Abstract Inference (Probability & Mathematical Statistics) - download pdf or read online

By Ulf Grenander

ISBN-10: 0471082678

ISBN-13: 9780471082675

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A random variable τ : Ω → {0, 1, . . , it does not depend on the future. Using sequence (fn )N n=0 and a stopping time τ we define the following contingent claim N fτ (ω) ≡ fτ (ω) (ω) = fn (ω) I{ω: τ =n} . n=0 It is clear from the definition that this claim is determined by all trading information up to time N , but it is exercised at a random time τ , which is therefore called the exercise time. According to the described earlier methodology of managing risk associated with a contingent claim in the framework of a binomial market (B, S)-market, we can © 2004 CRC Press LLC price this claim using averaging with respect to a risk-neutral probability P ∗ : fτ Bτ C(fτ ) = E ∗ = E∗ fτ (1 + r)τ .

Consider an investment portfolio π with βn representing investment in a bank account and γn equal to the number of shares of S traded via futures contracts. Then N π ∆Dk Xnπ XN = + γn+1 . BN Bn Bk k=n+1 From the no-arbitrage condition we have E∗ N k=n+1 ∆Dk Fn Bk = 0, which is equivalent to the fact that (Dn )n≤N is a martingale with respect to P ∗ , and hence Dn = E ∗ (DN |Fn ). Taking into account the equalities DN = FN∗ = SN and © 2004 CRC Press LLC Dn = δ0 + δ1 + · · · + δn = Fn∗ , we obtain SN Fn BN Fn∗ = E ∗ (SN |Fn ) = BN E ∗ = BN Sn = Fn .

B a The case when γN +1 = γN +1 = 0 is trivial: γN = β N = 0 , a b γN +1 ≤ γN ≤ γN +1 . Finally, we give examples of typical investment strategies that are based on options (see, for example, [42]). Straddle is a combination of call and put options on the same stock with the same strike price K and the same expiry date N . The function V(SN ) := f (SN ) − CN is called the gain-loss function. For a buyer we have V(SN ) = |SN − K| − CN . Strangle is a combination of call and put options on the same stock with the same expiry date N but different strike prices K1 and K2 .

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Abstract Inference (Probability & Mathematical Statistics) by Ulf Grenander

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