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A note on the valuation of contingent claims
Economics Letters North-Holland
467
39 (1992) 467-471
A note on the valuation of contingent claims S.J. Dilworth Uniwrsity
of South Carolina,
Received Accepted
15 June 1992 27 July 1992
Columbia
SC, USA
We present an approach, which is derived from functional analysis, to the valuation of contingent claims on a stock whose price is a random variable with a given distribution. When the price of the stock is a geometric Wiener process the Black-Scholes price is obtained. Following this approach, formulas are derived in a general setting for the price of a call-option and its sensitivity to changes in the price of the underlying stock.
1. A functional Let S be Let B be a denote the functions 4
analysis
approach
a stock whose value at time T is a random variable S, with law p supported on [O, ~1. discount bond maturing at time T with a face value of one unit of money. S, and B, present values of S and B respectively. Let 27 be a cone of non-negative continuous on [O, a> satisfying the following:
(a) ‘Z’ contains (b) ‘57contains (c) 55’contains
1 (the function that is identically one); x (the identity function); go (the cone of non-negative continuous
functions
vanishing
For each 4 E ‘Z”, let +r denote a contingent claim which can be exercised equal to the random variable 4(S,). We wish to define a price function following natural conditions: (9 (ii> (iii) (iv)
7r(4) ~(a$ 77(l) 57(x)
2 0 for all 4 E %; + b$) = arr(4) + brr($) = B,; = S,,.
at infinity). at time T for a payout r on 5% satisfying the
for all 4, I,!JE $9 and for all a, b 2 0;
Given such a function r, we regard ~44) as being a possible price for the claim 4T for each 4 E 29. Conditions (i> and (ii) then have a standard interpretation: they say simply that r is a non-negative linear functional. Condition (iii) is also clear: B, is the present value of the discount bond B, which is equivalent, of course, to the claim corresponding to 1. Only condition (iv> requires some justification. The claim corresponding to the identity function is clearly equivalent to the right to receive delivery at time T of one share of S. If 4.x) < S,, then a hypothetical investor would buy Correspondence 0165.1765/92/$05.00
to: S.J. Dilworth,
University
0 1992 - Elsevier
of South Carolina,
Science
Publishers
Columbia,
SC 29208, USA
B.V. All rights reserved
468
S.J. Dilworth
/ The ~duation
ofcontingent claims
the claim and sell S short. Assuming that the stock does not pay a dividend, at time T he will receive delivery of the stock, cover his short position, and make a riskless arbitrage profit of S,, - ~4x1. On the other hand, if r(x) > S,, then the investor will sell the claim and buy the stock for a riskless profit of r(x) - S,. So r(x) = S, is the only price which does not present an opportunity for arbitrage. Suppose that r is any price which satisfies 6) to (iv). Since ‘27 contains ‘Z’(,, the Riesz Representation Theorem [e.g. Royden (1988>] guarantees the existence of a finite Bore1 measure v such that
for all 4 E go. We shall restrict attention to those price functions 7~ for which (1) extends to all $I E %?. Call such prices and their associated measures admissible. Condition (iii) and (1) imply that v is a probability measure. Condition (iv) implies that
/x dv(x)
= ;, 0
that is, that S,/B, is the barycenter of ZJ. Conversely, any probability measure v with barycenter S,/B, determines via (1) a price r on the cone E’,, of all continuous 4 which are integrable with respect to V; this cone satisfies (a) to (cl above. For example, the simplest admissible measure is the point-mass measure located at S,/B,, for which the corresponding price is B,,4(S,/B,). But usually the latter will not be a realistic candidate for a price function, because it takes no account of our expectations of the future price of S. The problem is to select an admissible measure which appropriately reflects our expectations in some manner. An obvious candidate is p, the law of S,, since this contains all the information about S, that is presently available. But since
/
x dp(x)
= IE[S,.] = &, 0
where A = S,/(B,E[S,]), condition (iv) will not be satisfied unless A = 1. (Here E[S,] denotes, as usual, the expected value of the random variable S,.) The quantity A is the ratio of the stock price to the present value of its expected return at time T. Because risky assets are priced at a discount from their expected return, one would normally expect A < 1 with riskier stocks having correspondingly lower values of A. How, then, can we modify p in such a way that the new measure is admissible? The most natural mathematical transformation to apply is to dilute p by a factor of A to obtain a measure V, which is defined by v(E) = ~(h-‘E) for every Bore1 set E. This measure v is admissible, and the corresponding price function, r(4), is given by each of the three equivalent formulas:
B&(X)
dv(x)
=B&(Ax)
d/4x)
=&E[4(A&)1.
(2)
The justification for selecting v from the collection of all admissible measures is somewhat heuristic and therefore open to question. Indeed, for a particular distribution w there may be compelling reasons for making a different choice. But without the benefit of additional information, v appears to be the most plausible candidate as well as the most natural one from a mathematical point of view.
S.J. Dilworth
2. An example: The Black-Scholes
/ The duation
of contingent
469
claims
price
Suppose that S,, the price of S at time t, is a geometric Wiener process with parameters p and (TV: dS, = pSt + cr2dZ,, where Z, is the standard Brownian Motion. Suppose also that B,, the price density function (p.d.f.) p(x) given by of B at time t, is e Pr(T-r). Then the law of Sr has probability
p(x)
and GS,l by
=
&
= S,, exp((p
q(x)
exp[ - (In( x/S,,)
+ u2/2>T>.
P(X/A)
= 7
- ~T)‘/2a’7]
!?, X
Thus A = exp((r - (p + a2/2))T),
1 = ~ J2xra
[ - (ln( x/S,)
and so v has p.d.f. q(x) given
- (r - ~‘/2)T)~/2rr’T]
$.
exp
Using (1) to price the claim corresponding Black-Scholes formula [Black and Scholes strike price X and expiration date T) on a In the following section we consider the relationships proved below necessarily hold
to C,(x) = max(x -X, O), one obtains the classical (197311 for the theoretical price of a call option (with stock which pays no dividend. call option in a more general setting, In particular, the for the Black-Scholes model.
3. An example: The call option We return now to the general case, in which p is the distribution To compute the price of a call option, we use (2) to evaluate rr(C,): r(Cx)
=B,/C,(x)
dv(x)
=B,,/max(hx-X,
of S, and A = S,/(B,,E[S,.]).
0) dp(x)
=%/CX,A(X) ‘b.(x) =+%~[(‘&A),]). This formula has an obvious interpretation. In pricing a call option with strike price X, a hypothetical investor should: (i) compute the risk factor A; (ii) compute the present value of the expected return on a call option with the higher strike price X/A; and finally, (iii) decrease the result of (ii) by a factor A. Suppose now that early exercise of the option is possible. Let us show that if the price is r(C,), then it will never be rational to exercise the option before expiration. To see this, we use the fact that C,(x) is a conuex function. So by Jensen’s inequality [e.g. Royden (1988, p. 11.5>], we have
~B,,C,(jx do(x)) =WAWB,) = max( S, - XB,,
0) 2 max( S, - X, 0)
by condition
(iv) since v is admissible]
S.J. Dilworth / The ualuation of contingent claims
470
Thus the theoretical price is always at least as great as the intrinsic value of the option. So early exercise would not be rational. Finally, we consider the important special case in which r*.(S,) has p.d.f. p(x, T I S,) = of S,. This occurs when S,/S, has a distribution (l/S&(x/S,, T), where p(x, T) is independent which is independent of S,. It follows that 1 - = B, “xp( x, T) dx. /0 A The theoretical
price of a claim
=B,, /
C#J~is given by
o~4(~&,*)p(x, T) dx.
Provided C$is sufficiently regular [e.g. if 4’(x) is bounded], and noting that h is independent we obtain the following formula for the hedge ratio (or delra) &~(c$)/&S,:
w4>
as,, Evaluating
=hB,~mx~‘( hS,x)p(
for the call option,
wcx> =AB as,,
we have
m 0
/
x, T) dx.
XP(X, T) dx
X/LLs,,)
=W,( (x-W(AS,))P(X~ “‘X/(AS,,) \ ~B”qG/,),I -+7
L 0
r(C.Y) =-+SC, where
F,(x)
is the cumulative
B,X
B,X
T) dx+
=
WC,Y>
as,, ’
0
0
m
P(X, T) sn/X/LLs,,)
dx
BnX (1-Fp(x/w”)))~ S” distribution
function
of p. Rearranging
where N,
m
~/h/(~~ pk T) dx
4
=X(1 - ~,(X/W,,))).
this formula,
we obtain
of S,,,
S.J. Dilworth
/ The valuation of contingent
471
claims
Thus one obtains a ‘portfolio’, consisting of N, shares of S and IV2 shares of B sold short, which instantaneously has the same value as (C,), and the same sensitivity to changes in the price of S. This is the analogue, in our setting, of the replicating portfolio, which occurs in the derivation of the Black-Scholes formula. Observe, finally, that the option’s gamma is given by
a*wa as,2
a4 Bo - =p(X/(ASo), qg. = as,s,
0
References Black, F. and M. Scholes, 1973, The pricing of options and corporate liabilities, Royden, H.L., 1988, Real analysis, Third edition (Macmillan, London).
Journal
of Political
Economy
81. 637-654.
467
39 (1992) 467-471
A note on the valuation of contingent claims S.J. Dilworth Uniwrsity
of South Carolina,
Received Accepted
15 June 1992 27 July 1992
Columbia
SC, USA
We present an approach, which is derived from functional analysis, to the valuation of contingent claims on a stock whose price is a random variable with a given distribution. When the price of the stock is a geometric Wiener process the Black-Scholes price is obtained. Following this approach, formulas are derived in a general setting for the price of a call-option and its sensitivity to changes in the price of the underlying stock.
1. A functional Let S be Let B be a denote the functions 4
analysis
approach
a stock whose value at time T is a random variable S, with law p supported on [O, ~1. discount bond maturing at time T with a face value of one unit of money. S, and B, present values of S and B respectively. Let 27 be a cone of non-negative continuous on [O, a> satisfying the following:
(a) ‘Z’ contains (b) ‘57contains (c) 55’contains
1 (the function that is identically one); x (the identity function); go (the cone of non-negative continuous
functions
vanishing
For each 4 E ‘Z”, let +r denote a contingent claim which can be exercised equal to the random variable 4(S,). We wish to define a price function following natural conditions: (9 (ii> (iii) (iv)
7r(4) ~(a$ 77(l) 57(x)
2 0 for all 4 E %; + b$) = arr(4) + brr($) = B,; = S,,.
at infinity). at time T for a payout r on 5% satisfying the
for all 4, I,!JE $9 and for all a, b 2 0;
Given such a function r, we regard ~44) as being a possible price for the claim 4T for each 4 E 29. Conditions (i> and (ii) then have a standard interpretation: they say simply that r is a non-negative linear functional. Condition (iii) is also clear: B, is the present value of the discount bond B, which is equivalent, of course, to the claim corresponding to 1. Only condition (iv> requires some justification. The claim corresponding to the identity function is clearly equivalent to the right to receive delivery at time T of one share of S. If 4.x) < S,, then a hypothetical investor would buy Correspondence 0165.1765/92/$05.00
to: S.J. Dilworth,
University
0 1992 - Elsevier
of South Carolina,
Science
Publishers
Columbia,
SC 29208, USA
B.V. All rights reserved
468
S.J. Dilworth
/ The ~duation
ofcontingent claims
the claim and sell S short. Assuming that the stock does not pay a dividend, at time T he will receive delivery of the stock, cover his short position, and make a riskless arbitrage profit of S,, - ~4x1. On the other hand, if r(x) > S,, then the investor will sell the claim and buy the stock for a riskless profit of r(x) - S,. So r(x) = S, is the only price which does not present an opportunity for arbitrage. Suppose that r is any price which satisfies 6) to (iv). Since ‘27 contains ‘Z’(,, the Riesz Representation Theorem [e.g. Royden (1988>] guarantees the existence of a finite Bore1 measure v such that
for all 4 E go. We shall restrict attention to those price functions 7~ for which (1) extends to all $I E %?. Call such prices and their associated measures admissible. Condition (iii) and (1) imply that v is a probability measure. Condition (iv) implies that
/x dv(x)
= ;, 0
that is, that S,/B, is the barycenter of ZJ. Conversely, any probability measure v with barycenter S,/B, determines via (1) a price r on the cone E’,, of all continuous 4 which are integrable with respect to V; this cone satisfies (a) to (cl above. For example, the simplest admissible measure is the point-mass measure located at S,/B,, for which the corresponding price is B,,4(S,/B,). But usually the latter will not be a realistic candidate for a price function, because it takes no account of our expectations of the future price of S. The problem is to select an admissible measure which appropriately reflects our expectations in some manner. An obvious candidate is p, the law of S,, since this contains all the information about S, that is presently available. But since
/
x dp(x)
= IE[S,.] = &, 0
where A = S,/(B,E[S,]), condition (iv) will not be satisfied unless A = 1. (Here E[S,] denotes, as usual, the expected value of the random variable S,.) The quantity A is the ratio of the stock price to the present value of its expected return at time T. Because risky assets are priced at a discount from their expected return, one would normally expect A < 1 with riskier stocks having correspondingly lower values of A. How, then, can we modify p in such a way that the new measure is admissible? The most natural mathematical transformation to apply is to dilute p by a factor of A to obtain a measure V, which is defined by v(E) = ~(h-‘E) for every Bore1 set E. This measure v is admissible, and the corresponding price function, r(4), is given by each of the three equivalent formulas:
B&(X)
dv(x)
=B&(Ax)
d/4x)
=&E[4(A&)1.
(2)
The justification for selecting v from the collection of all admissible measures is somewhat heuristic and therefore open to question. Indeed, for a particular distribution w there may be compelling reasons for making a different choice. But without the benefit of additional information, v appears to be the most plausible candidate as well as the most natural one from a mathematical point of view.
S.J. Dilworth
2. An example: The Black-Scholes
/ The duation
of contingent
469
claims
price
Suppose that S,, the price of S at time t, is a geometric Wiener process with parameters p and (TV: dS, = pSt + cr2dZ,, where Z, is the standard Brownian Motion. Suppose also that B,, the price density function (p.d.f.) p(x) given by of B at time t, is e Pr(T-r). Then the law of Sr has probability
p(x)
and GS,l by
=
&
= S,, exp((p
q(x)
exp[ - (In( x/S,,)
+ u2/2>T>.
P(X/A)
= 7
- ~T)‘/2a’7]
!?, X
Thus A = exp((r - (p + a2/2))T),
1 = ~ J2xra
[ - (ln( x/S,)
and so v has p.d.f. q(x) given
- (r - ~‘/2)T)~/2rr’T]
$.
exp
Using (1) to price the claim corresponding Black-Scholes formula [Black and Scholes strike price X and expiration date T) on a In the following section we consider the relationships proved below necessarily hold
to C,(x) = max(x -X, O), one obtains the classical (197311 for the theoretical price of a call option (with stock which pays no dividend. call option in a more general setting, In particular, the for the Black-Scholes model.
3. An example: The call option We return now to the general case, in which p is the distribution To compute the price of a call option, we use (2) to evaluate rr(C,): r(Cx)
=B,/C,(x)
dv(x)
=B,,/max(hx-X,
of S, and A = S,/(B,,E[S,.]).
0) dp(x)
=%/CX,A(X) ‘b.(x) =+%~[(‘&A),]). This formula has an obvious interpretation. In pricing a call option with strike price X, a hypothetical investor should: (i) compute the risk factor A; (ii) compute the present value of the expected return on a call option with the higher strike price X/A; and finally, (iii) decrease the result of (ii) by a factor A. Suppose now that early exercise of the option is possible. Let us show that if the price is r(C,), then it will never be rational to exercise the option before expiration. To see this, we use the fact that C,(x) is a conuex function. So by Jensen’s inequality [e.g. Royden (1988, p. 11.5>], we have
~B,,C,(jx do(x)) =WAWB,) = max( S, - XB,,
0) 2 max( S, - X, 0)
by condition
(iv) since v is admissible]
S.J. Dilworth / The ualuation of contingent claims
470
Thus the theoretical price is always at least as great as the intrinsic value of the option. So early exercise would not be rational. Finally, we consider the important special case in which r*.(S,) has p.d.f. p(x, T I S,) = of S,. This occurs when S,/S, has a distribution (l/S&(x/S,, T), where p(x, T) is independent which is independent of S,. It follows that 1 - = B, “xp( x, T) dx. /0 A The theoretical
price of a claim
=B,, /
C#J~is given by
o~4(~&,*)p(x, T) dx.
Provided C$is sufficiently regular [e.g. if 4’(x) is bounded], and noting that h is independent we obtain the following formula for the hedge ratio (or delra) &~(c$)/&S,:
w4>
as,, Evaluating
=hB,~mx~‘( hS,x)p(
for the call option,
wcx> =AB as,,
we have
m 0
/
x, T) dx.
XP(X, T) dx
X/LLs,,)
=W,( (x-W(AS,))P(X~ “‘X/(AS,,) \ ~B”qG/,),I -+7
L 0
r(C.Y) =-+SC, where
F,(x)
is the cumulative
B,X
B,X
T) dx+
=
WC,Y>
as,, ’
0
0
m
P(X, T) sn/X/LLs,,)
dx
BnX (1-Fp(x/w”)))~ S” distribution
function
of p. Rearranging
where N,
m
~/h/(~~ pk T) dx
4
=X(1 - ~,(X/W,,))).
this formula,
we obtain
of S,,,
S.J. Dilworth
/ The valuation of contingent
471
claims
Thus one obtains a ‘portfolio’, consisting of N, shares of S and IV2 shares of B sold short, which instantaneously has the same value as (C,), and the same sensitivity to changes in the price of S. This is the analogue, in our setting, of the replicating portfolio, which occurs in the derivation of the Black-Scholes formula. Observe, finally, that the option’s gamma is given by
a*wa as,2
a4 Bo - =p(X/(ASo), qg. = as,s,
0
References Black, F. and M. Scholes, 1973, The pricing of options and corporate liabilities, Royden, H.L., 1988, Real analysis, Third edition (Macmillan, London).
Journal
of Political
Economy
81. 637-654.