Asset prices with jumps Geometric Brownian motion has continuous - - PowerPoint PPT Presentation

Asset prices with jumps

• Geometric Brownian motion has continuous paths.
• Stock prices, and prices of other assets, often show jumps

caused by unpredictable events or news items.

• So geometric Brownian motion can only be an approximation

to the real behavior of asset prices.

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• The times at which jumps occur are often modeled by a

Poisson process.

• That is, if Nt jumps occur in (0, t], then for some λ > 0

– for each s ≥ 0 and t > 0, Ns+t−Ns ∼ Poisson with mean λt; – for each n ≥ 1 and times 0 ≤ t0 ≤ · · · ≤ tn, the increments {Ntr − Ntr−1} are independent; – N0 = 0; – Nt is right-continuous in t ≥ 0.

• Note the parallel with the definition of Brownian motion.

2

• The parameter λ is the intensity of the process; that is, the

expected number of jumps per unit time: E[Ns+t − Ns] t = λ.

• If τi is the time of the ith jump, then the inter-jump times

τ1, τ2 − τ1, τ3 − τ2, . . . are independently exponentially dis- tributed with mean 1/λ.

3

• Suppose that an asset price generally follows a GBM, but

each jump reduces the asset price by a fraction δ: St = S0 exp

• µ − 1

2σ2

• t + σWt
• (1 − δ)Nt.
• Then St should satisfy a differential equation like

dSt St = µdt + σdWt − δdNt.

• As always, the meaning of this SDE is the corresponding

stochastic integral equation St − S0 =

t

0 µSudu +

t

0 σSudWu −

t

0 δSudNu.

4

• The first integral is conventional, and the second is a stochas-

tic integral.

• For a continuous function f, the conventional Stieltjes inte-

gral is

t

0 f(u)dNu = Nt

• i=1

f (τi) .

• Because St may have discontinuities at the times τi when Nt

increases, the third integral is not a conventional Stieltjes integral.

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• We define it by

t

0 f(u, Su)dNu = Nt

• i=1

f

• τi−, Sτi−
• .
• With this definition, we have a generalized Itˆ
• formula: if

dYt = µtdt + σtdWt + νtdNt and f is twice continuously differentiable, then f(Yt) − f(Y0) =

t

0 f′(Yu−)dYu + 1

2

t

0 f′′(Yu−)du

−

Nt

• i=1

f′ Yτi− Yτi − Yτi−

• +

Nt

• i=1
• f

Yτi − f

• Yτi−
• .

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Girsanov’s Theorem with jumps

• Let {Wt}t≥0 be a standard P-Brownian motion and {Nt}t≥0 a

(possibly time-inhomogenous) Poisson process with intensity {λt}t≥0 under P. That is, Mt = Nt −

t

0 λudu

is a P-martingale.

• We write Ft for the σ-field generated by FW

t

∪ FN

t .

• Suppose that {θt}t≥0 and {φt}t≥0 are {Ft}t≥0-previsible pro-

cesses with φt > 0, such that

t

0 θ2 s ds < ∞ and

t

0 φsλsds < ∞.

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• Further, let Q be the measure whose Radon-Nikodym deriva-

tive with respect to P is dQ dP

• Ft

= Lt, where L0 = 1 and dLt Lt− = θtdWt − (1 − φt)dMt.

• Then, under Q, the process {Xt}t≥0 defined by

Xt = Wt −

t

0 θsds

is a standard Brownian motion and {Nt}t≥0 has intensity {φtλt}t≥0.

8

• Can we use this theorem to find an equivalent martingale

measure?

• Suppose again that

dSt St = µdt + σdWt − δdNt,

• Then the discounted process {˜

St}t≥0 satisfies d˜ St ˜ St = (µ − r)dt + σdWt − δdNt = (µ − r + σθt − δλφt)dt + σdXt − δdMt.

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• Here {Xt}t≥0 is Q-Brownian motion, and {Mt}t≥0 defined by

Mt = Nt −

t

0 λsφsds

is a Q-martingale.

• So {˜

St}t≥0 is a Q-martingale for any choice of {θt}t≥0 and {φt}t≥0 for which µ − r + σθt − δλφt = 0.

• Thus there is no unique equivalent martingale measure, and

the market is incomplete: there exist FT-measurable claims CT that cannot be replicated, and cannot be hedged.

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• If we have a second asset whose price is driven by the same

{Wt}t≥0 and {Nt}t≥0, then both discounted processes become martingales under a unique Q, provided the equations µ(i) − r + σ(i)θt − δ(i)λφt = 0, i = 1, 2 are nonsingular for all t ≥ 0.

• The extended market is therefore complete, and all claims

can be replicated.

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Stochastic Volatility

• The time-dependent volatility σt may have its own dynamics:

dSt = µStdt + σtStdW (1)

t

where σt = a(St, σt)dt + b(St, σt)

• ρdW (1)

t

+

• 1 − ρ2dW (2)

t

• .
• Here
• W (1)

t

• t≥0

and

• W (2)

t

• t≥0

are independent Brownian motions, and ρ ∈ (−1, 1) is the correlation between the in- crements in the two equations.

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• As in the case of jumps, equivalent martingale measures may

be found but are not unique.

• Again, if we have a second asset whose price is governed

by the same

• W (1)

t

• t≥0

and

• σ(1)

t

• t≥0

, such as an option

• n ST, the extended market may have a unique equivalent

martingale measure and hence be complete.

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