Mathematical finance: continuous time by Ewald C.-O.

By Ewald C.-O.

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For the whole section let Mm,n = ((Xt , Ft )t∈[0,T ] , Φ) a financial market. 1. Let g : Ω → R be a contingent claim in Mm,n . A price process (gt )t∈[0,T ] for g is a non negative (Ft )t∈[0,T ] adapted stochastic process such that gT = g almost sure. 38 Before thinking about the question whether the extended financial market is arbitrage free or not, we must give a precise mathematical formulation of how to extend financial markets at all. 2. Let Mm,n = ((Xt , Ft )t∈I , Φ) and N k,l = ((Yt , Gt )t∈I , Ψ) be financial markets.

The pay-out of a digital Call option is given as g(ω) := 1{XT1 ≥K} . 2. Compute a fair price for the Butterfly-Spread option with strike price K in the standard Black-Scholes model. 19) Why is this option been called Butterfly-Spread ? 8 Why is the Black-Scholes model not good enough ? The Black-Scholes model certainly is still the most famous and one of the most applied models. On the other side, it is known that the reality at financial markets looks different. In this short section we 46 will demonstrate why this is so.

We will later see that from the probabilistic point of view, its existence is very natural. Let us now state ( without proof ) some properties of the stochastic integral. 3. Let X be an adapted lcrl process and Y ∈ S. Then τ ∞ 1. For any stopping time τ we have 0 XdY = 0 X · 1[0,τ ] dY = ∞ ∞ XdYτ where 0 means integration over the whole parameter 0 set I.

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