Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
MFE6516 Stochastic Calculus for Finance
William C. H. Leon
Nanyang Business School
December 11, 2017
1 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
1 Stochastic CalculusIto IntegralIto-Doeblin FormulaSome Applications
2 Multivariate Stochastic CalculusMultivariate Brownian MotionMultivariate Ito-Doeblin Formula
3 Black-Scholes-Merton ModelBlack-Scholes-Merton EquationParabolic Partial Differential Equation
2 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Ito IntegralIto-Doeblin FormulaSome Applications
Ito Integral for Simple Integrands
Let(Ω,F ,
(Ft
),P
)be a filtered probability space, W be a Brownian
motion, h be an adapted process and T be a positive constant.
Let Π = {t0, t1, t2, . . . , tm} be a partition of [0,T ] and consider a simpleprocess
h(t) =
m∑i=1
h(ti−1) 1[ti−1,ti )(t).
3 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Ito IntegralIto-Doeblin FormulaSome Applications
For 0 ≤ t ≤ T , define
I (t) =
∫ t
0
h(u) dW (u)
as follow:
I (t) =
k−1∑i=1
h(ti−1)(W (ti )−W (ti−1)
)+ h(tk−1)
(W (t)−W (tk−1)
),
if tk−1 ≤ t < tk , and
I (T ) =
m∑i=1
h(ti−1)(W (ti )−W (ti−1)
).
4 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Ito IntegralIto-Doeblin FormulaSome Applications
Some Properties of Ito Integral
I (t) =
∫ t
0
h(u) dW (u)
1 Ito integral is a martingale.
2 E (I (t)) = 0 for all t ≥ 0.
3 (Ito isometry) E(I 2(t)
)= E
(∫ t
0h2(u) du
)for all t ≥ 0.
4 (Quadratic variation) [I , I ](t) =∫ t
0h2(u) du for all t ≥ 0.
5 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Ito IntegralIto-Doeblin FormulaSome Applications
Proof
Let 0 ≤ s < t ≤ T where s ∈ [tl−1, tl ), t ∈ [tk−1, tk) and l ≤ k . Then
I (t) =
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩
I (s) + h(tl−1)(W (t)−W (s)
)if l = k ;
I (s)+h(tl−1)(W (tl )−W (s)
)+
k−1∑i=l+1
h(ti−1)(W (ti )−W (ti−1)
)+ h(tk−1)
(W (t)−W (tk−1)
)if l < k .
Thus,
E( I (t) | F(s)) = I (s) and E (I (t)) = 0.
6 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Ito IntegralIto-Doeblin FormulaSome Applications
Let Di = W (ti )−W (ti−1), for i = 1, 2, . . . , k − 1, andDk = W (t)−W (ti−1). Then
I 2(t) =
(k∑
i=1
h(ti−1)Di
)2
=
k∑i=1
h2(ti−1)D2i + 2
∑∑1≤i<j≤k
h(ti−1)h(tj−1)DiDj .
Note that
1 h2(ti−1) ∈ F(ti−1) and Di is independent of F(ti−1), for 1 ≤ i ≤ k .
2 h(ti−1)h(tj−1)Di ∈ F(tj−1) and Dj is independent of F(tj−1), for1 ≤ i < j ≤ k .
7 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Ito IntegralIto-Doeblin FormulaSome Applications
E(I 2(t)
)=
k∑i=1
E(h2(ti−1)D
2i
)+ 2
∑∑1≤i<j≤k
E (h(ti−1)h(tj−1)DiDj)︸ ︷︷ ︸=0
=k−1∑i=1
E(h2(ti−1)
)(ti − ti−1) + E
(h2(tk−1)
)(t − tk−1)
= E
(k−1∑i=1
h2(ti−1)(ti − ti−1) + h2(tk−1)(t − tk−1)
)
= E
(∫ t
0
h2(u) du
).
8 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Ito IntegralIto-Doeblin FormulaSome Applications
Let ΠS = {s0, s1, s2, . . . , sm} be a partition of [tj−1, tj ]. Then
lim‖ΠS‖→0
m∑i=1
(I (si )− I (si−1)
)2
= lim‖ΠS‖→0
h2(tj−1)
m∑i=1
(W (si )−W (si−1)
)2
= h2(tj−1)(tj − tj−1
)=
∫ tj
tj−1
h2(u) du.
Thus,
[I , I ](t) =
∫ t
0
h2(u) du.
9 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Ito IntegralIto-Doeblin FormulaSome Applications
Ito Integral for General Integrands
Suppose the adapted process h is square integrable, i.e.
E
(∫ T
0
h2(u) du
)< ∞.
Choose a sequence of simple process hn such that
limn→∞
E
(∫ T
0
∣∣hn(u)− h(u)∣∣2 du
)= 0.
For 0 ≤ t ≤ T , define∫ t
0
h(u) dW (u) = limn→∞
∫ t
0
hn(u) dW (u).
10 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Ito IntegralIto-Doeblin FormulaSome Applications
Properties of Ito Integral
Fix T > 0. Let h be an adapted and square integrable process. Then theIto integral I (t) =
∫ t
0h(u) dW (u) as defined earlier has the following
properties:
1 (Continuity) Paths of I (t), as a function of t, are continuous.
2 (Adaptivity) I (t) is F(t)-measurable for all t.
3 (Linearity) Ito integral of a linear combination is the linearcombination of Ito integrals.
4 (Martingale) Ito integral is a martingale.
5 (Ito isometry) E(I 2(t)
)= E
(∫ t
0h2(u) du
)for all t ≥ 0.
6 (Quadratic variation) [I , I ](t) =∫ t
0h2(u) du for all t ≥ 0.
11 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Ito IntegralIto-Doeblin FormulaSome Applications
Exercise
Compute ∫ T
0
W (u) dW (u).
Hint: Let n ∈ Z+ and ti =
Tni , for i = 0, 1, 2, . . . , n. Consider the simple
process
hn(t) =
n∑i=1
W (ti−1) 1[ti−1,ti )(t).
12 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Ito IntegralIto-Doeblin FormulaSome Applications
Solution
∫ T
0
W (u) dW (u) = limn→∞
∫ T
0
hn(u) dW (u)
= limn→∞
n∑i=1
W (ti−1)(W (ti )−W (ti−1)
)
= limn→∞
1
2
(W 2(T )−
n∑i=1
(W (ti )−W (ti−1)
)2)
=1
2
(W 2(T )− [W ,W ](T )
)=
1
2
(W 2(T )− T
).
13 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Ito IntegralIto-Doeblin FormulaSome Applications
Remark
Recall
Var (W (t)) = t = [W ,W ](t).
Let
I (t) =
∫ t
0
W (u) dW (u) =1
2
(W 2(t)− t
).
Then
Var (I (t)) =1
4Var
(W 2(t)
)=
1
2t2
[I , I ](t) =
∫ t
0
W 2(u) du.
14 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Ito IntegralIto-Doeblin FormulaSome Applications
Ito-Doeblin Formula for Brownian Motion
Let f (t, x) be a C 1,2 function and W be a Brownian motion. Then, forevery T ≥ 0,
f(T ,W (T )
)=f
(0,W (0)
)+
∫ T
0
ft(t,W (t)
)dt
+
∫ T
0
fx(t,W (t)
)dW (t)︸ ︷︷ ︸
Ito integral
+1
2
∫ T
0
fxx(t,W (t)
)dt;
and in differential form
df(T ,W (T )
)=ft(T ,W (T )) dT
+ fx(T ,W (T )) dW (T ) +1
2fxx(T ,W (T )) dT .
15 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Ito IntegralIto-Doeblin FormulaSome Applications
Sketch of Proof
Let Π = {t0, t1, t2, . . . , tn} be a partition of [0,T ]. Then
f(T ,W (T )
)− f(0,W (0)
)=
n∑i=1
(f(ti ,W (ti )
)− f(ti−1,W (ti−1)
)).
By Taylor’s Theorem,
f(ti ,W (ti )
)− f(ti−1,W (ti−1)
)=ft
(ti−1,W (ti−1)
)(ti − ti−1
)+ fx
(ti−1,W (ti−1)
)(W (ti )−W (ti−1)
)+
1
2ftt
(ti−1,W (ti−1)
)(ti − ti−1
)2+ ftx
(ti−1,W (ti−1)
)(ti − ti−1
)(W (ti )−W (ti−1)
)+
1
2fxx
(ti−1,W (ti−1)
)(W (ti )−W (ti−1)
)2+ ◦(higher order).
16 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Ito IntegralIto-Doeblin FormulaSome Applications
Since
lim‖Π‖→0
n∑i=1
ft(ti−1,W (ti−1)
)(ti − ti−1
)=
∫ T
0
ft(t,W (t)
)dt,
lim‖Π‖→0
n∑i=1
fx(ti−1,W (ti−1)
)(W (ti )−W (ti−1)
)=
∫ T
0
fx(t,W (t)
)dW (t),
lim‖Π‖→0
n∑i=1
ftt(ti−1,W (ti−1)
)(ti − ti−1
)2= 0,
lim‖Π‖→0
n∑i=1
ftx(ti−1,W (ti−1)
)(ti − ti−1
)(W (ti )−W (ti−1)
)= 0,
lim‖Π‖→0
n∑i=1
fxx(ti−1,W (ti−1)
)(W (ti )−W (ti−1)
)2=
∫ T
0
fxx(t,W (t)
)dt,
we have the Ito-Doeblin formula.
17 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Ito IntegralIto-Doeblin FormulaSome Applications
Example
Let f (x) = 12x
2. Then
1
2W 2(T ) = f
(W (T )
)− f(W (0)
)=
∫ T
0
fx(W (t)
)︸ ︷︷ ︸W (t)
dW (t) +1
2
∫ T
0
fxx(W (t)
)︸ ︷︷ ︸1
dt
=
∫ T
0
W (t) dW (t) +1
2T .
18 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Ito IntegralIto-Doeblin FormulaSome Applications
Example
Let s, r and σ be positive constants, and(W (t)
)t≥0
be a Brownian
motion. Define
f(t,W (t)
)= s e
(r− 12σ
2)t+σW (t), for t ≥ 0.
Show that
f(t,W (t)
)= s +
∫ t
0
r f(u,W (u)
)du +
∫ t
0
σ f(u,W (u)
)dW (u).
19 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Ito IntegralIto-Doeblin FormulaSome Applications
Solution
Consider f(t, x
)= s e
(r− 12σ
2)t+σx ∈ C 1,2. Since
ft(t, x
)=
(r − 1
2σ2)f(t, x
),
fx(t, x
)= σ f
(t, x
), and fxx
(t, x
)= σ2 f
(t, x
),
the Ito-Doeblin formula says
df(u,W (u)
)=
(r − 1
2σ2)f(u,W (u)
)︸ ︷︷ ︸=ft
(u,W (u)
) du + σ f(u,W (u)
)︸ ︷︷ ︸=fx
(u,W (u)
) dW (u) +1
2σ2 f
(u,W (u)
)︸ ︷︷ ︸=fxx
(u,W (u)
) du
= r f(u,W (u)
)du + σ f
(u,W (u)
)dW (u).
20 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Ito IntegralIto-Doeblin FormulaSome Applications
Ito Process
Let W be a Brownian motion. An Ito process is a stochastic process ofthe form
X (t) = X (0) +
∫ t
0
g(u) du +
∫ t
0
h(u) dW (u)
where X (0) is non-random, and g and h are adapted stochasticprocesses.
1 Ito process are continuous.
2 Almost all stochastic processes, except those with jumps, are Itoprocess.
3 [X ,X ](t) =∫ t
0h2(u) du.
21 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Ito IntegralIto-Doeblin FormulaSome Applications
Proof
Let R(t) =∫ t
0g(u) du and I (t) =
∫ t
0h(u) dW (u). Then
[X ,X ](t) = [R ,R](t)︸ ︷︷ ︸=0
+ 2 [R , I ](t)︸ ︷︷ ︸=0
+ [I , I ](t)
=
∫ t
0
h2(u) du.
22 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Ito IntegralIto-Doeblin FormulaSome Applications
Integration w.r.t. Ito Process
Let X be an Ito process where
X (t) = X (0) +
∫ t
0
g(u) du +
∫ t
0
h(u) dW (u)
and Γ be an adapted process. Define the integral with respect to an Itoprocess ∫ t
0
Γ(u) dX (u) =
∫ t
0
Γ(u)g(u) du +
∫ t
0
Γ(u)h(u) dW (u)︸ ︷︷ ︸Ito process
.
23 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Ito IntegralIto-Doeblin FormulaSome Applications
Ito-Doeblin Formula for Ito Process
Let f (t, x) be a C 1,2 function, W be a Brownian motion and X be an Itoprocess where
X (t) = X (0) +
∫ t
0
g(u) du +
∫ t
0
h(u) dW (u).
Then for every T ≥ 0
f(T ,X (T )
)= f
(0,X (0)
)+
∫ T
0
ft(t,X (t)) dt
+
∫ T
0
fx(t,X (t)) dX (t) +1
2
∫ T
0
fxx(t,X (t)) d [X ,X ](t)
= f(0,X (0)
)+
∫ T
0
ft(t,X (t)) dt +
∫ T
0
fx(t,X (t))g(t) dt
+
∫ T
0
fx(t,X (t))h(t) dW (t) +1
2
∫ T
0
fxx(t,X (t))h2(t) dt.
24 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Ito IntegralIto-Doeblin FormulaSome Applications
In differential form,
dX (T ) = g(T ) dT + h(T ) dW (T )
and
df(T ,X (T )
)= ft(T ,X (T )) dT
+ fx(T ,X (T )) dX (T ) +1
2fxx(T ,X (T )) d [X ,X ](T )
= ft(T ,X (T )) dT + fx(T ,X (T ))g(T ) dT
+ fx(T ,X (T ))h(T ) dW (T ) +1
2fxx(T ,X (T ))h2(T ) dT .
25 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Ito IntegralIto-Doeblin FormulaSome Applications
Integration by Parts Formula
Let X and Y be Ito processes. Then for every T ≥ 0
X (T )Y (T ) = X (0)Y (0)
+
∫ T
0
X (t) dY (t) +
∫ T
0
Y (t) dX (t) + [X ,Y ](T ).
In particular,
X 2(T ) = X 2(0) + 2
∫ T
0
X (t) dX (t) + [X ,X ](T ).
26 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Ito IntegralIto-Doeblin FormulaSome Applications
In differential form,
d(X (T )Y (T )
)= X (T ) dY (T ) + Y (T ) dX (t) + dX (T ) dY (T )
and
d(X 2(T )
)= 2X (T ) dX (T ) + dX (T ) dX (T )
27 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Ito IntegralIto-Doeblin FormulaSome Applications
Proof
Using the Ito-Doeblin formula for f (x) = x2, we have
X 2(T ) = X 2(0) + 2
∫ T
0
X (t) dX (t) + [X ,X ](T ).
The integration by parts formula then follows from
X (T )Y (T ) =1
2
((X (T ) + Y (T )
)2 − X 2(T )− Y 2(T )).
28 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Ito IntegralIto-Doeblin FormulaSome Applications
Ito Integral of Deterministic Integrand
Let h be a non-random function of time t and
I (t) =
∫ t
0
h(u) dW (u).
For each t ≥ 0, the random variable
I (t) ∼ N(0,
∫ t
0
h2(u) du
).
29 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Ito IntegralIto-Doeblin FormulaSome Applications
Proof
Since I is a martingale,
E (I (t)) = I (0) = 0,
and by Ito isometry,
Var (I (t)) = E(I 2(t)
)=
∫ t
0
h2(u) du.
30 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
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Black-Scholes-Merton Model
Ito IntegralIto-Doeblin FormulaSome Applications
Let u ∈ R, and define
X (t) = −1
2
∫ t
0
(u h(s)
)2
ds︸ ︷︷ ︸=u2 Var(I (t))
+
∫ t
0
u h(s) dW (s)︸ ︷︷ ︸=u I (t)
.
From the Ito-Doeblin formula,
d(eX (t)
)= u h(t) eX (t) dW (t),
eX (t) = 1 +
∫ t
0
u h(s) eX (s) dW (s),
therefore,
E
(eX (t)
)= 1 ⇒ E
(eu I (t)
)= e
12 u
2Var(I (t)).
31 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
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Black-Scholes-Merton Model
Ito IntegralIto-Doeblin FormulaSome Applications
Generalized Geometric Brownian Motion
Let W be a Brownian motion, and μ and σ be adapted stochasticprocesses. The stochastic differential equation
dS(t) = S(t)(μ(t) dt + σ(t) dW (t)
)has the solution
S(t) = S(0) e∫
t
0
(μ(u)− 1
2σ2(u)
)du+
∫t
0σ(u) dW (u).
If μ and σ are deterministic functions, then
ln
(S(t)
S(0)
)∼ N
(∫ t
0
(μ(u)− 1
2σ2(u)
)du,
∫ t
0
σ2(u) du
).
32 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
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Black-Scholes-Merton Model
Ito IntegralIto-Doeblin FormulaSome Applications
Verification
Let
X (t) =
∫ t
0
(μ(u)− 1
2σ2(u)
)du +
∫ t
0
σ(u) dW (u).
Then
dX (t) =
(μ(t)− 1
2σ2(t)
)dt + σ(t) dW (t)
and
d [X ,X ](t) = σ2(t) dt.
33 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
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Black-Scholes-Merton Model
Ito IntegralIto-Doeblin FormulaSome Applications
Let f (x) = S(0) ex . Then
fx(x) = f (x) and fxx(x) = f (x).
From the Ito-Doeblin formula,
df(X (t)
)= fx
(X (t)
)dX (t) +
1
2fxx
(X (t)
)d [X ,X ](t)
dS(t) = S(t)(μ(t) dt + σ(t) dW (t)
).
34 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Ito IntegralIto-Doeblin FormulaSome Applications
Ornstein-Uhlenbeck or Mean Reverting Process
Let W be a Brownian motion, and α, β and σ be positive constants.The stochastic differential equation
dR(t) = β(α− R(t)
)dt + σ dW (t)
has the solution
R(t) = e−β tR(0) + α
(1− e
−β t)+ σ e−β t
∫ t
0
eβ u dW (u)
∼ N(e−β tR(0) + α
(1− e
−β t),σ2
2β
(1− e
−2β t))
.
35 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
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Black-Scholes-Merton Model
Ito IntegralIto-Doeblin FormulaSome Applications
Verification
Let
X (t) =
∫ t
0
eβ u dW (u) ∼ N
(0, e
2βt−12β
).
Then
dX (t) = eβ t dW (t) and d [X ,X ](t) = e
2β t dt.
Let
f (t, x) = e−β tR(0) + α
(1− e
−β t)+ σ e−β tx
Then
ft(t, x) = β(α− f (t, x)
),
fx(t, x) = σ e−β t and fxx(t, x) = 0.
36 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Ito IntegralIto-Doeblin FormulaSome Applications
From the Ito-Doeblin formula,
df(t,X (t)
)= ft(t,W (t)) dt + fx(t,X (t)) dX (t)
dR(t) = β(α− R(t)
)dt + σ dW (t).
37 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Ito IntegralIto-Doeblin FormulaSome Applications
Solving Linear Stochastic Differential Equation
Let W be a Brownian motion, and α, β, γ and σ be deterministicfunctions. The linear stochastic differential equation
dX (t) =(α(t) + β(t)X (t)
)dt +
(γ(t) + σ(t)X (t)
)dW (t)
has the solution
X (t) = e
∫t
0
(β(u)− 1
2σ2(u)
)du+
∫t
0σ(u) dW (u) ×
[X (0)
+
∫ t
0
e−
∫u
0
(β(v)− 1
2σ2(v)
)dv−
∫u
0σ(v) dW (v)
(α(u)− γ(u)σ(u)
)du
+
∫ t
0
e−
∫u
0
(β(v)− 1
2σ2(v)
)dv−
∫u
0σ(v) dW (v)γ(u) dW (u)
].
38 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Ito IntegralIto-Doeblin FormulaSome Applications
Proof
Consider solutions of the form X (t) = X1(t)X2(t) where
dX1(t) = β(t)X1(t) dt + σ(t)X1(t) dW (t),
dX2(t) = a(t) dt + b(t) dW (t)
for some functions a(t) and b(t). Then
dX (t) = X1(t) dX2(t) + X2(t) dX1(t) + d [X1,X2](t)
=( =α(t)︷ ︸︸ ︷(
a(t) + σ(t)b(t))X1(t) + β(t)X (t)
)dt
+(b(t)X1(t)︸ ︷︷ ︸
=γ(t)
+ σ(t)X (t))dW (t)
39 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Ito IntegralIto-Doeblin FormulaSome Applications
Since
a(t) =α(t)− γ(t)σ(t)
X1(t)and b(t) =
γ(t)
X1(t),
therefore,
X1(t) = e
∫t
0
(β(u)− 1
2σ2(u)
)du+
∫t
0σ(u) dW (u)
and
X2(t) = X (0) +
∫ t
0
α(u)− γ(u)σ(u)
X1(u)du +
∫ t
0
γ(u)
X1(u)dW (u).
40 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Ito IntegralIto-Doeblin FormulaSome Applications
Mean Reverting Process
Let W be a Brownian motion, and α, β and σ be positive constants.The stochastic differential equation
dR(t) = β(α− R(t)
)dt + σ
√R(t) dW (t)
has the “solution”
R(t) = e−β tR(0) + α
(1− e
−β t)+ σ e−β t
∫ t
0
eβ u
√R(u) dW (u),
and
E (R(t)) → α and Var (R(t)) → ασ2
2β
as t → ∞.
41 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Ito IntegralIto-Doeblin FormulaSome Applications
Verification
Let f (t, x) = eβ tx . Then
ft(t, x) = β f (t, x), fx(t, x) = eβ t and fxx(t, x) = 0.
From the Ito-Doeblin formula,
df(t,R(t)
)= ft(t,R(t)) dt + fx(t,R(t)) dR(t),
d(eβ tR(t)
)= αβ e
β t dt + σ eβ t√
R(t) dW (t).
Hence,
eβ tR(t) = R(0) + α
(eβ t − 1
)+ σ
∫ t
0
eβ u
√R(u) dW (u).
42 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Ito IntegralIto-Doeblin FormulaSome Applications
Since
R(t) = e−β tR(0) + α
(1− e
−β t)+ σ e−β t
∫ t
0
eβ u
√R(u) dW (u)︸ ︷︷ ︸
Ito integral
,
thus
E (R(t)) = e−β tR(0) + α
(1− e
−β t)
and
limt→∞
E (R(t)) = α.
43 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Ito IntegralIto-Doeblin FormulaSome Applications
Var (R(t)) = E
((σ e−β t
∫ t
0
eβ u
√R(u) dW (u)
)2)
= σ2e−2β t
E
(∫ t
0
e2β uR(u) du
)
= σ2e−2β t
∫ t
0
e2β u
E (R(u)) du
=ασ2
2β
(1− 2 e−β t + e
−2β t)+ R(0)
σ2
β
(e−β t − e
−2β t)
and
limt→∞
Var (R(t)) =ασ2
2β.
44 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Multivariate Brownian MotionMultivariate Ito-Doeblin Formula
Definition of Multivariate Brownian Motion
A d-dimensional Brownian motion is a process
W (t) =(W1(t),W2(t), . . . ,Wd (t)
)�,
for t ≥ 0, such that:
1 Each Wi (t) is a 1-dimensional Brownian motion.
2 Wi (t) and Wj(t) are independent for i = j .
Associated with{W (t)
}t≥0
is a filtration{F(t)
}t≥0
such that:
3 F(s) ⊂ F(t) for 0 ≤ s < t.
4 W (t) is F(t)-measurable for each t ≥ 0.
5 W (t)−W (s) is independent of F(s) for 0 ≤ s < t.
45 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Multivariate Brownian MotionMultivariate Ito-Doeblin Formula
Cross Variation
For a multivariate Brownian motion W ,
dWi (t) dWj (t) =
{dt if i = j ;
0 if i = j .
46 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Multivariate Brownian MotionMultivariate Ito-Doeblin Formula
2-Dimensional Ito-Doeblin Formula
Let f (t, x , y) be a C 1,2,2 function, W = (W1,W2)� be a two dimensional
Brownian motion, and X and Y be Ito processes where
X (t) = X (0) +
∫ t
0
α1(u) du +
∫ t
0
σ11(u) dW1(t) +
∫ t
0
σ12(u) dW2(t)
Y (t) = Y (0) +
∫ t
0
α2(u) du +
∫ t
0
σ21(u) dW1(t) +
∫ t
0
σ22(u) dW2(t).
47 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Multivariate Brownian MotionMultivariate Ito-Doeblin Formula
Then
f(t,X (t),Y (t)
)= f
(0,X (0),Y (0)
)+
∫ t
0
ft(u,X (u),Y (u)
)du
+
∫ t
0
fx(u,X (u),Y (u)
)dX (u) +
∫ t
0
fy(u,X (u),Y (u)
)dY (u)
+1
2
∫ t
0
fxx(u,X (u),Y (u)
)d [X ,X ](u)
+1
2
∫ t
0
fyy(u,X (u),Y (u)
)d [Y ,Y ](u)
+
∫ t
0
fxy(u,X (u),Y (u)
)d [X ,Y ](u).
48 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Multivariate Brownian MotionMultivariate Ito-Doeblin Formula
In differential form,
dX (t) = α1(t) dt + σ11(t) dW1(t) + σ12(t) dW2(t),
dY (t) = α2(t) dt + σ21(t) dW1(t) + σ22(t) dW2(t)
and
df(t,X (t),Y (t)
)= ft
(t,X (t),Y (t)
)dt + fx
(t,X (t),Y (t)
)dX (t)
+ fy(t,X (t),Y (t)
)dY (t) +
1
2fxx
(t,X (t),Y (t)
)d [X ,X ](t)
+1
2fyy
(t,X (t),Y (t)
)d [Y ,Y ](t) + fxy
(t,X (t),Y (t)
)d [X ,Y ](t).
49 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Multivariate Brownian MotionMultivariate Ito-Doeblin Formula
df (·) =[ft(·) + α1(t)fx(·) + α2(t)fy (·)
+1
2
(σ211(t) + σ2
12(t))fxx(·) + 1
2
(σ221(t) + σ2
22(t))fyy (·)
+(σ11(t)σ21(t) + σ12(t)σ22(t)
)fxy (·)
]dt
+(σ11(t)fx(·) + σ21(t)fy (·)
)dW1(t)
+(σ12(t)fx(·) + σ22(t)fy (·)
)dW2(t).
50 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Multivariate Brownian MotionMultivariate Ito-Doeblin Formula
Multi-Dimensional Ito-Doeblin Formula
Let f (t, x) : R+ × Rm �→ R be a function with continuous partial
derivatives ft , fxi and fxi xj , for 1 ≤ i , j ≤ m, and X be a m-dimensional Itoprocess with
dX(t) = α
(t,X(t)
)dt + σ
(t,X(t)
)dW(t)
where α = (αi )m×1, σ = (σij)m×d and W = (Wi )d×1 is a d-dimensionalBrownian motion. Then
df =
⎛⎝ft +
m∑i=1
αi fxi +1
2
m∑i=1
m∑j=1
d∑k=1
σikσjk fxi xj
⎞⎠ dt +
m∑i=1
d∑k=1
σik fxidWk
=
(ft + (∇Xf )
�α+
1
2Tr
(σ
�HX f σ
))dt + (∇Xf )
�σ dW(t).
51 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Multivariate Brownian MotionMultivariate Ito-Doeblin Formula
Levy Characterization of Brownian Motion
Let the stochastic processes Mi be martingales, for i = 1, 2, . . . , d .Suppose Mi (0) = 0, Mi (t) has continuous paths, and
[Mi ,Mj ](t) =
{t if i = j ,
0 otherwise,
for 1 ≤ i , j ≤ d and t ≥ 0. Then M =(Mi
)is a d-dimensional Brownian
motion.
52 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Multivariate Brownian MotionMultivariate Ito-Doeblin Formula
Proof
Suppose d = 1. Let f (t, x) = eux− 1
2 u2t . Then
df(t,M(t)
)=
=− 12 u
2f (t,x)︷ ︸︸ ︷ft(t,M(t)
)dt +
=u f (t,x)︷ ︸︸ ︷fx(t,M(t)
)dM(t) + 1
2
=u2f (t,x)︷ ︸︸ ︷fxx
(t,M(t)
) =dt︷ ︸︸ ︷d [M,M](t),
= u f(t,M(t)
)dM(t),
f(t,M(t)
)= 1 +
∫ t
0
u f(s,M(s)
)dM(s)︸ ︷︷ ︸
Martingale
.
Hence,
E
(eu M(t)
)= e
12 u
2tE(f(t,M(t)
))= e
12 u
2t .
53 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Multivariate Brownian MotionMultivariate Ito-Doeblin Formula
Let 0 ≤ τ < t. Then
d(euM(t)
)= u e
u M(t) dM(t) +1
2u2 eu M(t) dt
eu M(t) = e
u M(τ) + u
∫ t
τ
eu M(s) dM(s)︸ ︷︷ ︸Martingale
+1
2u2
∫ t
τ
eu M(s) ds.
Therefore,
E
(eu
(M(t)−M(τ)
) ∣∣∣∣ F(τ)
)= 1 +
1
2u2
∫ t
τ
E
(eu
(M(s)−M(τ)
) ∣∣∣∣ F(τ)
)ds
d
dt
(E
(eu
(M(t)−M(τ)
) ∣∣∣∣ F(τ)
))=
1
2u2 E
(eu
(M(t)−M(τ)
) ∣∣∣∣ F(τ)
).
Hence,
E
(eu
(M(t)−M(τ)
) ∣∣∣∣ F(τ)
)= e
12 u
2(t−τ).
54 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Multivariate Brownian MotionMultivariate Ito-Doeblin Formula
Let 0 ≤ τ1 < t1 ≤ τ2 < t2. Then
E
(eu2
(M(t2)−M(τ2)
)+u1
(M(t1)−M(τ1)
))
= E
(E
(eu2
(M(t2)−M(τ2)
) ∣∣∣∣ F(τ2)
)eu1
(M(t1)−M(τ1)
))
= e12 u
22(t2−τ2) E
(E
(eu1
(M(t1)−M(τ1)
) ∣∣∣∣ F(τ1)
))
= e12 (u
22(t2−τ2)+u2
1(t1−τ1))
Hence increments over non-overlapping intervals are independent.
55 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Multivariate Brownian MotionMultivariate Ito-Doeblin Formula
Suppose d = 2. From the one-dimensional result, both M1 and M2 areBrownian motions. It remains to show that they are independent. Letf (t, x , y) = e
u1x+u2y−12 (u
21+u2
2)t . Then
df(t,M1(t),M2(t)
)=
=− 12 (u
21+u2
2)f (t,x,y)︷ ︸︸ ︷ft(t,M1(t),M2(t)
)dt
+
=u1 f (t,x,y)︷ ︸︸ ︷fx(t,M1(t),M2(t)
)dM1(t) +
=u2 f (t,x,y)︷ ︸︸ ︷fy(t,M1(t),M2(t)
)dM2(t)
+ 12
=u21 f (t,x,y)︷ ︸︸ ︷
fxx(t,M1(t),M2(t)
) =dt︷ ︸︸ ︷d [M1,M1](t) +
=u1u2f (t,x,y)︷ ︸︸ ︷fxy
(t,M1(t),M2(t)
) =0︷ ︸︸ ︷d [M1,M2](t)
+ 12
=u22 f (t,x,y)︷ ︸︸ ︷
fyy(t,M1(t),M2(t)
) =dt︷ ︸︸ ︷d [M2,M2](t),
= f(t,M(t)
)(u1 dM1(t) + u2 dM2(t)
).
56 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Multivariate Brownian MotionMultivariate Ito-Doeblin Formula
Since
f(t,M1(t),M2(t)
)= 1 +
Martingale︷ ︸︸ ︷∫ t
0
u1 f(s,M1(s),M2(s)
)dM1(s)
+
∫ t
0
u2 f(s,M1(s),M2(s)
)dM2(s)︸ ︷︷ ︸
Martingale
,
thus,
E
(eu1 M1(t)+u2 M2(t)
)= e
12 (u
21+u2
2)t E(f(t,M1(t),M2(t)
))= e
12 (u
21+u2
2)t .
57 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Black-Scholes-Merton EquationParabolic Partial Differential Equation
Risk-Free Asset Price Process
Let M(t), t ≥ 0, denotes the risk-free asset price process that is modeledby
dM(t) = r M(t) dt
where r is a constant. The ordinary differential equation has the solution
M(t) = er t
assuming M(0) = 1.
58 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Black-Scholes-Merton EquationParabolic Partial Differential Equation
Risky Asset Price Process
Let S(t), t ≥ 0, denotes the risky asset price process that is modeled bya geometric Brownian motion
dS(t) = S(t)(μ dt + σ dW (t)
)where W is a Brownian motion, and μ and σ > 0 are constants. Thestochastic differential equation has the solution
S(t) = S(0) e
(μ− 1
2σ2)t+σW (t)
and
ln
(S(t)
S(0)
)∼ N
((μ− 1
2σ2)t, σ2t
).
59 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Black-Scholes-Merton EquationParabolic Partial Differential Equation
Portfolio Value Process
Consider a portfolio strategy(h(t), v(t)
), t ≥ 0, where h(t) and v(t)
represent the unit holdings at time t in the risky and risk-free assetsrespectively.
Let X (t), t ≥ 0, denotes the portfolio value process of the abovestrategy. Then
X (t) = h(t) S(t) + v(t)M(t)
and
dX (t) = h(t) dS(t) + v(t) dM(t)
= X (t) r dt︸ ︷︷ ︸Riskless Return
+ h(t) S(t)(μ− r
)dt︸ ︷︷ ︸
Risk Compensation
+ h(t) S(t)σ dW (t)︸ ︷︷ ︸Risk / Volatility
.
60 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Black-Scholes-Merton EquationParabolic Partial Differential Equation
Verification
Since
X (t +Δt)− X (t)
= h(t)(S(t +Δt)− S(t)
)︸ ︷︷ ︸=S(t)
(μΔt+σΔW (t)
) + v(t)(M(t +Δt)−M(t)
)︸ ︷︷ ︸
=r M(t)Δt
,
letting Δt → 0, we have
dX (t) = h(t) dS(t) + v(t) dM(t)
= h(t) S(t)(μ dt + σ dW (t)
)+ v(t)M(t)︸ ︷︷ ︸
=X (t)−h(t)S(t)
r dt
= X (t) r dt + h(t) S(t)(μ− r
)dt + h(t) S(t)σ dW (t).
61 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Black-Scholes-Merton EquationParabolic Partial Differential Equation
Self-Financing
The continuous-time self-financing condition
dX (t) = h(t) dS(t) + v(t) dM(t)
can equivalently be stated as
S(t) dh(t) + dS(t) dh(t) +M(t) dv(t) + dM(t) dv(t)︸ ︷︷ ︸=0
= 0.
62 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Black-Scholes-Merton EquationParabolic Partial Differential Equation
Proof
Using Ito-Doeblin formula,
dX (t)︸ ︷︷ ︸=h(t) dS(t)+v(t) dM(t)
= h(t) dS(t) + S(t) dh(t) + d [S , h](t)
+ v(t) dM(t) +M(t) dv(t) + d [M, v ](t),
hence
S(t) dh(t) + dS(t) dh(t) +M(t) dv(t) + dM(t) dv(t) = 0.
63 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Black-Scholes-Merton EquationParabolic Partial Differential Equation
Verification
Since(h(t +Δt)− h(t)
)S(t +Δt) +
(v(t +Δt)− v(t)
)M(t +Δt) = 0,
S(t)(h(t +Δt)− h(t)
)+
(S(t +Δt)− S(t)
)(h(t +Δt)− h(t)
)+M(t)
(v(t +Δt)− v(t)
)+
(M(t +Δt)−M(t)
)(v(t +Δt)− v(t)
)= 0,
letting Δt → 0, we have
S(t) dh(t) + dS(t) dh(t) +M(t) dv(t) + dM(t) dv(t) = 0.
64 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Black-Scholes-Merton EquationParabolic Partial Differential Equation
Self-Financing Condition
For t ≥ 0, the value of a portfolio at time t is
X (t) =
m∑i=0
hi (t) Si (t),
where Si (t) is the price of asset i and hi (t) is the unit of asset i in theportfolio at time t, for i = 0, 1, . . . ,m.
The trading strategy(h0(t), h1(t), . . . , hm(t)
)t≥0
is self-financing if
dX (t) =m∑i=0
hi (t) dSi (t),
for t ≥ 0.
65 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Black-Scholes-Merton EquationParabolic Partial Differential Equation
Option Value Process
Consider a simple option that pays g(S(T )
)at time T .
Suppose the value of the option at time t is given by C(t, S(t)
), for
0 ≤ t ≤ T . Then
dC(t, S(t)
)=
(Ct
(t, S(t)
)+ μS(t)CS
(t, S(t)
)+
1
2σ2S2(t)CSS
(t, S(t)
))dt
+ σS(t)CS
(t, S(t)
)dW (t).
66 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
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Verification
Since
dS(t) = S(t)(μ dt + σ dW (t)
)and d [S , S ](t) = σ2S2(t) dt,
from the Ito-Doeblin formula,
dC(t, S(t)
)= Ct
(t, S(t)
)dt + CS
(t, S(t)
)dS(t) +
1
2CSS
(t, S(t)
)d [S , S ](t)
=
(Ct
(t, S(t)
)+ μ S(t)CS
(t, S(t)
)+
1
2σ2S2(t)CSS
(t, S(t)
))dt
+ σ S(t)CS
(t, S(t)
)dW (t).
67 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Black-Scholes-Merton EquationParabolic Partial Differential Equation
Equating Evolutions
Consider a hedging portfolio that consists of a short one unit of optionand h(t) units of the underlying at time t, 0 ≤ t ≤ T .
The portfolio value at time t is
X (t) = h(t) S(t)− C(t, S(t)
)and
dX (t) = h(t) dS(t) +
=0 Self-Financing︷ ︸︸ ︷S(t) dh(t) + dS(t) dh(t)− dC
(t, S(t)
)=
(μ h(t) S(t)− Ct
(t, S(t)
)− μ S(t)CS
(t, S(t)
)− 1
2σ2S2(t)CSS
(t, S(t)
))dt
+ σ S(t)(h(t)− CS
(t, S(t)
))dW (t).
68 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Black-Scholes-Merton EquationParabolic Partial Differential Equation
Choose h(t) = CS
(t, S(t)
). Then
dX (t) =
(−Ct
(t, S(t)
)− 1
2σ2S2(t)CSS
(t, S(t)
))dt
= r X (t) dt.
Therefore
−Ct
(t, S(t)
)− 1
2σ2S2(t)CSS
(t, S(t)
)= r
(S(t)CS
(t, S(t)
)− C(t, S(t)
)).
69 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Black-Scholes-Merton EquationParabolic Partial Differential Equation
Black-Scholes-Merton Partial Differential Equation
The value of the option C (t, s) satisfies the Black-Scholes-Merton partialdifferential equation
Ct(t, s) + r s Cs(t, s) +1
2σ2s2Css(t, s) = r C (t, s),
for 0 ≤ t < T and 0 ≤ s < ∞, with terminal condition
C (T , s) = g(s).
70 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Black-Scholes-Merton EquationParabolic Partial Differential Equation
Second-Order Partial Differential Equation
General second-order partial differential equation in two variables, τ andx , has the form
A fττ + B fτx + C fxx + · · · = 0.
The partial differential equation is
1 Elliptic if B2 − 4AC < 0;
2 Parabolic if B2 − 4AC = 0;
3 Hyperbolic if B2 − 4AC > 0.
The Black-Scholes-Merton partial differential equation is an example of abackward parabolic partial differential equation that can be transformedinto a form analogous to the heat equation.
71 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Black-Scholes-Merton EquationParabolic Partial Differential Equation
Homogeneous Heat Equation
The following initial value problem on (−∞,∞):
fτ (τ, x) = a fxx(τ, x), 0 < τ < ∞, −∞ < x < ∞,
f (0, x) = g(x);
has the solution
f (τ, x) =1√4πaτ
∫ ∞
−∞
e−
(x−y)2
4aτ g(y) dy .
72 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Black-Scholes-Merton EquationParabolic Partial Differential Equation
The following initial value problem on [0,∞) with homogenous Dirichletboundary conditions:
fτ (τ, x) = a fxx(τ, x), 0 < τ < ∞, 0 ≤ x < ∞,
f (0, x) = g(x),
f (τ, 0) = 0;
has the solution
f (τ, x) =1√4πaτ
∫ ∞
0
(e−
(x−y)2
4aτ − e−
(x+y)2
4aτ
)g(y) dy .
73 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Black-Scholes-Merton EquationParabolic Partial Differential Equation
European Call Option
The value of a European call option C (t, s), with strike price K andmaturity time T , satisfies the Black-Scholes-Merton partial differentialequation
Ct(t, s) + r s Cs(t, s) +1
2σ2s2Css(t, s)− r C (t, s) = 0
for 0 ≤ t < T and 0 ≤ s < ∞, with terminal condition
C (T , s) = (s − K )+.
74 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Black-Scholes-Merton EquationParabolic Partial Differential Equation
Transformation
Under the following transformation:
x = ln s and τ = T − t;
the Black-Scholes-Merton partial differential equation becomes
Cτ (τ, x)−(r − 1
2σ2
)Cx(τ, x)− 1
2σ2Cxx(τ, x) + r C (τ, x) = 0
for 0 < τ ≤ T and −∞ < x < ∞, with initial condition
C (0, x) = ( ex − K )+.
75 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Black-Scholes-Merton EquationParabolic Partial Differential Equation
Let
C (τ, x) = eατ+βx f (τ, x)
where
α = −r − 1
2σ2
(r − 1
2σ2
)2
and β =1
2− r
σ2.
Then the Black-Scholes-Merton partial differential equation reduces tothe heat equation
fτ (τ, x) =1
2σ2fxx(τ, x)
for 0 < τ ≤ T and −∞ < x < ∞, with initial condition
f (0, x) = e−βx( ex − K )+.
76 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance
Stochastic CalculusMultivariate Stochastic Calculus
Black-Scholes-Merton Model
Black-Scholes-Merton EquationParabolic Partial Differential Equation
Solution
f (τ, x) =1√
2πσ2τ
∫ ∞
lnK
e−
(x−y)2
2σ2τ
(e(1−β)y − K e
−βy)dy
and
C (τ, x) = eατ+βx f (τ, x)
= exN(d+)− e
−rτK N(d−)
where
d± =x − lnK + (r ± 1
2σ2)τ
σ√τ
.
77 / 77 William C. H. Leon MFE6516 Stochastic Calculus for Finance