carr, bandyopadhay-how to derive bsm correctly
TRANSCRIPT
1
How to Derive Black-Scholes Equation Correctly?
Peter P. Carr1
Banc of America Securities
9 West 57th Street, 40th Floor
New York, NY 10019
and
Akash Bandyopadhyay2
Loomis Laboratory of Physics
University of Illinois at Urbana-Champaign
1110 West Green Street
Urbana, IL 61801
Abstract
This paper points out the mathematical deficiencies in the derivation of the Black-
Scholes partial differential equation as found in many MBA level textbooks. In addition to
correcting the analysis, we present a financial justification for Black and Scholes original
hedging arguments.
1 E.Mail: [email protected], (212) 583-8529 (Voice), (212) 583-8569 (FAX)
2 E.Mail: [email protected], (217) 332-2465 (Voice), (217) 333-9819 (FAX)
2
I. Introduction
The Black-Scholes-Merton [1,2] analysis is the central tool for pricing and hedging
options and other derivative securities. Many option pricing textbooks aimed at MBA's
derive the Black-Scholes partial differential equation (pde) in a mathematically
inconsistent manner [3,4,5,6]. The mathematically correct evaluation of the total
derivative of their hedged portfolio renders the portfolio neither riskless nor self-financing.
Such inconsistencies make it difficult for mathematically inclined newcomers to
appreciate the derivation of the central tenet of derivative pricing theory.
There are now many good derivations of the Black-Scholes equation [7,8,9].
While mathematically rigorous, these derivations generally do not focus on explaining the
financial insights behind the great success of Black-Scholes-Merton theory. The purpose
of this paper is three-fold. First, to re-construct Black-Scholes analysis in a
mathematically sound fashion from Merton’s self-financing portfolio. It is done in the
next section. Second, in section III, we pinpoint the mathematical and financial problems
in the original derivation which is blindly repeated with meaningless or no arguments in
several textbooks. Third, in section IV, we present a financial justification for Black and
Scholes original hedging technique. We believe that Black and Scholes original thoughts
were more on financial perspectives of option valuation than on precise mathematical
analysis of portfolios.
II. Mathematical Foundation to Black-Scholes Analysis
3
The existence of self-financing continuous trading strategy, and the existence of
an unique riskfree interest rate in the market are the two fundamental building blocks of
the Black-Scholes-Merton analysis. Uniqueness of riskfree interest rate is a consequence
of the absence of arbitrage opportunity in the market. It states that all riskless assets obey
the same deterministic price process. Any riskfree portfolio can be replicated by the
riskless asset under this condition. To quantify the ideas, let us assume the standard
continuous time price processes for a risky asset St (stock), its dividend Dt , and the
riskfree asset Bt (zero coupon bond):
( ) ( )[ ] ( )dS S t S t S dt S t S dW S tt t t t t t t= − + > > ≥µ δ σ µ, , , , , , ;0 0 0 0 (2.1a)
( )dD S t S dt D tt t t= ≥ ≥ ≥δ δ, , , , ;0 0 0 0 (2.1b)
( )dB r B t B dt B r tt t t= ≥ ≥ ≥, , , , ;0 0 0 0 (2.1c)
where ( )µ S tt , and ( )σ S tt , are the expected rate of return and volatility respectively. The
unique riskfree interest rate is ( )r B tt , , and ( )δ S tt , is the dividend yield on the stock. Wt
is a Wiener process. We assume that the time t value of a path-independent derivative
security Vt is a [ ]( )C T2 1 0, ,ℜ ×+ function ( )V S tt , . Its payoff is given by a continuous
deterministic terminal boundary condition:
( ) ( )V S T g S S TT T T, , , .= ∈ ℜ >+ 0 (2.2)
Consider the value of a self-financing portfolio Π t at time [ ]t T∈ 0, consisting of
α t ≠ 0 derivative securities, ∆ t shares of stock and β t bonds:
[ ]Π ∆ Πt t t t t t tV S B t T= + + ≥ ∈α β , , , ;0 0 0 (2.3a)
4
[ ]Π Π ∆ ∆ Πt u u
t
u u
t
u u
t
u u
t
dV dS dD dB t T= + + + + ≥ ∈∫ ∫ ∫ ∫00 0 0 0
0 0 0α β , , , . (2.3b)
The differential form of the self-financing condition (2.3b) sets a constraint on the trading
strategy ( )α βt t t, ,∆ :
( ) ( ) ( ) [ ]d V dV d S dS d B dB dD t Tt t t t t t t t t t tα β+ + + + + = ∈∆ ∆ , , .0 (2.4)
It allows the following formula for the dynamics of this self-financing portfolio:
( ) [ ]d dV dS S dt dB t Tt t t t t t t tΠ ∆= + + + ∈α δ β , , .0 (2.5)
It is crucial to realize that eqn. (2.5) cannot be obtained without the self-financing
condition (2.4).
Applying It⊥ ’s lemma to ( )V V S tt t= , and setting the hedge ratio
[ ]∆ t
t t
VS
t Tα
∂∂
= − ∀ ∈ 0, we find:
[ ]dVt
SV
SS
VS
rB dt t Tt t t tt
t tt
t tΠ = + − +
∈α ∂
∂α σ ∂
∂α δ ∂
∂β1
202 2
2
2 , , . (2.6)
Uniqueness of riskfree interest rate means that this riskless portfolio can be replicated by a
bond position. Therefore:
[ ]d r dt t Tt tΠ Π Π= ≥ ∈, , , .0 0 0 (2.7)
Equating the last two equations we obtain the Black-Scholes pde:
( ) [ ]∂∂
σ ∂∂
δ ∂∂
Vt
SV
Sr S
VS
rV S t Ttt
tt
t+ + − − = ∈ ℜ ∈+12
0 02 22
2 , , , . (2.8)
5
It is important to notice that the self-financing condition and the hedge ratio set
two constraints on the trading strategy ( ) [ ]α βt t t t T, , ,∆ ∀ ∈ 0 . Therefore only one
portfolio weight can be chosen arbitrarily. Once a portfolio weight is fixed for its life, the
other two weights are uniquely determined from the hedge ratio, self-financing condition
(2.4), initial value of the portfolio Π 0 , and the terminal boundary condition on the
derivative security (2.2). Since dynamic trading strategies in derivatives are expansive in
practice, without any loss of generality we set [ ]α t t T= − ∀ ∈1 0, in the subsequent
analysis.
The standard hedging argument given in Black and Scholes paper [1], and in many
MBA level textbooks [3,4,5,6] over-constrain the portfolio by fixing two portfolio weights
for its life. In the next section we will see that this is the root of the inconsistencies in the
traditional analysis [7]. In section IV, we discuss a method to circumvent this problem.
III. Problems in Black-Scholes Analysis
The hedging argument given in Black and Scholes paper [1] fixed the number of
shares of stock held at one and varied the number of call options written. Since dynamic
trading strategies in options are expensive in practice, the usual textbook derivation
instead fixes the number of options written at one and then varies the number of shares of
stock held long. Both derivations assume that there is no riskfree asset in the portfolio for
its life. Without loss of generality, let us focus on the textbook derivation. Thus, consider
6
the value of a portfolio at t ≥ 0 consisting of one written derivative security Vt and ∆ t
shares of dividend-free stock held long:
[ ]H V S t Tt t t t= − + ∈∆ , , .0 (3.1)
Following the Black-Scholes derivation, textbooks generally offer that the total derivative
of the portfolio is given by:
[ ]dH dV dS t Tt t t t= − + ∈∆ , , .0 (3.2)
Applying It⊥ ’s lemma, and blindly substituting the hedge ratio [ ]∆ tt
VS
t T= ∀ ∈∂∂
0,
yields:
[ ]dHVt
SV
Sdt t Tt t
t
= − −
∈∂
∂σ ∂
∂12
02 22
2 , , . (3.3)
It thus appears that the differential change in value of this portfolio is deterministic and
therefore riskless, and hence it is claimed that from the absence of arbitrage:
dH rH dt H tt t= ≥ ≥, , .0 0 0 (3.4)
Setting expressions (3.3) and (3.4) equal to each other yields the Black-Scholes pde (2.8)
at δ = 0 . This analysis suffers from two major drawbacks. It is a great favor of luck that
the following two inconsistencies exactly cancel each other, and thereby one gets the right
partial differential equation for the valuation of derivative securities [10].
Mathematical Problem : The total derivative of the portfolio was computed in a
meaningless manner. While:
( ) [ ]d S d S dS d dS t Tt t t t t t t t∆ ∆ ∆ ∆= + + ∀ ∈ 0, ; (3.5)
7
this analysis assumes:
( ) [ ]d S dS t Tt t t t∆ ∆= ∀ ∈ 0, ; (3.6)
which, in fact, demands [ ]∆ t constant t T= ∀ ∈ 0, -- a direct contradiction to the
definition of hedge ratio as [ ]∆ tt
VS
t T= ∀ ∈∂∂
0, . The portfolio was constructed by
helding two portfolio weights constant for its life, namely α t = − 1 and β t = 0
[ ]∀ ∈t T0, . Since ∆ t is not a constant, it violates the self-financing condition (2.4).
Therefore, the traditional hedge portfolio is in fact a non-self-financing position.
Financial Problem : Riskfree price process (2.1c) has been applied to a portfolio whose
strictly correct differential change in value is not deterministic. Since ∆ t randomly
changes with time, the traditional hedge portfolio is a risky position.
The derivation becomes much worse for dividend paying stocks where the text
books usually offer ( ) [ ]dH dV dS dD t Tt t t t t= − + + ∈∆ , ,0 from eqn. (3.1). In the next
section we discuss a method to circumvent these problems.
IV. Financial Foundation to Black-Scholes Analysis
The proof given in the Back-Scholes paper [1] can be made strictly correct simply
by replacing the mathematical operation of taking a total derivative with the financial
operation of computing the “gain” on a portfolio. We believe that Black and Scholes did
8
not wanted to compute the total derivative of the hedged portfolio consisting options and
stock. Instead, they were interested in the financial gain on the hedged portfolio.
Consider the value of a portfolio Ht at time t consisting of one written derivative
security and ∆ t shares of stock held long:
[ ]H V S t Tt t t t= − + ∈∆ , , .0 (4.1)
We define the financial gain on the portfolio Ht at time t as:
( ) ( ) [ ]g H dV dS S du t Tt u
t
u u u
t
= − + + ∀ ∈∫ ∫0 0
0∆ δ , . (4.2)
Applying It⊥ ’s lemma on ( )V V S uu u= , , and hedging the portfolio by setting
[ ]∆ uu
VS
u t= ∀ ∈∂∂
0, yields:
( ) [ ]g HVu
SV
SS
VS
du t Tt uu
uu
t
= − − +
∀ ∈∫ ∂
∂σ ∂
∂δ ∂
∂12
02 22
20
, . (4.3)
Since this financial gain is deterministic for all time [ ]t T∈ 0, , absence of arbitrage
requires that it be the same as the interest gain on a dynamic position ( )B B S tt t= , in the
riskless asset chosen so as to finance all trades in the derivative and the stock:
( ) [ ]g B r V SVS
du t Tt u uu
t
= − +
∀ ∈∫ ∂∂
00
, . (4.4)
Equating the financial gains (4.3) and (4.4) then leads to the Black-Scholes pde (2.9).
Given the non-self-financing portfolio Ht , we can always construct the following
self-financing riskless portfolio:
9
[ ]Π t t t tH t T= ≠ ∀ ∈α α, , ;0 0 (4.5a)
( ) ( )( ) [ ]− + + + = ∈d V dV d S dS dD t Tt t t t t t t t t tα α α∆ ∆ , , ;0 (4.5b)
[ ]d r dt t Tt tΠ Π= ∈, , .0 (4.5c)
Thus, financial gain on Ht is a self-financing trading strategy on a multiplicative portfolio.
This idea of financial gain justifies the original hedging argument proposed by
Black and Scholes [1] and reproduced in many textbooks [3,4,5,6]. Equations (4.5a,b,c)
illustrates that Black-Scholes portfolio can be dynamically modified to a self-financing
strategy, which provides a solid foundation to the long debated analysis.
It is worth noting that the portfolio consisting of the option and stock is not self-
financing. Similarly, positions in the riskless asset are not self-financing. Nonetheless, by
showing that the trading gains between two non-self-financing strategies are always equal
under no arbitrage, the value of the derivative security can be determined.
V. Conclusion
We discussed several financial and mathematical issues for basic option pricing
theory in continuous time. These fundamental points are often overlooked in MBA level
treatments. Our analysis provides a rigorous foundation to Black and Scholes original
hedging arguments from the perspective of financial gain on an arbitrary portfolio.
More than twenty five years has passed since Black, Scholes, and Merton
developed the present theoretical framework of option valuation. At present, the
10
derivatives business has become a more than US$15 trillion market. Unfortunately, many
subtle issues at the theoretical foundation of option valuation are still not well-explained
clearly in standard business school literature. We believe that this paper will provide some
insight on the deep question, “Why does the Black-Scholes-Merton analysis work?”
ACKNOWLEDGMENT
Akash would like to thank Prof. Yoshi Oono for his tremendous support. His
work is supported by National Science Foundation grant NSF-DMR99-70690.
11
References
[1] Black, F., and M. Scholes. (1973). “The Pricing of Options and Corporate
Liabilities.” Journal of Political Economy 81: 637-654.
[2] Merton, R. (1973). “The Theory of Rational Option Pricing.” Bell Journal of
Economics and Management Science 4: 141-183.
(1977). “On the Pricing of Contingent Claims and the Modigliani-Miller Theorem.”
Journal of Financial Economics 5: 241-250.
[3] Hull, J. (1999). Options, Futures, and Other Derivatives (4th Ed.). Upper Saddle
River, New Jersey: Prentice-Hall.
[4] Wilmott, P., J. Dewynne, and S. Howison. (1993). Option Pricing: Mathematical
Models and Computation . Oxford: Oxford Financial Press.
(1995). The Mathematics of Financial Derivatives: A Student Introduction . New
York: Cambridge University Press.
(1998). Derivatives, The Theory and Practice of Financial Engineering . West
Sussex, England: John Wiley & Sons.
[5] Shimko, D. (1992). Finance in Continuous Time: A Primer . Miami, Florida: Kolb
Publishing Company.
[6] Kwok, Y. (1998). Mathematical Models of Financial Derivatives . Singapore:
Springer-Verlag.
[7] Beck, T. (1993). “Black-Scholes Revisited: Some Important Details.” The Financial
Review 28: 77-90.
12
[8] Duffie, D. (1996). Dynamic Asset Pricing Theory (2nd Ed.). Princeton, New Jersey:
Princeton University Press.
[9] Bjrk, T. (1998). Arbitrage Theory in Continuous Time . New York: Oxford
University Press.
[10] Duffie, D. (1988). Security Markets: Stochastic Models . Boston: Academic Press.