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Lecture 17
Overture to continuous models
Before rigorously deriving the acclaimed Black-Scholes pricing formula for the value of a European
option, we developed a substantial body of material, in continuous time. Then, we developed the
risk-neutral asset pricing theory in discrete time. We can use our discrete techniques to see what
form our results must take in the continuous world.
It is easy to believe that we should be able to use a discrete model with very small time periods
to approximate a continuous model. The Black-Scholes model is based in the lognormal model
(geometric Brownian motion). With this in mind, we choose our approximation to have constant
growth rate and constant ‘noise’.
The model is parametrized by the time period, δt, and three fixed constant parameters, v, σ and
the riskless rate r.
• The cash bond has the form Bt = ert , which does not depend on the interval size.
• The stock price process follows the nodes of a binomial tree. If the current value of the stock is
s, then over the next time period it moves to the new value
(
√
s exp(vδt + σ δt), if up,
√
s exp(vδt − σ δt), if down,
Suppose our belief is that the jumps are equally likely to be up or down. So under the market
measure, P [up jump] = 1/2 = P [down jump] at each time step.
For a fixed time t, set N to be the number of time periods until time t, that is N = t/δt. Then
√ 2XN − N
St = S0 exp(vt + σ t( √
))
N
where XN is the total number of the N separate jumps which were up jumps. To see what happens
as δt −→ 0 (or equivalently N −→ ∞) we call on the Central limit Theorem.
Theorem (Central Limit Theorem)
Let ξ1 , ξ2 , ... be a sequence of independent identically distributed random variables under the probability measure P with finite mean µ and finite non-zero variance σ 2 and let Sn = ξ1 + ... + ξn .
Then
Sn − nµ
√
nσ 2
converges in distribution to an N (0, 1) random variable as n −→ ∞.
Now XN is the sum of N independent random variables {ξi }1≤i≤N taking the value +1 with
probability 1/2 and 0 otherwise. This means E[ξi ] = 1/2 and var[ξi ] =√1/4 so that by the Central
Limit Theorem, the distribution of the random variable (2XN − N )/ N converges to that of a
1
normal random variable with mean zero and variance one. In other words, as δt gets smaller (and so
N gets larger), the distribution of St converges to that of a lognormal distribution. More precisely,
in the limit, log St is normally distributed with mean log S0 + vt and variance σ 2 t.
This is what happens under the original measure P . What happens under the martingale measure,
Q, that we use for pricing ?
Under the martingale measure, the probability of an up jump is
√
exp(rδt) − exp(vδt − σ δt)
√
√
q=
exp(vδt + σ δt) − exp(vδt − σ δt)
which is approximately
√ v + 1 σ2 − r
1
2
(1 − δt(
)).
2
σ
So under the martingale measure, Q, XN is still binomially distributed, but now has mean N q and
variance N q(1 − q).
√
√
Thus, under Q, (2XN − N )/ N has mean that tends to − t(v + 21 σ 2 − r)/σ and variance that
approaches √
one as δt tends to zero. Again using the Central Limit Theorem the √
random variable
(2XN −N )/ N converges to a normally distributed random variable, with mean − t(v+ 12 σ 2 −r)/σ
and variance one. Under Q then, St is lognormally distributed with mean log S0 + (r − 12 σ 2 )t and
variance σ 2 t. This can be written
√
1
St = exp(σ tZ + (r − σ 2 )t).
2
where, under Q, the random variable Z is normally distributed with mean zero and variance one.
If our discrete theory carries over to the continuous limit, then in our continuous model the price at
time zero of a European call option with strike price K at time T will be the discounted expected
value of the claim under the martingale measure, that is
EQ [e−rT (ST − K)+ ]
where r is the riskless rate. Substituting, we obtain
√
1
E Q [(S0 exp(σ T Z − σ 2 T ) − Kexp(−rT ))+ ].
2
We’ll derive this pricing formula rigorously later and we will also show that this equation can be
evaluated as
log SK0 + (r + 12 σ 2 )T
log SK0 + (r − 12 σ 2 )T
√
√
S0 Φ(
) − Ke−rT Φ(
)
σ T
σ T
where Φ is the standard normal distribution function,
Z z
1
2
√ e−x /2 dx
Φ(z) = Q[Z ≤ z] =
2π
−∞
2
The Girsanov Theorem
In order to price and hedge in the Black-Scholes framework we shall need two fundamental results.
The first will allow us to change probability measure so that the discounted asset prices are martingales. Recall that in our discrete time world, once we had such a martingale measure, the pricing
of options was reduced to calculating expectations under that measure. In the continuous world
it will no longer be possible to find the martingale measure by linear algebra. Nonetheless, before
stating the continuous time result, we revert to our binomial trees for guidance.
Suppose that, under the probability measure P, if the value of an asset at time iδt is known to be
Si then its value at time (i + 1)δt is Si u with probability p and it is Si d with probability 1 − p.
As we saw before, if we let Q be the probability measure under which the probability of an up
jump is q = (1 − d)/(u − d) and of a down jump is (u − 1)(u − d), then the process {Si }0≤i≤N is a
Q−martingale.
We can regard the measure Q as a reweighting of the measure P. For example, consider a path
S0 , S1 , ..., Si through the tree. Its probability under P is pN (i) (1−p)i−N (i) , where N (i) is the number
of up jumps that the path makes. Under Q its probability is Li pN (i) (1 − p)i−N (i) where
q
1 − q i−N (i)
Li = ( )N (i) (
)
.
p
1−p
Evidently Li depends on the path that the stochastic process takes through the tree and can itself
be thought of as a stochastic process adapted to the filtration {Fi }1≤i≤N . Moreover, Li /Li−1 is
q/p if Si /Si−1 = u and is (1 − q)/(1 − p) if Si /Si−1 = d, so that
q
1−q
EP [Li |Fi−1 ] = Li−1 (p + (1 − p)
) = Li−1 .
p
1−p
In other words, {Li }0≤i≤N is a (P, {Fi }1≤i≤N )−martingale with E[Li ] = L0 = 1.
If we wish to calculate the expected value of a claim in the Q-measure, it is given by
EQ [C] = EP [LN C].
Notation:
We have obtained the Radon-Nikodym derivative of Q with respect to P. It is customary to write
Li =
dQ
|F .
dP i
We have shown that the process of changing to the martingale measure can be viewed as a reweighting of the probabilities of paths under our original measure P according to a positive, mean one,
P-martingale. This procedure of reweighting according to a positive martingale can be extended
to the continuous setting. Our aim now is to investigate the effect of such a reweighting on the
distribution of the P-Brownian motion. Later this will enable us to choose the right reweighting so
that under the new measure obtained in this way the discounted stock price is a martingale.
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Theorem(Girsanov’s Theorem)
Suppose that {Wt }t≥0 is a P−Brownian motion with the natural filtration {Ft }t≥0 and that {θt }t≥0
is an {Ft }t≥0 -adapted process such that
Z
1 T 2
E[exp(
θ dt)] < ∞
2 0 t
.
Define
t
Z
1
θs dWs −
2
Lt = exp(−
0
and let
P(L)
Z
t
θs2 ds)
0
be the probability measure defined by
Z
(L)
P [A] =
Lt (ω)P(dω).
A
(L)
Then under the probability measure P(L) , the process {Wt }0≤t≤T , defined by
Z t
(L)
Wt = Wt +
θs ds,
0
is a standard Brownian motion.
Notation:
We write
dP(L)
|Fi = Lt .
dP
(Lt is the Radon-Nikodym derivative of P(L) with respect to P.)
Remark
1. The condition
1
E[exp(
2
Z
T
θt2 dt)] < ∞,
0
known as Novikov’s condition, is enough to guarantee that {Lt }t≥0 is a (P, {Ft }t≥0 )-martingale.
Since Lt is clearly positive and has expectation one, P(L) really does define a probability measure.
2. Just as in the discrete world, two probability measures are equivalent if they have the same sets
of probability zero. Evidently P and P(L) are equivalent.
3. If we wish to calculate an expectation with respect to P(L) we have
EP
(L)
[φt ] = E[φt Lt ].
Moreover generally,
EP
(L)
[φt |Fs ] = EP [φt
4
Lt
|Fs ].
Ls
This will e fundamental in option pricing.
Example
Let {Xt }t≥0 be the drifting Brownian motion process
Xt = σWt + µt,
where {Wt }t≥0 is a P-Brownian motion and σ and µ are constants. Find a measure under which
{Xt }t≥0 is a martingale.
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