lecture_slides-TermStructure_Binomial_#1

Will use risk neutral pricing to price every security

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Unformatted text preview: ¨ qu¨¨ ¨¨ ¨¨ r1,1 ¨ d r2,1 ¨ r3,1 ¨ ¨ q ¨¨ ¨ ¨¨ ¨¨ ¨ ¨¨ r0,0 ¨ r1,0 ¨ r2,0 ¨ r3,0 ¨ ¨ ¨ ¨ ¨ r3,3 t=0 t=1 t=2 t=3 t=4 Let Zi ,j be the date i , state j price of a non-coupon paying security. Will use risk-neutral pricing to price every security so that: Zi ,j = 1 [qu × Zi +1,j +1 + qd × Zi +1,j ] 1 + ri ,j (1) – where qu and qd are the risk-neutral probabilities of an up- and down-move – so qd + qu = 1 and must have qd > 0 and qu > 0. There can be no arbitrage when we price using (3). Why? 6 Binomial Models for the Short Rate If the security pays a “coupon”, Ci +1,j , at date i + 1 and state j then Zi ,j = 1 [qu (Zi +1,j +1 + Ci +1,j +1 ) + qd (Zi +1,j + Ci +1,j )] 1 + ri ,j (2) – where Zi +1,. is now the ex-coupon price at date i + 1. If we use (3) or (2) to price securities in the lattice model then arbitrage is not possible - Regardless of what probabilities we use! Why is this? In fact it is very common to simply set qu = qd = 1/2 - and to calibrate other parameters to market prices. We will assume qu = qd = 1/2 in our examples. 7 Financial Engineering & Risk Management The Cash Account and Pricing Zero-Coupon Bonds M. Haugh G. Iyengar Department of Industrial Engineering and Operations Research Columbia University Binomial Models for the Short-Rate ¨¨ ¨ ¨¨ ¨ ¨ ¨ r2,2 ¨ r3,2 ¨ ¨ ¨ ¨ qu¨¨ ¨¨ ¨¨ r1,1 ¨ d r2,1 ¨ r3,1 q ¨ ¨ ¨ ¨¨ ¨¨¨ ¨¨¨ ¨¨ r0,0 ¨ r1,0 r2,0 r3,0 ¨ ¨ ¨ ¨ ¨ r3,3 t=0 t=1 t=2 t=3 t=4 We use risk-neutral pricing to price every non-coupon paying security: Zi ,j = 1 [qu × Zi +1,j +1 + qd × Zi +1,j ] 1 + ri ,j (3) – qu > 0 and qd > 0 are the risk-neutral probabilities of an up- and down-move, respectively, of the short-rate. There can be no arbitrage when we price using (3). Why? 2 The Cash-Account The cash-account is a particular security that in each period earns interest at the short-rate - we use Bt to denote its value at time t and assume that B0 = 1. The cash-account is not risk-free since Bt +s is not known at time t for any s>1 - it is locally risk-free since Bt +1 is known at time t . Note that Bt satisfies Bt = (1 + r0,0 )(1 + r1 ) . . . (1 + rt −1 ) - so that Bt /Bt +1 = 1/(1 + rt ). Risk-neutral pricing for a “non-coupon” paying security then takes the form: Zt ,j = = = 1 [qu × Zt +1,j +1 + qd × Zt +1,j ] 1 + rt ,j Zt +1 EQ t 1 + rt ,j Bt EQ Zt +1 t Bt +1 (4) 3 Risk-Neutral Pricing with the Cash-Account Therefore for a non-coupon paying security, (4) is equivalent to Zt Zt +1 = EQ t Bt Bt +1 (5) We can iterate (5) to obtain Zt +s Zt = EQ t Bt Bt +s (6) for any non-coupon paying security and any s > 0. 4 Risk-Neutral Pricing with the Cash-Account Risk-neutral pricing for a “coupon” paying security takes the form: Zt ,j 1 [qu (Zt +1,j +1 + Ct +1,j +1 ) + qd (Zt +1,j + Ct +1,j )] 1 + rt ,j Zt +1 + Ct +1 (7) = EQ t 1 + rt ,j = We can rewrite (7) as Zt Ct +1 Zt +1 = EQ + t Bt Bt +1 Bt +1 (8) More generally, we can iterate...
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This note was uploaded on 01/09/2014 for the course FIN 347 taught by Professor Martinhaugh during the Fall '13 term at Bingham University.

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