Mathematical Foundations for Finance Exercise 12 Martin Stefanik - - PowerPoint PPT Presentation

Mathematical Foundations for Finance Exercise 12

Martin Stefanik ETH Zurich

Itô Representation Theorem

Let FW = (F W

t )0≤t≤∞ denote the filtration generated by a Brownian motion

W augmented by P-nullsets in F 0

∞ = σ(Ws, s ≥ 0).

Theorem 1 (Itô representation theorem) Suppose that W = (Wt)t≥0 is a Brownian motion in Rm, m ∈ N. Then every random variable H ∈ L1(F W

∞, P) has a unique representation as

H = E [H] + ∫ ∞ ψsdWs P-a.s. for an Rm-valued integrand ψ ∈ L2

loc(W) with the additional property that

∫ ψdW is a (P, FW)-martingale on [0, ∞].

• We can also make this work for a finite time horizon T > 0 by replacing

∞ by T > 0.

• Note that this is precisely of the form V0 + G(ψ) that we saw in discrete

time.

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Itô Representation Theorem

Corollary 2

• 1. Every (real-valued) local (P, FW)-martingale L is of the form

L = L0 + ∫ γdW for some Rm-valued process γ ∈ L2

loc(W)

• 2. Every local (P, FW)-martingale is continuous.
• This provides a simple characterization of every local (P, FW)-martingale

that can live in our probability space.

• It also gives an indication that our probability space needs to be richer

if we want to be able to define say (compensated) Poisson process on that space.

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Introduction to Black-Scholes Model

The Black-Scholes model is a continuous-time analogue of the symmetric multiplicative binomial model (also called the Cox-Ross-Rubinstein model). This model can be described by the following set of SDEs d S0

t =

S0

t rdt,

• S0

0 = 1

d S1

t =

S1

tµdt +

S1

tσdWt

• S1

0 =

s1 for some constants r, µ ∈ R and s1

0, σ > 0.

• r ∈ R corresponds to continuously compounded interest rate, so if we

take r′ > −1 as the simple interest rate from the CRR model, we obtain that r = log(1 + r′).

• µ ∈ R and σ > 0 correspond to the mean growth rate and volatility of

the relative change in the price over an infinitesimal time step “dt”.

• An alternative interpretation is that

( µ − 1

2σ2)

and σ correspond to the mean and volatility of the logarithmic return of the stock over one unit

• f time, respectively.

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Introduction to Black-Scholes Model

If SDEs admit explicit solutions at all, they can sometimes be found by applying Itô’s formula to some f : (x, t) → f(x, t) in C2,2. For SDEs of the above form, f(x) = log(x) is a good guess. We obtain

• S0

t = exp(rt)

• S1

t =

s1

0 exp

(( µ − 1 2σ2) t + σWt ) Defining logarithmic returns for t ∈ N as Lt = log ( S1

t

• S1

t−1

) , we have that Lt are i.i.d random variables with Lt ∼ N ( µ − 1 2σ2, σ2 ) , so the mean is µ − 1

2σ2 and standard deviation (volatility) is σ.

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Introduction to Black-Scholes Model

It is mentioned in the script that the Black-Scholes model as well as the CRR are too simple to be realistic. Why?

500 1000 1500 2000 −0.05 0.00 0.05 0.10 0.15 500 1000 1500 2000 −0.05 0.00 0.05 0.10 0.15 5 / 6

Introduction to Black-Scholes Model

−0.06 −0.04 −0.02 0.00 0.02 0.04 0.06 10 20 30 40 Empirical Black−Scholes

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Thank you for your attention!