zero-coupon bond with maturity T and par F.Thus,the market value of the forward contract is $100x D(t,T+T)-Fx D(t,T).Consequently the forward price is given by F=$100xD D(t,T) A Floating Payment:a floating payment indexed to a riskless rate with tenor r,with coupon rate reset at T and payment made at T+r, can be replicated by a portfolio of spot instruments consisting of long a zero-coupon bond with maturity T and par $100 and short a zero- coupon bond with maturity T+r and par $100.Thus,the price of the floating payment is $100 x [D(t,T)-D(t,T+)].This implies immediately that a floating rate note with payment in arrears is always priced at par on any reset date. A Plain Vanilla Interest Rate Swap:a plain-vanilla interest rate swap with the tenor of the floating index matching the payment fre- quency can be perfectly replicated by a portfolio of spot instruments consisting of long a floating rate note with the same floating index, payment frequency,and maturity and short a coupon bond with the same maturity and payment frequency,and with coupon rate equal to the swap rate.It follows that,at the inception of the swap,the swap rate is equal to the par rate: s(t,T)= 1-D(t,T) ∑0i,Dt,T where t =To<T<...<TN =T,6j Tj+1-Tj is the length of the accrual payment period indexed by j,0<j<N-1,based on an appropriate day-count convention,N is the number of payments,and T is the maturity of the swap. In the presence of default risk,the above pricing results may not hold ex- cept under specific conditions(see,e.g.,Section 6.5 for pricing of Eurodollar swaps). 2.4 FIS with Stopping Times For some fixed-income securities,including American options and defaultable securities,the cash flow payoff dates are also random.A random payoff date is typically modeled as a stopping time,that may be exogenously given or 7zero-coupon bond with maturity T and par F. Thus, the market value of the forward contract is $100×D(t, T +τ )−F ×D(t, T). Consequently the forward price is given by F = $100 × D(t,T +τ) D(t,T) . • A Floating Payment: a floating payment indexed to a riskless rate with tenor τ , with coupon rate reset at T and payment made at T + τ , can be replicated by a portfolio of spot instruments consisting of long a zero-coupon bond with maturity T and par $100 and short a zero￾coupon bond with maturity T + τ and par $100. Thus, the price of the floating payment is $100 × [D(t, T) − D(t, T + τ )]. This implies immediately that a floating rate note with payment in arrears is always priced at par on any reset date. • A Plain Vanilla Interest Rate Swap: a plain-vanilla interest rate swap with the tenor of the floating index matching the payment fre￾quency can be perfectly replicated by a portfolio of spot instruments consisting of long a floating rate note with the same floating index, payment frequency, and maturity and short a coupon bond with the same maturity and payment frequency, and with coupon rate equal to the swap rate. It follows that, at the inception of the swap, the swap rate is equal to the par rate: s(t, T) = 1 − D(t, T) PN−1 j=0 δjD(t, Tj ) , where t ≡ T0 < T1 < ... < TN ≡ T, δj = Tj+1 − Tj is the length of the accrual payment period indexed by j, 0 ≤ j ≤ N − 1, based on an appropriate day-count convention, N is the number of payments, and T is the maturity of the swap. In the presence of default risk, the above pricing results may not hold ex￾cept under specific conditions (see, e.g., Section 6.5 for pricing of Eurodollar swaps). 2.4 FIS with Stopping Times For some fixed-income securities, including American options and defaultable securities, the cash flow payoff dates are also random. A random payoff date is typically modeled as a stopping time, that may be exogenously given or 7