Study Notes prepared by Carlos Brioso
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Brownian Motion and Ito’s Lemma
1
- Black-Scholes model assumes that stock prices follow a process called geometric Brownian
motion.
dS (t )
= αdt + σdZ (t )
α : cont. comp. expected return on the stock
S (t )
σ : cont. comp. standard deviation (volatility)
Z(t): normally distributed random variable that follows
a process called Brownian motion
Brownian Motion (BM)
- BM is a stochastic process (function of time) that is a random walk occurring in continuous time
(i.e. flip coin)
- Z(t) represents the value of a Brownian motion at time t.
Z(0)=0
Z(t+s)-Z(t) ~ N (0,s)
Z(t+s1)-Z(t) is independent of Z(t+s2)-Z(t)
Z(t) is continuous
- Z(t) is a martingale: E(Z(t+s)|Z(t))=Z(t)
- Z(T)-Z(0)→N(0,T) (T=n*h : h is the small time length where the stock goes up or down)
T
- Stochastic integral: Z (t ) → Z (0) + ∫ dZ (t )
0
Properties
- The quadratic variation of a Brownian process from time 0 to T is T.
- The absolute length of a Brownian process is infinite over any finite interval.
Arithmetic Brownian Motion
- We allow a non zero mean and arbitrary variance in the relationship:
X (t + h) − X (t ) = αh + σY (t + h) h ~ N (αT , σ 2T )
It’s integral representation is:
T
T
0
0
X (T ) = X (0) + ∫ αdt + ∫ σdZ (t )
- X(t) is normally distributed – scaled Brownian process
- Random term. Since dZ(t) has a variance of 1 per unit of time, σdZ (t ) has a variance of σ 2 per
unit of time
- “non random drift”: effect of adding α per unit of time to X(0)
- Drawbacks: X(t) can be negative. The mean and variance of changes in dollar terms are
independent of the level of price (in practice that is not true)
- Ornsteing-Uhlembeck process: by modifying the drift term we allow mean reversion (if a price is
far from the mean is likely to reverse to the mean)
λ measures the speed of reversion
dX (t ) = λ[α − X (t )]dt + σdZ (t )
Example (SOA Spring 2007 # 24)
Info
{Z(t)} is standard BM
dX (t ) = λ[α − X (t )]dt + σdZ (t )
X(0) is known. Find the solution.
Study Notes prepared by Carlos Brioso
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2
Sol
dX (t ) + λX (t ) dt = λα dt + σdZ (t ) (using the integrating factor e λt )
e λt dX (t ) + e λt λX (t )dt = e λt λαdt + e λt σdZ (t )
e λt dX (t ) + e λt λX (t )dt = e λt λαdt + e λt σdZ (t )
d [e λt X (t )] → d [e λv X (v)] = λαe λv dv + σe λv dZ (v)
Solving by integrating both sides we get:
t
X (t ) = X (0)e −λt + α (1 − e −λt ) + σ ∫ e −λ (t − s ) dZ (v)
(E)
0
Geometric Brownian Motion
- The drift and volatility are function of X (Ito process). The percentage change in an asset is
value is normally distributed with instantaneous mean α and instantaneous variance σ 2 .
dX (t )
= αdt + σdZ (t ) (differential representation)
X (t )
T
T
0
0
X (T ) − X (0) = ∫ αX (t )dt + ∫ σX (t )dZ (t )
- A variable that follows geometric Brownian is lognormally distributed
(α − 0.5σ 2 )t + σ t Z
and E[X(t)]= X (0)eαt
X (t ) = X ( 0 ) e
- Over short periods of time the motion is determined almost entirely by the random component.
The ratio of the standard deviation to the drift
σ
α h
becomes infinite as h becomes smaller.
- Multiplication rules:
(dt ) 2 = 0
(dZ ) 2 = dt
dZxdZ ' = ρ ( dt )
dX (t )
- By using the multiplication rules: if X follows a geometric BM [
= αdt + σdZ (t ) ] the
X (t )
( dt )( dZ ) = 0
Log[X(t)] follows an arithmetic BM ( d [ln X (t )] = (α − 0.5σ 2 )dt + σdZ (t ) ).
The Sharpe ratio
Sharpe ratio=
αi − r
(risk premium per unit of volatility)
σi
Two assets that have the same correlation with the market they will have the same Sharpe ratios.
Example (SOA Spring 2007 # 18)
Info:
Assets X,Y (no div) driven by BM Z.
Study Notes prepared by Carlos Brioso
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dY (t )
dX (t )
= Gdt + HdZ (t )
= 0.07 dt + 1.12dZ (t )
Y (t )
X (t )
d {ln[Y (t )]) = 0.06dt + σdZ (t )
r = 0.04; σ < 0.25 ; and G,H are constants
G=?
Sol:
If the price follows a geometric BM, the log(price) follows arithmetic BM.
dY (t )
= Gdt + HdZ (t ) → d [ln Y (t )] = (G − 0.5H 2 )dt + HdZ (t )
Y (t )
Then H= σ and G = 0.06 + 0.5H 2
X and Y are driven by the same BM→ they must have the same Sharpe ratio
0.07 − 0.04 G − 0.04 0.06 + 0.5 H 2 − 0.04
→ H=0.1→G=0.065 (A)
=
=
0.12
H
H
Example (SOA 2009 QA # 11)
Info:
BS framework. Three statements.
Sol
(i)
(ii)
(iii)
True. Since S(t) follows geometrical BM, ln[S(t)] is lognormally distributed with
Var= σ 2 h
True. Since BS framework, S(t) follows geometrical BM.
True. Prices follow geometrical BM.
Var[ S (t + dt ) | S (t )] = Var (dS (t ) | S (t )) = Var (αS (t )dt + σS (t )dZ (t ) | Z (t )) = σ 2 [ S (t )]2 dt
Example (SOA 2009 QA # 12)
Info:
Two assets X and Y (no div) share a source of uncertainty BM {Z(t)}
dX (t )
dY (t )
= 0.07dt + 0.12dZ (t )
= Adt + BdZ (t )
A,B are constants
X (t )
Y (t )
d [ln Y (t )] = μdt + 0.085dZ (t ) ; r = 0.04
A=?
Sol
dY (t )
= Adt + BdZ (t ) → d [ln Y (t )] = ( A − 0.5B 2 )dt + BdZ (t ) →B=0.085
Y (t )
Since they share the source of uncertainty they must have the same Sharpe ratio
α − r 0.07 − 0.04 A − 0.04
→ A=0.06125 (A)
=
=
σ
0.12
0.085
3
Study Notes prepared by Carlos Brioso
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4
The risk neutral process
dS (t )
- Given
= (α − δ )dt + σdZ (t ) (we expect S to appreciate at rate α − δ and with a deviation
S (t )
from this return of 0 - martingale). For risk-adverse investor (values $1 of gain less than $1 loss)
the expected value will be negative in utility terms.
~
- We create a new process Z (t ) that is martingale to the investor in utility terms. The we have a
risk-neutral price process:
~
dS (t )
= (r − δ )dt + σd Z (t )
S (t )
~
Z (t ) is called risk neutral measure. The transformation was possible thanks to the Girsanov’s
Theorem using a transformed probability distribution.
Ito’s Lemma
^
^
Given a stock with an expected instantaneous return of α , dividend yield of δ and instantaneous
^
volatility σ follows geometric Brownian motion:
^
^
^
dS (t ) = {α [ S (t ), t ] − δ [ S (t ), t ]}dt + σ [ S (t ), t ]dZ (t ) (*) 1
(Ito’s Lemma)
If C[S(t),t] is twice-differentiable function of S(t) the:
^
^
^
1
1^
dC(S , t ) = C S dS + C SS (dS ) 2 + Ct dt = {[α[S , t ] − δ [S , t ]}CS + σ (S , t ) 2 CSS + Ct }dt + σ (S , t )CS dZ
2
2
Note: over a short period of time the delta-gamma approximation is the Ito’s Lemma.
Example (SOA Spring 2007 # 12)
Info:
S(t) : value of one British pound in U.S. dollars at time t.
rUS = 0.08 ; rbrit = 0.10 ; G (t ) = S (t )e ( rusa −rbrit )(T −t ) forward price in U.S. dollars per British
pound.
Based on Itô’s Lemma, dG(t) ?
1
When S(t) follows geometric Brownian motion we have:
^
^
^
α [ S (t ), t ] = αS (t ); δ [ S (t ), t ]} = δS (t );σ [ S (t ), t ] = σS (t )
Study Notes prepared by Carlos Brioso
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Sol:
G (t ) = S (t )e −0.02(T −t )
GS = e −0.02(T − t )
GSS = 0
Gt = S (t )(0.02)e −0.02(T − t )
Using Ito’s Lemma:
1
dG (t ) = G S dS (t ) + G SS [dS (t )]2 + Gt dt
2
1
dG (t ) = e −0.02(T −t ) dS (t ) + (0)[ dS (t )] 2 + S (t )(0.02)e −0.02(T −t ) dt
2
G (t ) = S (t )e −0.02(T −t )
dG(t ) = GT [0.12dt + 0.4dZ (t )] (A)
Example (SOA QA 2009 # 13)
Info:
{Z(t)} standard BM
What processes have zero drift?
Sol
Applying Ito’s Lemma:
(i) U Z = 2 ; U ZZ = 0 ; U t = 0 → dU (t ) = 2dZ (t ) + 0 or dU (t ) = 0 + 2dZ (t ) (zero drift)
1
(ii) V z = 2Z (t ) ; VZZ = 2 ; Vt = −1 → dV (t ) = 2 Z (t )dZ (t ) + (2)[dZ (t )] 2 − dt and
2
2
(dZ ) = dt (multiplication rule) → dV (t ) = 2 Z (t ) dZ (t ) (zero drift)
(iii) dW (t ) = d [t 2 Z (t )]dt − 2tZ (t )dt
d [t 2 Z (t )] = t 2 dZ (t ) + 2tZ (t )dt → dW (t ) = t 2 dZ (t ) (zero drift)
Example (CAS Spring 2007 # 20)
Info
S (t ) = S (0)e
dS(t)=?
(α −δ − 1 σ 2 +σZ (t ))
2
;
Z(t) follows BM
Sol
A lognormal stock price implies that the stock follows geometric BM. Then:
dS (t )
= (α − δ )dt + σdZ (t ) or dS (t ) = (α − δ ) S (t )dt + σS (t ) dZ (t )
S (t )
5
Study Notes prepared by Carlos Brioso
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The expression is equivalent to B. Since:
(α − δ − 1 / 2σ 2 ) S (t )dt + σS (t )dZ (t ) + 1 / 2σ 2 S (t )dt
6
Valuing a Claim on Sa
- If S follows a process described by *. The value of a claim paying S (T ) a (pre-paid fwd price) is:
F
P
( S (T ) a )
0,T
=e
−rT
S (0)
a
1
[ a ( r −δ )+ a ( a −1)σ 2 ]T
2
e
1
2
a
- The process followed by S is geometric Brownian motion describe by:
dS a
1
= [a (α − δ ) + a(a − 1)σ 2 ]dt + aσdZ
a
2
S
The lease rate is:
δ * = r − a (r − δ ) − a (a − 1)σ 2
Drift
Risk
expected return: a (α − r ) + r ; risk premium: a (α − r )
Example (SOA QA 2009 # 16)
Info
BS framework. S(t) no div. σ = 20% ; r = 4%
P
F t ,T [S (T )]
x
x
= S (t ) x (pre-paid forward price of a claim that pays S (T ) at time T.
A solution for the equation is x=1. Determine another x that solves the equation.
Sol
F
P
( S (T ) x )
0,T
=e
1
[ a ( r −δ ) + x ( x −1)σ 2 ](T −t )
2
e
= S (t )
1
[ −0.04+( 0.04−0) + (1−1)(0.20) 2 ](T −t )
2
e
= S (0)
−r (T −t )
S (0)
x
x
1
[ −0.04+ x ( 0.04−0) + x ( x −1)(0.20)2 ](T −t )
2
e
If x = 1 then
F
P
( S (T ) x )
0,T
We can solve
1
= S (t )
[ −0.04 + x ( 0.04 − 0 ) +
x 2 + x − 2 = 0 the x=1,-2 (B)
1
2
x ( x − 1)( 0.20 ) ] = 0 to
2
find another x that solves the equation.