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By Mark A. Pinsky, Samuel Karlin

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10) holds. 10) is satisfied we have t rs ds V˜t dVt = d exp 0 t t rs ds V˜t dt + exp = rt exp 0 rs ds dV˜t 0 t ˆt rs ds V˜t dt + σt ηt St dB = rt exp 0 ˆt = Vt rt dt + σt ηt St dB ˆt = ζt At rt dt + ηt St rt dt + σt ηt St dB = ζt dAt + ηt dSt , hence the portfolio is self-financing. In the next proposition we compute a self-financing hedging strategy leading to an arbitrary square-integrable random variable F admitting a predictable representation of the form T ˆt , ξt d B F = IEQ [F ] + 0 where (ξt )t∈[0,t] is a square-integrable adapted process.

3) can be written as At dζt + St dηt = 0, 0≤t≤T provided one neglects the bracket d S, η t . 3 PDE Method In this standard Black-Scholes model it is possible to determine a portfolio strategy for the hedging of European claims. 4) = rt Vt dt + (µt − rt )ηt St dt + σt ηt St dBt , t ∈ R+ . Assume now that the value Vt of the portfolio at time t is given by a function C(t, x) as Vt = C(t, St ), t ∈ R+ . 8) leads to dC(t, St ) = ∂C 1 ∂2C 2 2 ∂C + µt St + S σ (t, St )dt ∂t ∂x 2 ∂x2 t t ∂C (t, St )dBt .

In the remaining of this chapter we focus on the stochastic case (c). The pricing of the bond P0 (t, T ) will follow the following steps, previously used in the case of Black-Scholes pricing. Pricing bonds with non-zero coupon is not difficult in the case of a deterministic continuous-time coupon yield at rate c > 0. In this case the price Pc (t, T ) of the coupon bound is given by Pc (t, T ) = ec(T −t) P0 (t, T ), 0 ≤ t ≤ T. In the sequel we will only consider zero-coupon bonds, and let P (t, T ) = P0 (t, T ), 0 ≤ t ≤ T .

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