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FM4521g/9521b

Today’s Topics1. Review

2. Existence of the Risk-Neutral Measure

3. Uniqueness of the Risk-Neutral Measure

4. Summary of risk-neutral measure approach

5. Dividend-Paying Stocks

Review

• If the portfolio X(t) is used to replicate a contingent claim

P (t) with delta hedging, then

∆(t) =Γ̃(t)

σ(t)D(t)S(t), 0 ≤ t ≤ T

by Martingale representation theorem and Girsanov’s theorem

extension

• Definition: A probability measure Q is said to be risk-neutral

if

1. P and Q are equivalent (i.e., for any A ∈ F, P (A) = 0

iff Q(A) = 0,

2. Under Q, the discounted stock price D(t)S(t) is a

martingale

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• For a stock following a specific SDE, Girsanov’s Theorem can

be used to find a risk-neutral measure Q, i.e., the discounted

stock must be a martingale under Q

• Definition: An arbitrage is a portfolioX(t) satisfyingX(0) =

0 and also satisfying for some T > 0

P (X(T ) ≥ 0) = 1, P (X(T ) > 0) > 0

Existence of the Risk-Neutral Measure

• Theorem (First fundamental theorem of asset pricing) If a

market model has a risk-neutral probability measure, then it

does not admit arbitrage

• Proof

? Let Q be the risk-neutral measure. Then every discounted portfolio

must be a martingale under Q

EQ[D(T )X(T )|F0] = EQ[D(T )X(T )] = X(0) = 0

? If P (X(T ) < 0) = 0, and Q is equivalent to P , then

Q(X(T ) < 0) = 0

? Need to verify Q(X(T ) > 0) = 0. Otherwise there is an

arbitrage (why?)

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? If Q(X(T ) > 0) > 0, then

EQ

[D(T )X(T )|F0] = EQ

[D(T )X(T )] > 0

which is impossible

? Hence Q(X(T ) > 0) = 0 implies

P (X(T ) > 0) = 0

? That is, X(t) has no arbitrage

Uniqueness of the Risk-Neutral Measure

• Definition: A market model is complete if every derivative

security can be hedged

• Remember: If the portfolio X(t) is used to replicate the

contingent claim P (t) with delta hedging, then

∆(t) =Γ̃(t)

σ(t)D(t)S(t), 0 ≤ t ≤ T

• Theorem (Second fundamental theorem of asset pricing)

Consider a market model that has a risk-neutral probability

measure. The model is complete iff the risk-neutral probability

measure is unique

• Sketch of proof

? First we assume the model is complete. If there exist two

risk-neutral measures Q1 and Q2

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? Let A ∈ FT and consider a derivative security P (t) with

payoff

P (T ) =I(A)

D(T )? Let X(t) be the portfolio to hedge the P (t)

? Qi, i = 1, 2 are risk-neutral =⇒ D(t)X(t) must be a

martingale under both Qi, i = 1, 2

X(0) = EQ1[D(T )P (T )|F0] = E

Q1[D(T )P (T )]

X(0) = Q1(A)

? Similarly X(0) = Q2(A). Hence

Q1(A) = Q2(A) for all A ∈ FT

? Normally FT = F . Hence Q1 and Q2 are really the same

? Assume that there is a unique risk-neutral measure Q such

that D(t)S(t) is a martingale under Q

? For any portfolio X(t) with ∆(t) shares in S(t)

d(D(t)X(t)) = ∆(t) d(D(t)S(t))

? Hence D(t)X(t) is a martingale and can be used to hedge

any derivative security

? Thus the market model is complete

? If there are multiple stocks, the proof is much hard since

one must proof there is a unique solution of Θi(t), i =

1, . . . , d

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Summary of risk-neutral measure approach

• A stock S(t) follows a SDE with a Brownian motion W (t)

and a filtration Ft• A discount process D(t). It is possible that D(t) may contain

a diffusion term

• With the help of Girsanov’s Theorem, there is a risk-neutral

measure Q (if such one can be found) such that

d(D(t)S(t)) = σ(t)D(t)S(t) dW̃ (t)

That is, D(t)S(t) is a martingale under Q

• A security derivative P (t) (or a contingent claim) with a

random payment P (T )

• A portfolio X(t) with ∆(t) shares in S(t) so that

d((D(t)X(t))) = ∆(t) d(D(t)S(t))

and D(t)X(t) is a martingale under Q

• By the first fundamental theorem of asset pricing, if X(t) is

used to replicate P (t) so that

X(t) = P (t), 0 ≤ t ≤, T

then

D(t)P (t) = EQ

[D(T )P (T )|Ft]admits no arbitrage

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• On the other hand, if D(t)S(t) is a martingale under Q,

there must exist a Γ̃(t) such that

d(D(t)S(t)) = Γ̃(t) dW̃ (t)

and

∆(t) =Γ̃(t)

σ(t)D(t)S(t), 0 ≤ t ≤ T

• The value of P (t) can be computed either through its

martingale representation or by delta hedging

Dividend-Paying Stocks

• If a stock pays no dividend, then under a risk-neutral measure,

the discounted stock is a martingale. So is the a discounted

portfolio value

• Remember: It is the martingale property of a discounted

portfolio value that is used for valuing a contingent claim

• With dividend paying stock, the discounted stock may not be

a martingale

• Continuously paying dividend

? Let S(t) be a stock following a generalized geometric

Brownian motion that pays dividend continuously over time

at a rate A(t) per unit time

dS(t) = µ(t)S(t) dt+ σ(t)S(t) dW (t)

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− A(t)S(t) dt, 0 ≤ t ≤ T,where A(t) ≥ 0 is adapted to Ft

? Let X(t) be a portfolio value at time t with ∆(t) shares

in the stock S(t). Then

dX(t) = R(t)[X(t)−∆(t)S(t)] dt + ∆(t) dS(t) +

∆(t)A(t)S(t) dt

= R(t)X(t) dt + (µ(t) − R(t))∆(t)S(t) dt +

σ(t)∆(t)S(t) dW (t)

= R(t)X(t) dt+ ∆(t)S(t)σ(t)[Θ(t) dt+ dW (t)],

where

Θ(t) =µ(t)− R(t)

σ(t)

? With the discounted process D(t)

d[D(t)X(t)] = ∆(t)D(t)S(t)σ(t)[Θ(t) dt+dW (t)]

? By Girsanov’s Theorem, there exists a risk-neutral measure

Q such that

W̃ (t) =

∫ t

0

Θ(u) du+W (t)

is a Browian motion under Q

? It implies that

d[D(t)X(t)] = ∆(t)D(t)S(t)σ(t) dW̃ (t)

is a martingale

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? Notice that the discounted stock D(t)S(t) is no longer a

martingale under Q

? If X(t) is used to replicate a contingent claim P (t) with

payment P (T ) so that X(t) = V (t) for all 0 ≤ t ≤ T ,

then

D(t)P (t) = EQ

[D(T )P (T )|Ft], 0 ≤ t ≤ T

? The same formula as though there is no dividend payment

? However the distribution S(t) under Q is different

dS(t) = (µ(t)− A(t))S(t) dt+ σ(t)S(t) dW (t)

dS(t) = (R(t)− A(t))S(t) dt+ σ(t)S(t) dW̃ (t)

S(t) = S(0) exp{∫ t

0

σ(u) dW̃ (u)

+

∫ t

0

[R(u)− A(u)−σ2(u)

2] du

}? The process

e∫ t0 A(u) du

D(t)S(t) = exp{∫ t

0

σ(u) dW̃ (u)

−1

2

∫ t

0

σ2(u) du

}is a martingale

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? A task: try to derive the SDE of the discounted stock

D(t)S(t) under P and Q

• Continuously paying dividend with constant coefficients

? R(t) ≡ r;σ(t) ≡ σ;A(t) ≡ a? S(t) = S(0) exp

{σW̃ (t) + [r − a− σ2

2 ]t}

? To evaluate an European call option, we need

S(T ) = S(t) exp{σ(W̃ (T )− W̃ (t))

+[r − a−σ2

2](T − t)

}? The European call option with dividend payment formula

c(t, x) = P (t) = EQ

[e−r(T−t)

(S(T )−K)+|Ft]

? Detail computations are left as a question in the Assignment

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• Lump payments of dividends

? Time intervals: 0 = t0 < t1 < · · · < tn < tn+1 = T

? Prior the time tj, a dividend payment is ajS(tj−), where

S(tj−) denotes the stock price just prior to the dividend

payment

? So S(tj) = S(tj−)− ajS(tj−) = (1− aj)S(tj−)

? Each aj ∈ [0, 1] is a r.v. and is adapted to Ftj? Notice that S(t) is no longer continuous over [0, T ]

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? So its SDE must be defined within each interval

dS(t) = µ(t)S(t) dt+ σ(t)S(t) dW (t),

where tj ≤ t < tj+1, j = 0, 1, . . . , n

? Let X(t) be a portfolio value at time t with ∆(t) shares

in the stock S(t)

? At time tj, the stock portion of the portfolio value drops

by aj∆(tj)S(tj−)

? But the portfolio collects the dividend aj∆(tj)S(tj−)

? So the portfolio values does not jump

? X(t) follows the same SDE

dX(t) = R(t)[X(t)−∆(t)S(t)] dt+ ∆(t) dS(t)

? So the SDE of the discounted portfolio

d[D(t)X(t)] = ∆(t)D(t)S(t)σ(t)[Θ(t) dt+dW (t)]

? By Girsanov’s Theorem, there exists a risk-neutral measure

Q such that

W̃ (t) =

∫ t

0

Θ(u) du+W (t)

is a Browian motion under Q

? It implies that

d[D(t)X(t)] = ∆(t)D(t)S(t)σ(t) dW̃ (t)

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is a martingale

• Lump payments of dividends with constant coefficients

? R(t) ≡ r;σ(t) ≡ σ; aj are constant

? The SDE of S(t) under Q, for j = 0, 1, . . . , n

dS(t) = rS(t) dt+ σS(t) dW̃ (t), tj ≤ t < tj+1

? To price the European call option c(0, x), we need to find

the distribution of S(T )

?S(T )

S(0)=S(tn+1)

S(t0)=

n∏j=0

S(tj+1)

S(tj)

? S(tj+1−) = S(tj) exp{σ(W̃ (tj+1)− W̃ (tj)) + (r−

σ2

2 )(tj+1 − tj)}

? Remember: S(tj+1) = S(tj+1−) − ajS(tj+1−) =

(1− aj+1)S(tj+1−)

? So

S(tj+1)

S(tj)= (1− aj+1) exp

{σ(W̃ (tj+1)− W̃ (tj))

+(r −σ2

2)(tj+1 − tj)

}? You derive the rest in the Question 3