Week #2:
Review week #1:
Separation of process and measure
Measure(2) equivalent to measure (1) if measure (2) associates positive probability to all
events (1) that have positive probability under measure (1).
Risk neutral probabilities  equivalent martingale measure
Under the risk neutral measure the expected return of all assets in the system is the risk
free rate.
A stochastic process is said to be a martingale if the expected change in the value of the
process is always zero. Under the risk neutral measure normalized (discounted at the risk
free rate) asset prices behave as a martingale.
A model does not permit arbitrage opportunities if and only if the model admits at least
one risk neutral probability measure(s).
The Black-Scholes model, Black Fischer and Scholes Myron, (1973), price derivative
security through “replication.”
Price of derivative obtained by “replication” method can be recovered simply by
calculating discounted value (risk free rate) of expected payoff using risk neutral
probabilities. The replicating portfolios are specific to the claim being valued, the risk
neutral probabilities depend only on the primary assets.
The replicating portfolios in the binomial setting are a process of self-financing
replicating portfolios.
Risk neutral probability measure may be identified from information set, It
Further a market is said to be complete if a derivative security can be replicated using
only the primary securities. In a complete market any derivative security has a unique
arbitrage-free price. A market is complete if and only if the model admits a unique riskneutral probability measure.
Multiple risk-neutral probabilities can exist if and only if there are multiple state-price
vectors. This means that at least one Arrow-Debreu security has many possible prices
consistent with no-arbitrage, and this is possible only if the Arrow-Debreu security is not
replicable. Thus, the market must be incomplete.
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Development of the Black-Scholes option pricing model EMM approach (Baxter and
Rennie Chapter 3):
Let:
St  S0 exp(Wt  t )
Bt  exp( rt )
Notice that Bt accumulates interest at the risk-free rate, r, and is a pre-visible process with
respect to the information set, t .
Wt  Brownian motion is a Gaussian stochastic process:





Although W is continuous everywhere, it is (with probability one) differentiable
nowhere.
Brownian motion will eventually hit any and every real value no matter how
large, or how negative.
Once a Brownian motion hits a value, it immediately hits it again infinitely often,
and then again from time to time in the future.
Brownian motion is fractal.
Wt  N(0, t)
Ito’s Lemma – How to create differentials of functions of Brownian motion and how to
create stochastic processes from stochastic differential equations.
Ito’s Lemma (Baxter and Rennie page 59)
Let Xt be a stochastic process the solution to stochastic differential equation (SDE)
dX t   t dWt  ut dt
Let Yt be any twice differentiable function of Xt, Yt = F(Xt , t)
 F 1 2  2 F F 
F
dWt  u
 

 dt
X
X 2 t 
 X 2
This represents a fundamental difference between Newtonian calculus and stochastic
calculus.
Then: dYt   t
Application:
Let St be an exponential (geometric) Brownian motion such that:
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S t  S 0 exp(Wt  (  12  2 )t )
Use Ito’s Lemma to find dSt
X t  Wt  (u  12  2 )t
dXt = dWt + (u-1/22 )dt
St = S0*exp(Xt) = F(Xt)

F
F 1 2  2 F 
dWt  (u  12  2 )
 2
dt
2
X

X

X


dS t   t S t dWt  uS t dt
dS t   t
The marginal distribution of ST is given by S0 times the exponential of a normal with
mean (u-1/22)(T-t) and variance 2(T-t). ST is distributed log-normal.
Second pass at application – note different definition of parameter u and resulting
outcome.
Let St be an exponential (geometric) Brownian motion such that:
S t  S 0 exp(Wt  t )
Use Ito’s Lemma to find dSt
X t  Wt  ut
dXt = dWt + udt
St = exp(Xt) = F(Xt)
dS t   t
 F 1 2  2 F 
F
dWt  u
 2
 dt
X
X 2 
 X
dS t   t S t dWt  (u  12  2 ) S t dt
In the single period – two state setting of Neftci Chapter 2 we discovered that the risk
neutral measure is the probability measure that made the normalized stock price a
martingale.
Martingale definition: (Baxter and Rennie page 76)
Define the normalized (discounted) stock process and the discounted claim (derivative)
payoff at expiration as:
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Z t  Bt1 S t
BT1 X
Where S t  S 0 exp(Wt  t ) and Bt  exp(rt )
Notice a subtle difference from the one-period setting in Neftci Chapter 2, discounted
claim is discounted over (T-t) periods. Also notice the introduction of continuous
compounding,
What is the SDE for Zt ? Because the risk free rate is assumed constant;
Z t  Bt1St  S 0 exp(Wt  (u  r )t )
From Ito’s Lemma and following the example above:


dZ t  Z t dWt  Z t (u  r )  12  2 dt
The Girsanov Theorem states that if a technical condition is satisfied, it will be possible
to convert a P-Brownian motion with drift to a Q-Brownian motion with specified drift,
vt. Recall that the risk neutral measure introduced in Neftci Chapter 2 was a probability
measure that made the normalized stock process a martingale, i.e. with drift vt = 0.
Girsanov Theorem (Baxter and Rennie page 74)
Given dZt = F(dWt) with drift [(u-r)+1/22] find a measure Q such that
~
dZ t  F (dWt ) with drift of zero, vt = 0.
dZ t
(u  r) 
 Z (dW 
Let  t
t
1 2

2

t
(u  r) 

1 2

2
 vt
 vt
)dt  v dt
t


Does this pre-visible process satisfy the technical constraint of the Girsanov Theorem?
Then the Girsanov Theorem ensures, subject to the technical constraint, that under
measure Q;
~
~
dZ t  Z t dWt  vt dt  Z t dWt
~
Is dZ t  Z t dWt a martingale? Another technical constraint must be checked. (Baxter
and Rennie page 78-79)
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~
Satisfied that dZ t  Z t dW is a Q-martingale, define Et to be the expectation of the
discounted terminal value of the derivative under measure Q;
Et  EQ ( BT1 X t )
This expectation is by definition a Q-martingale.
Is there a means to utilize a pre-visible process to construct Et from Zt ?
Martingale Representation Theorem (Baxter and Rennie page 78)
The martingale representation theorem ensures that there is a pre-visible process, t, that
constructs Et from changes in Zt.
Et  E 0  0t  s dZ s
dE t   t dZ t
Is t a replicating self-financing strategy for claim X?
Replicating Strategy:
t – units of stock at time t
t = Et - tZt – units of the money market account
The value of the replicating portfolio;
Vt = tSt + tBt which after substituting for t is Vt = BtEt . Notice that this implies the
portfolio replicates the derivative X at time T;
VT = BTET = BT EQ ( BT1 X T )  X
To determine if this replicating portfolio is self-financing (Baxter and Rennie page 89);
For a replicating portfolio (t , t) to be self financing the change in the portfolio’s value
necessary to maintain the replicating quality must be exactly matched by the change in
the value of the assets held in the beginning of period replicating portfolio.
dVt = tdSt + tdBt
This is not an automatic property of a general portfolio (t , t) and must be checked.
From the definition of Vt;
dVt = d(BtEt)
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From the multiplication rules (Baxter and Rennie page 62);
dVt = BtdEt + EtdBt
From martingale representation theorem;
dEt  t dZ t
From definition of t for replicating strategy
Et = t + tZt
Substituting into dVt = BtdEt + EtdBt and rearranging
dVt = t(BtdZt + ZtdBt) + tdBt
From multiplication rules; d(BtZt) = (BtdZt + ZtdBt) = dSt
Hence, dVt = tdSt + tdBt and the replicating portfolio is self-financing.
What we have learned ?
Given constant r, u, 
St  S0 exp(Wt  t )
Bt  exp( rt )
All integrable derivatives of S have pre-visible, self financing, replicating strategies (t ,
t).
Consequently the market is complete. Although the number of possible states is
uncountable and there exist only two independent primary assets the market model built
from Brownian motions is complete.
The value of the derivative at time (t) must be Vt to prevent arbitrage;
Vt= BtEt = Bt EQ ( BT1 X t )  e  r (T t ) EQ ( X t )
Where Q is the martingale (risk-neutral) measure for the discounted price process Zt.
Note that in moving to the right most term of the equality that this last change is possible
because r is assumed constant.
To price a specific derivative it is necessary to take the expectation;
Vt= e  r (T t ) EQ ( X t )
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Which we will do next.
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