Changing the probability distribution Recall the binary tree with a - - PowerPoint PPT Presentation

Changing the probability distribution

• Recall the binary tree with a probability distribution P defined

by Si+1 = Si ×

  

u with probability p d with probability 1 − p

• The process {Si}0≤i≤N is not a P-martingale for general p.
• But if the distribution Q is defined by replacing p by

q = 1 − d u − d, then {Si}0≤i≤N is a Q-martingale.

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• Consider a path with N(i) up-steps in the first i steps, so

Si = S0uN(i)di−N(i).

• Its probability under P is

pN(i)(1 − p)i−N(i) and under Q is qN(i)(1 − q)i−N(i) = Li × pN(i)(1 − p)i−N(i) where Li =

• q

p

N(i)

1 − q 1 − p

i−N(i)

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• Notice that for a given i, Li depends on the path up to

time iδt, and is therefore a random variable; it is also Fi- measurable.

• So {Li}0≤i≤N is an adapted stochastic process.
• {Li}0≤i≤N is also:

– positive, with P-probability 1; – a P-martingale (needs proof!); – with expected value 1.

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• Also, if C is FN-measurable, such as a claim with maturity

Nδt, then EQ[C] = EP[LNC] .

• Li is the Radon-Nikodym derivative of Q with respect to P
• n Fi:

Li = dQ dP

• Fi

.

• The measure Q, equivalent to P, under which {Si}0≤i≤N is a

martingale, i.e. the equivalent martingale measure, is defined by the process {Li}0≤i≤N.

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Girsanov’s Theorem

• Suppose that {Wt}t≥0 is a P-Brownian motion with the nat-

ural filtration {Ft}t≥0 and that {θt}t≥0 is an {Ft}t≥0-adapted process such that for a given T > 0 E

• exp
• 1

2

T

0 θ2 t dt

• < ∞
• Define

Lt = exp

• −

t

0 θsdWs − 1

2

t

0 θ2 s ds

• 5

• Note that {Lt}0≤t≤T is a positive
• P, {Ft}0≤t≤T
• martingale

with expected value 1.

• Let P(L) be the measure on FT defined by

P(L)(A) = E[1ALT] =

• A LT(ω)P(dω).
• Then under P(L), the process {W (L)

t

}0≤t≤T defined by W (L)

t

= Wt +

t

0 θsds

is a standard Brownian motion.

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• Note that

dW (L)

t

= dWt + θtdt, so under P, {W (L)

t

}0≤t≤T has non-zero drift, and is not a standard Brownian motion.

• Expected values: if φt is Ft-measurable, then

EP(L)[φt] = E[φtLt] and more generally, for 0 ≤ s < t, EP(L)[φt| Fs] = E

• φt

Lt Ls

• Fs
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Brownian Martingale Representation Theorem

• As always,
• Ω, F, {Ft}t≥0, P
• is a filtered probability space,

and {Wt}t≥0 is a

• P, {Ft}t≥0
• Brownian motion.
• Recall that if {ft}t≥0 is predictable and square-integrable

(strictly, f ∈ HT), then Mt =

t

0 fsdWs

is a

• P, {Ft}t≥0
• martingale.

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• A partial converse exists: suppose that {Ft}t≥0 is the natural

filtration of {Wt}t≥0.

• Representation Theorem: if, on the same filtered probability

space, {Mt}t≥0 is a square-integrable continuous

• P, {Ft}t≥0
• martingale, then there exists a predictable process {θt}t≥0

such that Mt − M0 =

t

0 θsdWs

with probability 1.

• That is, every continuous square-integrable martingale is a

stochastic integral with respect to Brownian motion.

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• In some cases, we do not need to assume the existence of

the Brownian motion; instead, we can construct it:

• If [M]t is almost surely absolutely continuous, with

λt

△

= d[M]t dt > 0 with probability 1, we can define ˜ Wt =

t

1 √λs dMs.

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• Then { ˜

Wt}t≥0 is a

• P, {Ft}t≥0
• martingale, and

d[ ˜ W]t =

• 1

√λt

2

d[M]t =

• 1

√λt

2

λtdt = dt.

• So { ˜

Wt}t≥0 is a

• P, {Ft}t≥0
• Brownian motion, and

Mt − M0 =

t

√ λsd ˜ Ws.

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Feynman-Kac Representation

• The probabilistic approach to assigning a value to a contin-

gent claim is: – Find a measure Q under which the discounted price of the underlying asset is a martingale; – The value of the claim at time t is its discounted expected value under Q, conditional on Ft.

• The value also satisfies a partial differential equation, and

may be found by solving it.

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• Assume that the function F solves the boundary value prob-

lem ∂F ∂t (t, x) + µ(t, x)∂F ∂x (t, x) + 1 2σ(t, x)2∂2F ∂x2 (t, x) = 0, for 0 ≤ t < T, and F(T, x) = Φ(x).

• Define {Xt}0≤t≤T to be the solution of the stochastic differ-

ential equation dXt = µ(t, Xt)dt + σ(t, Xt)dWt, 0 ≤ t ≤ T, where {Wt}t≥0 is standard Brownian motion under the prob- ability measure P.

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• If

T

0 E

• σ(t, Xt)∂F

∂x (t, Xt)

2

dt < ∞, then F(t, x) = EP[Φ(XT)|Xt = x].

• Black and Scholes originally derived their expression for the

price of a European option by solving the corresponding pde.

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