitude and functional form. In theory, the above framework can be generalized to (n) different tax dates on the option, however, the longer the life of the option, the less realistic it is to assume that no taxes are paid on the underlying security. For example, if there are two different tax due-dates for the option, the functional form, in equation (7) becomes:

s z c - - (CL,, T - t - c; ) - i -erT + (4 - ci fY)7

As before, we compute the discounted expected cash flows using the risk neutral measure, the value of the car11 option becomes:

T - t

ct" e-r(T-t)X (11)

Clollecting terms on both sides we obtain:

which leads to:

1-7. T(e -4F) - 1 ) T--tct + T - t ct' =

1 - 7 . e - d T ) 1 - T e - r ( T )

Using the rules for conditional expectations. This leads to: ) Ct.

1 -- 27 + q-e-r(? )

1 - T e - d T ) c; = (-

Again, the value of the: call option in a world with taxes is less than the value in a taxless world. This would affect put options as well and would become more observable, the longer tlhe maturity of t h e option.

All of the above discussion will only apply when there is no capital gains tax on the underlying security. In a world with taxation of capital gains we are forced to resort to the symbolic formulation.*

*

5.1 NOTATION. We must first devise an algebraic system of notation i n order t o facilitate the computations in a path-dependent model. The Cartesian co-ordinates of a particular node in the non-recombining tree (NRT) will be denoted by the paired vector symbol ( i , j ) E R2. The unique ordi- nal ranking (from 1 to an) of any particular node will be computed via the function G(( i , j ) ) = ai -k j . Also) the generic (operator that maps the (two dimensional) Cartesian co-ordinates to the (G((i , j)) dimensional) bi- nary co-ordinate will be denoted by the symbol P ( ( z , j ) ) . This simple function converts t he numeric value of 2% + j from base t e n to (the suitable presented) base two. T h c inverse function (from binary t o Cartesian) will be de- noted by P1((?)), where the vector ? consists of a sequence of ones and zeros. Thus:

where the natural distinction between the two possible co-ordinate systems will depend on the number (two V.S.

one) of angle brackets. For example, referring t o Figure 1 and Figure 2, we

see that G((2,2)) = 22 + 2 = 6,

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Figure 1 - Ordinal Co-ordinates of the N.R.T.

p - w , L O ) ) ) = (%a). Of particular importance is the quantity:

i

k=l

which represents, using the indicator function, the to- tal number of up-moves embedded inkhe binary cod- ing. Thus: U(((1,1,1,0))) = 3 - 2 2 1. This conven- tion is useful for converting from the co-ordinates of the non-recombining tree t o the cc-ordinates in the binomial world of the recombining tree. It is trivial to show that U ( P ( ( i , O ) ) ) = i and U(P((2,2i - 1))) = 0. Using this approach we can see tha t by definition:

S(, . = su~(p((a,j)))d~-~(~((~,~))) v) For example, in Figure 2:

W P ( ( 3 , 6 ) ) d 3- U ( P ( ( 396) 1 = SUI d2. S(3,6) = su

Finally, we will use the symbol ( i , j , -k) to denote the unique Ic'th node prior to (i, j ) . This particular concept will become exceedingly important when we develop a dynamic hedging scheme. For example, referring to Fig- ure 1 and Figure 2 we can see that:

(3,6,-1) = (2,3), (2 ,2 ,11) = (1,l).

By definition, (Z , j , - i ) = (0 ,O) which means that p((i , j , - i)) = ((I)). Rom a computational point of view, one can go-backwards in a non-recombining tree by converting the Cartesian co-ordinates into binary (path) co-ordinates, via t he function P(( i , j ) ) , then truncating the appropriate number of digits in the binary co- ordinate vector and then converting back to the Carte- sian co-ordinates. Formally, if

P ( ( i , j ) ) = ( ( 1 , ~ 1 1 ~ 2 , - - - , ~ i ) ) ,

Figure 2 - Cartesian Co-ordinates of the N.R.T.

then

This provides us with a convenieQt algorithm for com- puting the historical path-dependent values in a NRT. Thus, when we a re at a particular node and would like to examine the evolutionary history of the security, delta or bonds we can use the symbols:

S(.,j, ) = {S(i,j,-I) 1 S( i , j , -2) 1 ...> S(i,j,-i) = S(0,O) }

d(iJ, ) = 7 A(i,j,-2) I ..-> A(i,j,-1) = A(0,O) } B(z,j,.) = { ~ ( Z , ~ , - l ) ~ ~ ( ~ , j , - 2 ) ,...,B( Z , j , - l ) = B(O,O)}.

to denote values throughout the NRT. The short-hand notation for the entire history, up to the present node, will be: ( z , j , .)

The symbol ( N , j) denotes the path-dependent node, ATP is the After Tax Payoff and c : ~ , ~ ) represents the tax adjusted price of the option at the first node of the binomial tree. The objective of our exercise is to find tax-adjusted values of the option prices, delta and the bonds, denoted by ? ,A* and B', tha t solve the follow- ing system of equations;

ATP(N, j )=O ' d j ; O < j < N ,

with the final goal of computing c ; ~ , ~ ) . We use the method of dynamic programing, however

we must deal with the problem of the Adjusted Cost Basis (ACB) and Realized Net Gain (RNG) that is not known at the end of the tree. Thus, we will 'carry' i t back through the tree and eventually eliminate it from the price. This is the essence of the need for a sym- bolic computational language. Dynamically we do the following:

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(1) Compute ATi,j) and Bii,j, by solving a one-period system of two equations and two unknowns, express- ing A;i,j) and B*% ) in terms of the current RNG(,,j), ACB(,,j), s(i,j) ,b/i,j, ) and the yet t o be determined

( 2 ) In the nodes ( N - 1,j); j = 0..2N-’ the next periods tax liability is explicitly incorporated into the system of equation. In all the nodes prior t o ( N - l,?), the tax liability is implicitly contained in the variables.

(3) Write RNG(;,j) and ACB(,,j) in terms of

(4) Re-solve for A;i,j) and Bii,jL in terms of the past

c;o,o).

*ATi,j,-l) , RNG(i,j,-q and ACB(i,j,-q.

,RNG(i,j,-l) , ACB(i,j,-I) 1 S(i,j) 1 B(i,j,,) and the yet to be determined c ; ~ , ~ ) .

( 5 ) Express the current c;,,~) = A;,,j) S(i,j) f Bii,j, . (6) Proceed backwards through the tree until the first

node in which we finally obtain that C T ~ , ~ ) = F ( c ; ~ , ~ ) ) , where the function F ( - ) should not contain - any terms besides S(O,O) , Rf , U, d, N and

(7) The final stage is t o solve for t he fked point of the function F(.) .

The heart of the algorithm is two embedded do-loops that proceed through the entire tree oae node at a time. At first, the program creates a tree of order N, in which there are 2N possible paths. In the first part of the algo- rithm, each node will be initialized with a symbolic value for the delta, the bonds, the realized net gain (RNG), the adjusted cost basis (ACB) and the final call option. These ‘cells’ will remain empty until they are filled dur- ing the backward induction through the tree. The next part of the program goes t o t h e last possible point on the tree, namely the maturity value, and computes the after-tax payoffs as a function of the yet to be deter- mined hedge variables. Once the maturity parameters have been set, the actual do loops begin as described in Figure . The main computational problem is to ‘push’ tlhe realized net gain (RNG) and the adjusted cost ba- s k (ACB) ‘back’ through the tree until we reach node zero. The RNG and ACB are functions of the entire path up to and including the present node. Thus, we can not determine the actual RNG and ACB until we have terminated both do-loops. On the other hand, we ci3n not go t o the next stage of the do-loop until we have computed the appropriate current hedge ratio, which in itself depends on the RNG and ACB! Consequently, at each node, we end up solving for the delta and the bonds -- twice, the first time as an implicit function of the current delta and bonds and the second time as the solution of a fked point problem in which the delta and bond values are obtained without reference t o the RNG and ACB.

6 FINAL WORDS: What are the benefits of pricing derivative securities symbolica:lly, with a language such as MAPLE v, as

opposed to numerically? First and foremost, the ac- tual final expression for the price of the call option (or any other ,derivative security for that matter) can be ob- tained exactly, as opposed t o the approximate number that would be obtaiined from a numerical scheme. Sec- ond, the ability t o leave certain values as symbolic en- ables us to investigat,e the rates-of-change of the price of the call option relative t o any of its arguments. Indeed, a symbolic solution does not preclude using numerical methods, rather, the price can be solved symbolically and then translated into any programing language (for example, Maple’s ability t o translate into c) for the ac- tual values used in practice. A symbolic procedure can incorporate a wide variety of node dependent features, for example the non-deductibility of losses, with ease and versatility.

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P. H. Dybvig and S. A. Ross. “Tax Clienteles and Asset Pricing”. The Journal of Finance,

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R. C. Merton. “Theory of Rational Option Pric- ing”. Bell Journal of Economics and Manage- merit Science, 4:141-183, 1973.

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