The effects of transaction costs and different borrowing and lending rates on the option pricing model

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FACULTY WORKING PAPER NO. 874
College of Commerce and Business Administration
University of Illinois at Urbana-Champaign
June 1982

The Effects of Transaction Costs and

Different Borrowing and Lending Rates on the

Option pricing Model

John E. Gilster, Assistant Professor
Department of Finance

William Lee
Bendix Corporation

Acknowledgment: The authors would like to thank the University
of Illinois Investors in Business Education and the University
of Illinois Research Board for financial support.

Special Thanks go to George Constantinides who was very smart (and
kind enough) to point out a serious error in an earlier
version of the paper. Pvemaining errors are, of course, the
authors ' .

Abstract

This paper solves a stochastic differential equation to demonstrate
that the market imperfections of transaction costs and different borrowing
and lending rates partially offset each other to yield a range of
equilibrium prices for an option. The Black-Scholes model price is
shown to be in the lower portion of, or entirely below, the equilibrium
range. These observations are used to explain several of the mythical
anomalies found in the option pricing literature.

The paper also points out that under some conditions there may be
no equilibrium option price. Instead there may be a bounded disequilibrium
within which a single option will offer a -risk free return above the
Treasury bill rate, while simultaneously permitting borrowing below the
borrowing rate.

Digitized by the Internet Archive

in 2011 with funding from

University of Illinois Urbana-Champaign

http://www.archive.org/details/effectsoftransac874gils

I. Introduction

The Black-Scholes model requires an investor to create a risk free
hedge by taking a position in an option and the opposite position in
the underlying stock. The stocks and options are held in proportions
such that any price movement in the stock is perfectly offset by an
opposite movement in the option. These proportions are readjusted contin-
uously throughout the life of the hedge. The hedge is therefore risk free
and yields the risk free rate.

If the hedge consists of a long position in common stock and a short posi-
tion in options the hedge will require a net investment on which the investor
will earn the risk free rate (an "investment hedge"). If the hedge consists
of a long position in an option and a short position in the stock, the hedge
supplies funds to the investor for which he pays interest (a "borrowing hedge").

The original Black-Scholes (.1972) option pricing model assumes zero
transaction costs and implicitly assumes borrowing and lending at the risk
free rate. Under these assumptions the option price appropriate to an in-
vestment hedge is equal to the option price appropriate to a borrowing
hedge and this determines a unique equilibrium option price. This will be
shown to be a special case of a more general model.

If transaction costs are ignored, the effects of different borrowing
and lending rates are relatively obvious. The option price appropriate
to an investment hedge is the traditional Black and Scholes price (and
therefore earns the risk free rate) and the option price appropriate to
a borrowing hedge can be calculated from the Black-Scholes model equa-
tion with the investor's borrowing rate substituted for the Treasury bill
rate (the hedge therefore costs the borrowing rate). Obviously, the
option price appropriate to a borrowing hedge will be greater than the
option price appropriate to an investment hedge.

-2-

When transaction costs are considered, option prices must be adjusted
so as to earn the investor the risk free rate on an investment hedge or
cost him the borrowing rate on a borrowing hedge net of transaction costs.

In essence, the profit from an investment hedge comes from the de-
clining time premium of an option that has been short sold (or "written").
The additional revenue needed to pay transaction costs can be generated by
raising the initial option price to provide for a greater decline in time
premium and a greater profit for the hedge's short position in options.

In essence, the cost of a borrowing hedge results from the deteriorat-
ing time premium of the hedge's long position in options. To provide
for borrowing at the market rate after transaction costs the initial op-
tion price must be reduced so as to provide for less deterioration in time
premium and more funds available to pay transaction costs.

The reader will note that if transaction costs and the borrowing and
lending rate spread are of precisely the right size, the transaction cost
adjusted option price for an investment hedge can equal the transaction
cost adjusted option price for a borrowing hedge. In this case the market
imperfection of different borrowing and lending rates and the market imper-
fection of positive transaction costs cancel to produce a unique equilibrium
option price.

This precise cancellation is, of course, rare. Usually one of the
two imperfections dominates yielding a bounded range of option prices.
Section IV and the conclusion of the paper point out the potentially
bizarre nature of some of these situations.

Section I of the paper presents an introduction and a general
description of the problem. Section II modifies the Black-Scholes

-3-

option pricing model to include rebalancing transaction costs. Section
III suggests ways in which some investors may reduce the cost of acquir-
ing and terminating the hedge position. Section IV calculates option
prices with transaction costs and different borrowing and lending rates.
Section V presents a conclusion and summary.

II. Transaction Costs of Hedge Rebalancing

The Black-Scholes model assumes that the price of an option, w(x,t),
is a function of stock price x, and time, t. In this case, the equity
in an investing hedge of one stock share long and n = 1/w options
short is X - wn (where the subscript refers to the partial derivative
of w(x,t) with respect to its first argument). The equity change in a
short interval At can be expressed as:

A(x - w/w ) = Ax - A(wn) (1)

= Ax - [w(x+Ax, t+At)n(x+Ax, t+At) - w(x,t)n(x,t) ]
which can be expanded to:

= Ax - {[w+w Ax+^^^(Ax)^ + W2At][n+n^Ax+|n (Ax)^+w At] - wn}

Substituting v^ = (Ax/x) /At, and keeping only the terms of
Ax and At , equation (1) becomes :

12 2
A(x-w/w ) = (- -TV X w -w )nAt +

12 2 2 2

(-Axwn^/At-wn_- -rv x wo, -v x w n.)At

(2)

-4- I

The first term on the right side of equation (2) is the part of
the equity change which yields the risk-free rate as derived in the
Black and Schole model:

Ax - Aw/w, = Ax- [w(x+Ax,t+At) - w(x,t)]/w^

1 2

= Ax - [w+w^Ax+-w^^(Ax) + w^At-wJ/w

1 . ii<.

1 2 2
= C-^j_j_x V At-W2At)/w^

= C-|vVw^^-W2)nAt -. ^.; <^3)

The second term of equation (2) is the extra capital required to

2
maintain the hedge position. The extra capital is composed of the change

in the number of options An at the changed price -w(x+Ax, t+At) i.e.,

-wCx+Ax, t+At) An

= -[w+w^Ax+-iw^j_(Ax)2-Kj2At] [n+n^Ax+in^j_(Ax)2+n2At-n]

17 9 7 7 • r

= (-Axwn /At-wn - -rv x wn -v x w n )At (4)

Accompanying the extra capital, the transaction cost by which
the equity change should be reduced is a | - w(x+Ax, t+At)An |, where a is
the transaction cost rate for options. Therefore, the equity change Ax - Aw/w
yields the risk-free investing rate r on the equity x - w/w after the
cost, a I - w(x+Ax, t+At)An |, has been deducted. Therefore:

Ax

- Aw/w - a| - w(x+Ax,t+At)An| = (x-w/w^)r^At (5)

Substituting equations (3) and (4) into equation (5) and replacing

2 2 3 2

n, n^, v\.^^ and n^ by 1/w^, - w^^/w^ , (2w^^ ""l^lll^/'^l ^"*^ ~ "l2''^l

respectively, yields:

-5-

i i 1 2 2 ,..

r w - r xw - -TV X w - w„ = ctg (6)

where

19 9 9 9

g = E(w^ I - 5xwiL./At - WTU - -rv X wil. - v x w n^ |)

0 0 0 0 1

= E I Axww /(w At) + WW /w +v X (-w\>? /w^ +^^^^/w^+w^^) | (7)

Equation (6) is the differential equation for the option price
from an investing hedge. Similarly, one could be derived from a bor-
rowing hedge, i.e.,

b b 1 2 2 .-v

r w - r xw - -^ X w - w = -ag, (8)

where r is the appropriate borrowing rate.

Equation (7) makes it clear that as At -> 0, g approaches infinity,
yielding the horrifying (though not entirely unexpected) result that with
continuous rebalancing, transaction costs will be infinite for any posi-
tive transaction cost rate.

Fortunately, this result is more of a mathematical artifact than a
practical problem. As the tables presented in Section IV demonstrate,
short finite interval rebalancing (e.g., daily) results in very reason-
able transaction costs. '

To solve for equation (6), it is reasonable to assume the solution
w""- differs only slightly from the Black and Scholes model solution because
the transaction cost rate, a, is quite small. Let w^ be the Black and
Scholes solution and aw' the correction, then

vr^ = w° + aw' (w° >> aw') (9)

-6- I

Replacing w by w-"- in equation (6) :

i i ' 1 2 2 ' '
r w' - r xw - -^v X w - w- = g (10)

Since w = w , g can be approximated by

- . w, - -.

T7 1 A 0 0,, O,^. , 00/0^ 2 2/ 00 2, 02 ^100 ,0^ O.i „,.
g = E I Axw w^^/(w^At) + w w^,/w^ + V X (-w w^^ /w^ + -^w ^-^n'^l "n^ ' ^'^■^^

Because the hedge will not change when t = t*, x -> °° or x = 0,
there will be no transactions costs. Therefore, w = w and

w'(x,t*) = w'(",t) = w'(0,t) = 0 (12)

The same substitution for w' used in the Black and Scholes model
yields : , . ' '

w'(x,t) = e^'"(t*-t)2(u,s) (13)

where :

2 2

u = 2_(ri _ |_) [to J - (r^ - f-) (t-t*)], (14)

V - .

2 ,
s = - ^(r^ - f-r (t-t*) (15)

and c is the exercise price of the option.

Equation (15) implies :

2
t = t* - sv^/(2(r^ - 2~^^^

.- • I, ■ >

Equation (14) implies:

2

X V

c exp {(u-s)/[r^ - j") ] }

-7-

Substituting x and w' into equation (10) and (11):

Z2 - ^11 = h (16)

with h = h(u,s) being the function g after multiplying by

e V /(2(r - — ) ) and substituting in x and t. The boundary

conditions, equation (12), become:

Z (u,0) = Z (~,0) = Z(-",0) = 0

The solution for equation (16) is given by Butkov (1968, pp. 525-

526):

ZCu,s) =

|.oo ^_(u_u')2/4(s-s')
- (4Tr(s-s'))-'-^^

h (u',s') du'ds' (17)

Substituting (17) into (13) and then (9), yields the solution w .
Similarly, w can be solved for by using interest rate r and replacing
equations (9) and (10) with:

w = w + aw' and (18)

r w' - r X w - -jv^x w - w = -g (19)

III. Initiation and Termination Costs

In option hedging, the lowest cost market participant will usually be
the investor for whom acquiring (or short selling) the common stock portion
of the hedge is a by-product of other activities.

An investment hedge consists of a long position in common stock and
a short position in options. An investor who owns but wishes to sell the
common stock for which the hedge is to be written can form the hedge

-8-

without incurring a marginal cost for buying or selling the stock. In
this case, instead of selling the stock immediately the stock is retained
and the usual hedged position of w/w worth of options are written for
each share of stock held. This hedge is held until either:

1) The option price drops below 1/16 point at which time C.B.O.E.
trading in the option is halted. A hedge can no longer be
formed and the coinmon stock is (finally) sold.

2) The option is in the money at expiration and the stock is
called away. In this case transaction costs are calculated
as if the stock were sold at the exercize price, C rather
than the actual stock price X*.

As soon as the hedge is formed, stock price movements are neutral-
ized and the stock used in the hedge is in effect sold. The initial
cash flows (including the savings from not actually selling the stock)
are:

aw/w, - a X
1 X

where a is the transaction cost rate for common stock transactions.
When the stock is finally (actually) sold and the hedge position closed
out the flows are:

a >an(x^;c) = a (x^-w^) (20)

X X

where the superscript y indicates values at the time the hedge is closed
out (not necessarily at expiration; see contingency 1 above). ^

In equilibrium (ignoring dividends) the discounted present value
of the expected value of x^ is x. Therefore, when the hedge is con-
structed the risk adjusted present value of the total cash outflows
are:

-9-

aw/w - a e'^^*^E(w^) (21)

where K is the discoimt rate appropriate to the option and At is the
time until the option hedge is closed out. Equation (21) shows that
the investor has, in effect, paid a(w/w ) plus continuous rebalancing
costs to save the transaction costs on the amount by which x* might
exceed c at expiration (i.e., w*) . Under reasonable assumptions this
will involve a net outflow, but the costs will be small relative to
the size of the hedge. Moreover, these costs relate to more than one
option position. ^'Jhen the hedge begins it consists cf 1/w options and
if there are transaction cost savings at the dissolution of the hedge
it is because the option is in the money at expiration and therefore
has a hedge ratio of one (i.e., equation (21) then relates to only
one option) .

Similarly, a borrowing hedge can be formed without the marginal
cost of stock sales and purchases. In this case an investor who wishes
to purchase a stock does not purchase it immediately, instead he buys
the usual hedged position of w/w options and continuously rebalances
as if he actually held the stock. Eventually one of two things happens:

1) The option price drops below 1/16 point at which time trading
is halted. A hedge can no longer be formed and the stock is
finally, actually, purchased.

2) The option is in the money at expiration at which time the
option is exercised and the stock is (finally) acquired.

Since transaction costs are the same for buying and selling, equation
(21) will also describe the investor's costs for a borrowing hedge. The in-
vestor has, in effect, paid a(w/w ) per share plus continuous rebalancing

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-10-

costs to postpone the cost of acquiring the stock and save a w* (per
share) when the stock is finally acquired.

Since the continuously rebalanced option hedge position mimics
every price movement of the underlying stock, the investor has, in
effect bought the stock immediately without paying for the stock until
the option expires or becomes worthless. This procedure is therefore
a substitute for margin borrowing but without a margin requirement (or
collateral in the usual sense of the word).

IV. The Combined Effects of Transaction Costs and Different Borrowing
and Lending Rates

The effects of transaction costs and different borrowing and lending
rates are presented in Tables 1, 2, and 3. Column (1) is the Black-Scholes
option price calculated under the assumptions specified in the table.
Column (9) is the value the Black-Scholes model gives if the specified
borrowing rate is substituted for the risk free rate. The common stock
standard deviations listed in the tables roughly correspond to the 10th,
50th and 90th deciles of the standard deviations observed by Whaley
(1982).

Columns (2) and (8) ("Rebal Ad j ") are the difference between the
traditional B&S option price and the daily (i.e, At = 1/260) rebalancing
transaction cost price derived in Section II. In addition to the assump-
tions specified in the table, one way transaction costs for options are
assumed to be 2" and the underlying stock's expected return is 17% per year.

INSERT TABLES 1-3 ABOUT HERE

r t.

-11-

ColLimns (3) and (7) ("Init-End Adj") are estimates of the adjust-
ment to the option price required to cover the cost of acquiring the
hedge and finally liquidating it. These costs are based on the trading
techniques presented in the previous section and embodied in equation (21).
Common stock transaction costs (a ) are assumed to be 1.25% and option
transaction costs (a) are assumed to be 2%. The expected value of the
future value of the option is calculated from Sprenkle's equation (see
Smith, 1976, page 17) and this value is discounted back to the present
using a discount rate, K, derived from the CAPM under the assumptions
that the market return is 17% per year, the risk free rate is 12%, the
beta of the underlying stock is one and the beta of the option (as pointed

Q

out by Black and Scholes, 1972) is:

Columns (4) and (6) ("Net Adj Price") are Black & Scholes option
prices (columns (1) and (7)) with both types of transaction costs added
(for the investment hedge, columns (2) and (3)) or subtracted (for the
borrowing hedge, columns (7) and (8)). Columns (4) and (6) therefore show

the prices the options must sell for to net the specified borrowing

9

and lending rates after transaction costs. Needless to say, these

transaction costs would be different for different sets of assumptions.
The costs presented are illustrative and can be helpful in understanding
the nature of the phenomena. The reader is encouraged to analyse the
effects of his own assumptions.

Column (5) ("Net Price Spread") is the result of subtracting
column (4) from column (6). It should be interpreted as follows:

-12-

1) Negative values indicate that the option price is the bounded
range betv^7een the prices specified in colurans (4) and (6).
For option prices within this range neither investment hedges
nor borrowing hedges are particularly attractive. The reader
will note that the highest option price which produces an
attractive borrowing hedge (Column (6), Net Adj Price - Borrowing)
is frequently higher than the traditional Black-Scholes price
(Column (1)). When this occurs, the Black-Scholes price isn't

an equilibrium value. If an option sells for the Black-Scholes
price, excess profits can be made by forming a borrowing hedge
and (in effect) borrowing at below the market rate.

2) A zero value indicates a unique equilibrium option price (i.e.,
the value shown in both column (6) and column (4)). This oc-
curs when transaction cost effects and borrowing and lending
effects precisely cancel. This unique option price is never
the Black-Scholes price except in the trivial case where all
prices are zero.

3) A positive value indicates that the option hedge can be viewed
as a financial intermediary with a lower spread than traditional
intermediaries. If the option sells at a price between the prices
specified in columns (4) and (6) the option hedge is simultaneously
a higher return risk free investment than treasury bills and a
lower cost source of funds than traditional borrowing. In this
case, there is n£ price to which the option can adjust which will
eliminate excess profits. For example, if the option price were

to drop low enough for the investment hedge to no longer be attrac-
tive, this would only make a borrowing hedge even better. It may
be that the only thing that prevents all short term borrowing and
lending from being sucked into these financial "black holes" is the
limited number of investors in the special transaction cost situa-
tions described in the previous section.

This permanent disequilibrium is bounded between the column
(6) and column (4) prices. If the option price is above the
column (6) price the option is not attractive as a part of a
borrowing hedge but a short position in the option will be
very desirable as part of an investment hedge. This unbal-
anced selling pressure should drive the option price below
the column (6) price at which time the option is desirable as
both an investment hedge and a borrowing hedge. This presum-
ably results in a better balance between supply and demand for
the option.

Similarly, if the option were to sell below the column (4)
price it would be very attractive as a part of a borrowing
hedge but there would be no interest in forming investment

-13-

hedges . The resulting net buying pressure should push the
option price back above the column (4) value.

Therefore, when the column (5) value is positive, it indicates
a bizarre form of bounded disequilibrium.

Transaction costs and different borrowing and lending rates may help
to explain some empirical anomalies. Specifically:

Some empirical findings contradict each other. For example, Black
(1976) vs. Macbeth and Merville (1979) on the direction of the bias for
in the money, relative to out of the money options. This paper shows that
options usually have a range of equilibrium (or bounded disequilibrium)
values. It is therefore not surprising that studies which assume unique
equilibrium values sometimes contradict each other.

Merton (1976) says that practitioners believe that the B-S model under-
prices both in the money and out of the money options. Latane and
Rendleman (1976) conclude that "... the preponderance of evidence would
be toward options being over priced." (i.e., the B-S price is too low).
Tables 1, 2 and 3 show that when the transaction cost/interest rate effect
is considered, the B-S price (column (1)) is in the lower portion or
entirely below the equilibrium range of option values. Empirical tests
should show that the B-S model underprices options. It does.

Although empirical results conflict, the more modem, sophisticated
and exhaustive studies seem to show that the B-S model underprices in
the money options relative to out of the money options (see Macbeth and
Merville, 1979) and underprices options on low risk stocks relative to
high risk stocks (see Whaley, 1982).

This is the average effect a combination of transaction costs and
divergent interest rates should produce. If the effect of different

-14-

borrowing and lending rates is large relative to the size of transaction
costs (Tables 1, 2 and 3 and others calculated by the authors suggest
this is the case for options which are in the money and/or written on low
variance stocks) the B-S price will be near the bottom of, or below
the range of equilibrium prices. Therefore, the B-S model will be found
to underprice these options. On the other hand, options for which the
interest rate effect is small relative to the transaction cost effect
(out of the money options and options on high variance stocks), the B-S
price will be near the center of the equilibrium range and such options
may appear overpriced for one sample and underpriced for another; thus
explaining the conflicting results in the literature. However, for these
options the Black-Scholes price is near the middle of the equilibrium
range so, on average (and presumably for large sample sizes), the B-S
price may be a relatively unbiased description. Therefore, the transaction
cost/interest rate model may help to explain Macbeth and Mervilles' (1979)
and I'Jhaley's (1982) observed biases; but the transaction cost/interest
rate model suggests that the range of biases is more likely to extend
from the (roughly) correctly priced to the underpriced rather than from

the overpriced to the underpriced; a distinction their weighted implied

12
standard deviation techniques would have difficulty detecting.

V. Conclusion

When transaction costs and different borrowing and lending rates are
taken into consideration, options (in general) take on a range of equilib-
rium values. The traditional Black-Scholes price is in the bottom portion

-15-

of, or entirely below, this range. These observations are used to explain
empirical anomalies found in the option pricing literature.

The paper also makes the startling observation that for in the money
options on low variance stocks, there may be no equilibrium option price.
Any price these options might assume will offer a risk free investment
at above the risk free rate or borrowing at below the market rate or
both simultaneously. The option exchange becomes society's lowest

spread financial intermediary. All short term borrowing and lending

13
might be sucked into these financial "black holes" were it not

for the special nature of the situations needed to produce low enough

transaction costs.

i .

-16-

Appendix A

The Implications of Hedge Rebalancing Using Adjustments
to the Option Position

Throughout this paper the authors assume that rebalancing is done
by adjusting the option portion of the hedge. This is generally the
cheapest way to rebalance because option rebalancing involves smaller
dollar amounts than rebalancing with common stock. This cost advantage
is partially offset by the fact that the average bid-ask spread in the
options market is greater than in the stock market (see Phillips and
Smith, 1980).

The hedge acquisition and dissolution techniques described in
the text assume that the hedge contains the same number of shares of
stock at the beginning and end of the life of the hedge. Therefore,
equation (24) will only (usually) be an accurate description of costs
if all rebalancing is done with options (thus leaving the number of
shares in the hedge unchanged throughout the life of the hedge) .

Needless to say, an investor should not rebalance by buying an
overpriced option or selling an underpriced option. Therefore, the
assumption that all rebalancing is done with options is unrealistic.

However, one suspects that the advantage of being able to rebalance
with options when they are favorably priced and avoid them (with stock
rebalancing) when they are unfavorably priced more than offsets the
additional rebalancing cost of common stock rebalancing and the addi-
tional hedge dissolution cost which may result from acquiring an
unwanted common stock position to liquidate at the end of the life of
the hedge. Moreover, common stock rebalancing can also result in

-17-

the acquisition of part of the desired stock position prior to dis-
solution thus reducing costs below those assumed in equation (24).

Moreover, footnote 10 shows how some investors can reduce trans-
action costs to a level generally below those presented in this paper.
Finally, the reader may feel that the option rebalancing assumption
is unrealistic because it is not possible to trade options in odd lots.
The authors suggest that this is not a real problem because, if the
investor's hedge is so small that rebalancing involves trades of less
than several thousand dollars each, transaction costs will destroy the
investor no matter how he rebalances.

1 • ;.

-18-

Appendi:-: 3

A Demonstration of the Validity of the Proposed Solution to Equation (9).

The authors present this example in hopes of convincing the reader
that the proposed solution to equation (6) is correct. In order to pro-
vide a simple example, the authors have chosen parameters which are
realistic but relatively easy to calculate:
X = $50
c = $50
v^ = .25
Ct* - t) = .5 (years)
r = .10
6 = .15
a = .01
The g function (equation (11)) on the right side of equation (6)
is the dollar amount of rebalancing required. It can be calculated
from Black & Sholes pricing theory based on the parameters listed above.
This yields:

w = 8.1316

1- = (In ^+ (r + .5v2)(t* - t))/(v2(t* - t))^^^ = .
1 c

31820

w^ = N(d^) = .62483

For simplicity define

m = d^/{2)^^~ = .225

-19-

w-,-, can then be expressed as:

w^^ = 6-°^ /CxvCZirCt* - t))^'^^)

= .02145

Also:

2 2

"l2 " CCln^)/Ct* - t) - r - ^)(e"'" ) )/ C2vC2TrCt* - t))^' h

= -.12068

2
w^^^ = -e"^ Cv/C2Ct* - t))^''^ + m/Ct* - t))/ CvVtt^/^)

= -.00081523

Substituting these values into equation 0.1) of the paper and assuming
daily rebalancing (i.e., At = 1/260) yields:

g = 89.82 . ., ; ^ ,. ■ ^ ,:.:: ■
(the approximation listed in footnote 4 yields 89.76).
The right side of equation (6) is therefore: . ■,

ag = .8982

The left side of equation (6) includes partial derivatives of the
daily rebalancing transaction cost option price derived in this paper.
These derivatives must be approximated by taking small interval values
about the $50 stock price and the .5 year time to expiration:

-20-

Estimation Estimated

Parameter Interval Value

w 8.4503

$50 + $.50 .63184

$50 + $2.00 .02033

.5 year + .005 year -9.5952

"l

^1

^2

The width of the interval (column 2 above) used to approximate
each partial derivative is a function of the 5 to 6 significant digit
accuracy of the option price, w, as calculated from the numerical methods
solution to equation (17).

When the estimated values from column 3 above are substituted into
the left side of equation (6) they yield .9278. Considering the inherent
inaccuracy of small interval approximations, this seems to be a good
approximation of the value previously calculated for the right side of
equation (6) (i.e., they differ by less than 4%).

1

Footnotes
Thorpe (1973) demonstrated that option hedges can be sources of
funds despite restrictions on short selling.

These results are derived under the assumption that rebalancing
is done by buying or selling options (rather than stock). See Appendix
A for a discussion of the implications of this assumption.

3
Discrete rebalancing interval applications of the continuous

time option pricing model seem to pose no great problem. Boyle and
Bnanuel (1980) have demonstrated that the risk created by short interval
rebalancing is uncorrelated with the market; and Rubenstein (1976) and
Brennan (1979) have demonstrated that discrete interval applications
of the Black-Scholes model will be valid under assumptions of constant
proportional risk aversion and a bivariate lognormal distribution be-
tween market and underlying asset returns. , . ■ •

4

The RHS of (7) is evaluated by separating the term inside the

absolute value into a positive lognormal distribution truncated at zero
and a negative lognormal distribution truncated at zero. Sprenkle's
formula (see Smith (1976), pp. 17) is used to evaluate the expectation
of each truncated distribution and the absolute values of these ex-
pectations are then added. .. - ,,^

For rebalancing intervals of one day or less and common stock
standard deviations of .3 (annual) or more, the procedure can be greatly
simplified. In this case, the first term inside the absolute value
(which represents short term stock fluctuations) will be much greater
than the other terns (which represent longer term shifts in hedge ratio).

-22-

g can then be approximated by applying the formula for a full wave linear
detector:

g = E I Ax/x I xww,,/(w,dt) '= C2a dt/Tr) xww^^/w,dt

\^en an option is exercized the commission is based on the
exercise price not the stock price. Therefore the investor pays
a MinCx-^jc) when the hedge is terminated.

6

One problem with the Black Scholes model is its dependence on

the assumption that short positions in common stock are an immediate
source of funds. Thorpe (1973) has argued that if an investor currently
owns the stock for which a hedge is to be created, selling the stock
is equivalent to short selling and _i£ a source of funds. The procedure
for forming a minimum transaction cost borrowing hedge described in
this paper is another way in which a borrowing hedge can be a source
of funds.

The reader is encouraged to make his own transaction cost
assimiptions. The rebalancing adjustment is approximately proportional
to the transaction cost rate, a throughout the relevant range of values
and the initiation and ending transaction costs can be approximated
from equations (21) and (22).

The transaction costs used to create Tables 1, 2 and 3 were chosen
to illustrate the lower end of the reasonable range of costs. For
example, the 2% transaction cost rate for options can be interpreted
as the rate for an investor who has 1.1% (one way) transaction costs,
who is investing in an option series with an average bid-ask spread of

-23-

\

3% Cabout the 35th percentile of the option spreads observed by Phillips
and Smith, 1980) for which 40% of the trades require no market maker
involvement and a zero spread (Phillips and Roberts, 1979). Thus yielding
a one way spread cost of 3% x .6 x .5 = .9% and an average total trading
cost of 2%. Similarly, the 1.25% common stock transaction cost rate can
be interpreted as 1% transaction costs plus a one way average spread of
.25% i,±.e., the total spread, including the probability of specialist
involvement is .5%).

Equation C22) describes an option's beta at a specific instant in
time and is therefore only an approximation of the beta actually needed.
Fortunately, beta is only used to calculate the discount rate to be
applied to a small amount of money to be realized a short time in the
future. Therefore, more precision is probably unnecessary.

9

In effect Tables 1, 2 and 3 assume that the costs of initiating

and terminating the option hedge are prepaid by adding or subtracting
them from the initial option price. This change in initial option
price will slightly alter rebalancing transaction costs and is not
taken into consideration in the derivation of the continuous rebalancing
model (equations (1) through (19)).

Given the generally small size of initiation and termination costs
relative to the option price, and the approximate nature of other aspects
of the option pricing model, this should be acceptable for most purposes.

Some investors will be able to reduce transaction costs to levels
below those assumed in this paper by rebalancing less frequently. For

-24-

a borrowing hedge using the acquisition and dissolution technique sug-
gested in the paper, a failure to rebalance means that the equivalent
number of shares being mimicked by the option position changes as the
hedge ratio changes but is not immediately rebalanced (i.e., the
investor may have the option equivalent of 96 shares rather than the
100 shares he originally intended, etc.)* Some investors may not care
about the exact number of common stock equivalents his option position
equals. If not, he can save money by rebalancing only after the hedge
ratio has changed substantially. He can then avoid the transaction costs
resulting from all of the reversals of the hedge ratio occurring between
rebalancing. Moreover, this strategy allows the investor to save money
by rebalancing in larger amounts.

The same less frequent rebalancing strategy can be applied to the in-
vestment hedge strategy described in the paper except that in this
case a failure to rebalance means that the investor will have a small
long or short position in the stock he originally wished to sell (i.e.,
instead of having the equivalent of no stock, his equivalent position
might fluctuate between long and short positions amounting to a few
percent of his original holding.)

The existence of these bounded disequilibrium situations is not
particularly sensitive to the assumptions used to create Tables 1, 2 and
3. For example, the phenomenon does not disappear even if the transaction
cost rate is doubled. Moreover, if the bid-ask spread is positively cor-
related to the risk of the option, the options exhibiting bounded disequilib-
rium should have unusually low transaction costs because of their low vari-
ances (i.e., they are deep in the money options on low variance stocks).

-25-

12

The weighted implied standard deviation (WISD) technique

explicitly (as in Macbeth and Merville, 1979) or implicitly O-Jhaley,

1982) assumes the average option is correctly priced. This technique

is therefore excellent at detecting relative biases but ill suited to
detecting biases effecting all options.

13

No pun intended.

-26-

References

1. F. Black and M. Scholes, "The Pricing of Options and Corporate
Liabilities," Journal of Political Econoay (May-June, 1973), pp.
637-659.

2. P. Boyle and 0. Emanuel, "Discretely Adjusted Option Hedges,"
Journal of Financial Economics (September, 1980), pp. 259-282.

3. M. Brennan, "The Pricing of Contingent Claims in Discrete Time
Models," Journal of Finance (March, 1979), pp. 53-68.

4. E. Butkov, Mathematical Physics (Addison Wesley 1968).

5. H. Latane and R. Rendleman, "Standard Deviation of Stock Price
Ratios Implied by Option Prices," Journal of Finance, (May, 1976),
pp. 369-381.

6. J. Macbeth and L. Merville, "An Empirical Examination of the
Black-Scholes Call Option Pricing Model," Journal of Economics,
(December, 1979), pp. 1173-1186.

7. R. Merton, "Option Pricing When Underlying Stock Returns are Dis-
continuous ," j£umal_of_^|i^ian£ial_Ecor^^ (January /March, 1976),
pp. 125-144.

8. S. Phillips and D. Roberts, "Analysis of Dually Listed Options,"
Unpublished manuscript (University of Iowa, Iowa City, Iowa).

9. S. Phillips and C, Smith, "Trading Costs for Listed Options," Journal
of Financial Economics. Vol. 8 (June, 1980), pp. 179-201.

10. M. Rubenstein, "The Valuation of Uncertain Income Streams and the
Pricing of Options," Bell Journal of Economics and Management
Science (1976), pp. 407-425.

11. C. Smith, "Option Pricing (A Review)," Journal of Financial Economics,
Vol. 3 (March, 1976), pp. 3-51.

12. E. Thorpe, "Extensions of the Black-Scholes Option Model," 39th Session
of the International Statistical Institute (Vienna, Austria).

13. R. Whaley, "Valuation of American Call Options or Dividend- Paying
Stocks: Empirical Tests," Journal of Financial Economics (March,
1982), pp. 29-58.

M/E/228

ECKMAN

NDERY INC.

JUNSS

I -To-Plos? N MANCHESTER.
INDIANA 46962