stochastic.218

# stochastic.218 - CHAPTER 20 Pricing Exotic Options 217 For...

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Unformatted text preview: CHAPTER 20. Pricing Exotic Options 217 For 0  t  T , we compute the differential 1 d e,rt vt; S t = e,rt,rv + vt + rSvx + 2 2 S 2vxx  dt + e,rt Svx dB: Integrate from 0 to t ^ : e,rt^ v t ^ ; S t ^  = v0; S 0 Z t^ + e,ru ,rv + vt + rSvx + 1 2 S 2vxx  du 2 0 Z t^ + e,ru Svx dB: z |0 A stopped martingale is still a martingale Because e,rt^ v t ^ ; S t ^  is also a martingale, the Riemann integral Z t^ 0 1 e,ru ,rv + vt + rSvx + 2 2 S 2vxx  du is a martingale. Therefore, ,rvu; S u+ vtu; S u + rS uvxu; S u+ 1 2S 2uvxxu; S u = 0; 0  u  t ^ : 2 The PDE 1 ,rv + vt + rxvx + 2 2x2vxx = 0; 0  t  T; 0  x  L; then follows. d e,rtv t; S t = e,rt S tvxt; S t dBt; 0  t  : Let X t be the wealth process corresponding to some portfolio t. Then The Hedge de,rtX t = e,rt t S t dBt: We should take and X 0 = v 0; S 0 t = vx t; S t; 0  t  T ^ : Then X T ^  = vT ^ ; S T ^   vT; S T  = S T  , K + = v  ; L = 0 if if T  T. ...
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