Black-Scholes Formula
Brandon Lee
15.450 Recitation 2
Brandon Lee
Black-Scholes Formula
Expectation of a Lognormal Variable
Suppose X ∼ N µ , σ 2 . We want to know how to compute E e X .
This calculation is often needed (e.g., page 30 of Lecture Notes 1)
because we usually assume that log return is distributed normally.
h i Z∞
e x φ (x) dx
E eX =
−∞
Z ∞
1
1
=
exp (x) √
exp − 2 (x − µ)2 dx
2σ
−∞
2πσ 2
Z ∞
1
1
√
=
exp − 2 (x − µ)2 + x dx
2σ
−∞
2πσ 2
Z ∞
1
1
√
=
exp − 2 x 2 + −2µ − 2σ 2 x + µ 2 dx
2σ
−∞
2πσ 2
Z ∞
2 1
1 √
=
exp − 2 x 2 + −2 µ + σ 2 x + µ + σ 2
exp
2σ
−∞
2πσ 2
Z ∞
2
1 2
1
1
√
exp − 2 x − µ + σ 2
dx
= exp µ + σ
2σ
2
−∞
2πσ 2
1
= exp µ + σ 2
2
Brandon Lee
Black-Scholes Formula
Change of Measure
Given a probability measure (think probability distribution), a
random variable that is positive and integrates to one defines
a change of measure. In other words, suppose we have a
probability measure P and a random variable ξ such that
E P [ξ ] = 1. Then we can define a new probability measure Q
through ξ by
Z
Q (A) =
ξ dP
A
We can think of ξ as a redistribution of probability weights
from P to Q. Hence it’s called “change of measure” and
denoted dQ
dP .
Brandon Lee
Black-Scholes Formula
Normality-Preserving Change of Measure
Now, there is a special class of random variables called
exponential martingales that, as change of measures, preserve
normality. In more concrete terms, suppose probability measure P is given by the normal distribution N µ P , σ 2 .
Then, if dQ
dP is an exponential martingale, then the new
probability measure Q is also normally distributed, with
a
Q
2
different mean but with the same variance, N µ , σ .
Such exponential martingales take on the form
1
ξ = exp −ηε P − η 2
2
for arbitrary numbers η (later in Lecture Notes 2, we’ll see
that η can be stochastic processes as well).
Furthermore, we know the exact relationship between µ P and
µ Q : µ P − µ Q = ησ (the previous notes had a typo and had
σ 2 instead of σ ).
Brandon Lee
Black-Scholes Formula
Black-Scholes Formula
Suppose under Q (the risk-neutral measure), the stock return
is given by
1 2
St+1
Q
= exp r − σ + σ ε
St
2
where ε Q ∼ N (0, 1) under the Q-measure.
Let’s derive the Black-Scholes formula in this simple setting.
Suppose S0 = 1 and we have a call option that matures at
T = 1 with a strike price K . The price of this call option is
C = e −r E Q [max (S1 − K , 0)]
=e
−r
Z ∞
S =K
=e
−r
1
Z ∞
S1 =K
Brandon Lee
(S1 − K ) dQ
S1 dQ − e
−r
Z ∞
KdQ
S1 =K
Black-Scholes Formula
Continued
Call the first term C1 and the second term C2 .
Let’s calculate them separately.
C1 = e
−r
Z ∞
S1 =K
Z ∞
S1 dQ
σ2
1
1 Q 2
Q
2 exp
=e
r−
+σε
· √ exp − ε
dε
ln K −r + σ2
2
2
2π
σ
Z ∞
1
1 Q 2
Q
2
= ln K −r + σ 2 √ exp −
− 2σ ε + σ
ε
dε
2
2
2π
σ
Z ∞
2
1
1 Q
= ln K −r + σ 2 √ exp − ε − σ
dε
2
2
2π
σ
!
2
ln K − r + σ2
=Φ σ−
σ
!
2
− ln K + r + σ2
=Φ
σ
−r
Brandon Lee
Black-Scholes Formula
Continued
Now for C2
C2 = e
−r
Z ∞
KdQ
S1 =K
Z ∞
1
1 Q 2
2 K √
exp − ε
=e
dε
ln K −r + σ2
2
2π
σ
Z ∞
1
1 Q 2
−r
= e K ln K −r + σ 2 √ exp − ε
dε
2
2
2π
σ
!
σ2
−
ln
K
+
r
−
2
= e −r K Φ
σ
−r
So the option price is given by the Black-Scholes formula
C = C1 − C2
2
=Φ
− ln K + r + σ2
σ
Brandon Lee
2
!
− e −r K Φ
− ln K + r − σ2
σ
Black-Scholes Formula
!
Numerical Integration
Definite integrals can rarely be computed analytically. In those
cases, we need to resort to numerical methods. Here, we
present the simplest method using the Riemann sum
approximation.
As an example, let’s say we want to compute
Z ∞
1
1
x2
dx
We have to worry about two things: summation on the right
tail and fineness of approximating rectangles.
Refer to the MATLAB ® code.
Brandon Lee
Black-Scholes Formula
MIT OpenCourseWare
http://ocw.mit.edu
15.450 Analytics of Finance
Fall 2010
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