Stochastic Calculus II
Brandon Lee
15.450 Recitation 4
Brandon Lee
Stochastic Calculus II
Kolmogorov Backward Equation
Consider a stochastic process Xt . Xt is a martingale if for s > t,
Et [Xs ] = Xt
In other words, conditional expectation of future value is simply the
current value (example: fair gamble). The notion of a martingale
makes sense for both discrete and continuous time processes.
Now let’s look at this concept when Xt is an Ito process. Suppose
dXt = µ (t, Xt ) dt + σ (t, Xt ) dZt
If Xt were a martingale, then over a small time interval dt,
Et [Xt +dt ] = Xt
Et [Xt +dt − Xt ] = 0
Et [dXt ] = 0
This is another way of thinking about martingales: expected
changes are zero.
Brandon Lee
Stochastic Calculus II
Continued
But now recall the dynamics of dXt :
Et [dXt ] = Et [µ (t, Xt ) dt + σ (t, Xt ) dZt ]
= µ (t, Xt ) dt
So Xt is a martingale if and only if µ (t, Xt ) = 0. This is
intuitive: the drift term is responsible for expected change
whereas the diffusion term is responsible for variance of
change.
The Kolmogorov Backward Equation simply formalizes this
idea.
Brandon Lee
Stochastic Calculus II
Continued
Suppose we have an underlying process Xt where
dXt = µ (t, Xt ) dt + σ (t, Xt ) dZt
Consider Yt = f (t, Xt ). Suppose we know for some reason
that Yt is a martingale. We have seen two such examples: 1)
conditional expectation, 2) security price discounted at
risk-free rate under the risk-neutral measure. Then by the
above argument, we know that the drift term of
dYt = d (f (t, Xt )) should be zero.
By Ito’s Lemma,
df (t , Xt ) =
!
∂ f (t, Xt )
∂ f (t, Xt )
1 ∂ 2 f (t, Xt )
∂ f (t, Xt )
+
µ (t, Xt ) +
(σ (t, Xt ))2 dt +
σ (t, Xt ) dZt
2
∂t
∂ Xt
2
∂ Xt
∂ Xt
So if Yt = f (t, Xt ) is a martingale, then
∂ f (t, Xt ) ∂ f (t, Xt )
1 ∂ 2 f (t, Xt )
+
µ (t, Xt ) +
(σ (t, Xt ))2 = 0
∂t
∂ Xt
2 ∂ Xt2
Brandon Lee
Stochastic Calculus II
Option Replication
How do you dynamically replicate an European option in the
Black-Scholes setting? Answer: Match the “delta”.
Suppose you have the option price f (t, St ) on the one hand and the
value of a replicating portfolio consisting of the stock and the bond,
Wt . At time t, this replicating portfolio holds θt number of stocks
and the rest is invested in the risk-free bond. This means
dWt = θt dSt + r (Wt − θt St ) dt
To have perfect replication, we need
d (f (t, St )) = dWt
In particular we need the coefficient in front of dSt to match:
∂ f (t, St )
= θt
∂ St
This argument is exactly as in yesterday’s lecture where we tried to
replicate an option in an environment with stochastic volatility using
a stock and another option.
Brandon Lee
Stochastic Calculus II
State Price Density and Risk-Neutral Measure
We can always move from one to the other very easily in both
discrete time and continuous time settings.
In discrete time, if the state price density is given by
πt
1 2
= exp −rt−1 − ηt−1 εt − ηt−1
πt−1
2
then the change of measure from P to Q is given by
dQ
1 2
= exp −ηt−1 εt − ηt−1
2
dP t
In continuous time, if the state price density is given by
Zt
Z t
Z
1 t 2
πt = exp −
rs ds −
ηs dZs −
ηs ds
2 0
0
0
then the change of measure from P to Q is given by
Zt
Z
dQ
1 t 2
= exp −
ηs dZs −
ηs ds
dP t
2 0
0
We can always decompose the state price density into time discount
and change of measure from P to Q.
Brandon Lee
Stochastic Calculus II
Change of Measure in Continuous Time
Similar as in the discrete time case. If
Zt
Z
dQ
1 t 2
η ds
= exp − ηs dZs −
2 0 s
dP t
0
then the Brownian motion under P, ZtP acquires a drift but its
diffusion coefficient does not change. Therefore
dZtP = dZtQ − ηt dt
Suppose WtP is another Brownian motion under P and it has
correlation ρ with ZtP . How does WtP look under the
Q-measure?
Do the decomposition
p
WtP = ρZtP + 1 − ρ 2 VtP
where VtP is another Brownian motion under P that’s
independent of ZtP . Conclude
dWtP = dWtQ − ρηt dt
Brandon Lee
Stochastic Calculus II
MIT OpenCourseWare
http://ocw.mit.edu
15.450 Analytics of Finance
Fall 2010
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