© K. Cuthbertson, D. Nitzsche FINANCIAL ENGINEERING: DERIVATIVES AND RISK MANAGEMENT (J. Wiley, 2001) K. Cuthbertson and D. Nitzsche Lecture Pricing Interest Rate Derivatives

© K. Cuthbertson, D. Nitzsche Equilibrium Models pricing a zero, under stochastic interest rates No-Arbitrage Approach and the BOPM pricing coupon bonds, bond options, callable bonds, FRN’s, caps and FRA’s Pricing using Monte Carlo Simulation TOPICS

© K. Cuthbertson, D. Nitzsche Choose a continuous time formulation for the short rate process and estimate the unknown parameters of this process, from historic data on spot rates. Mathematically derive the whole term structure from the chosen short rate model. Use continuous time mathematics to establish the PDE for the interest rate derivative under consideration Solve this PDE to give a closed form solution for the price of the derivative in terms of the estimated parameters of the short rate model. EQUILIBRIUM MODELS

© K. Cuthbertson, D. Nitzsche RNV Model for short rate [18.31]dr =  (r,t) dt +  (r,t) Ho-Lee model [18.44]dr = a dt +  B-S, PDE EQUILIBRIUM MODELS

© K. Cuthbertson, D. Nitzsche Trial soln to BS, PDE [18.45] f(t,T) = k(t,T) exp{-r(T-t)} Substituting the derivatives of [18.45] into B-S, PDE and integrating, we obtain closed form solution [18.47] ln(k(t,T)) = -(1/2) (T-t) 2 a + (1/6) (T-t) 3  2 Sub soln for P(t,T) in long-rate equation: [18.48] Ho-Lee model only allows a parallel shift in the yield curve. Rather restrictive term structure: ameliorated by using alternative stochastic process for the short rate Price a Zero, under stochastic interest rates

© K. Cuthbertson, D. Nitzsche No-Arbitrage Approach and the BOPM

© K. Cuthbertson, D. Nitzsche Spot Rates (r i )Bond Prices (P i ) Volatility (  ij ) 1 st year5%  12 = 20% 2 nd year6%  23 = 19% 3 rd year7%  34 = 18% 4 th year8%  45 = 17% 5 th year9% T18.1 : Initial Term Structure (Spot Rates and Volatilities)

© K. Cuthbertson, D. Nitzsche Arbitrarily choose r d, solve for r u and interate until calculated price of 1-year zero V 0 equals P 0. [18.12] arbitrarily choose r d = 5.64%, then r u = e 2(0.20)(1) (5.64) = 1.49(5.64) = 8.41% [18.13a] [18.13b] [18.13c] No-Arbitrage and the BOPM

© K. Cuthbertson, D. Nitzsche Number of ups Time Table 18.2 : Short Term Interest Rate Lattice

© K. Cuthbertson, D. Nitzsche 10%, 5 year, $100 principal Number of ups Time Pricing Coupon Paying Bond (Excel Table 18.3)

© K. Cuthbertson, D. Nitzsche Payoff for the call (at T=4) = Max{ 0, P i B -K } Number of ups Time T18.4 : European Call Option on 5 Year Bond

© K. Cuthbertson, D. Nitzsche T18.5: American Call on Coupon Bond

© K. Cuthbertson, D. Nitzsche Condition for calling the bond is V i B (rec) >M+C then replace V i B (rec) with M+C in the lattice Number of ups Time Note : Bond can only be called in years 1 to 4 and is called at nodes (4,0), (4,1),(3,0),(2,0),(1,0). Value conventional bond + value written call = value of a callable bond (-3.0) = Table 18.6 : Pricing a 5 Year Callable Bond

© K. Cuthbertson, D. Nitzsche If r i >K cap then Coupon received = $100( K cap ) Value of the capped-FRN, V i = [$100 (1+ K cap )] / (1+r i ) If r i < = K cap then The coupon is (r i K cap ) and hence V i =100 See table 18.2 for the values of r i in the no-arbitrage lattice. Cap rate K cap = 16% Number of ups Time Table 18.7 : Pricing a 5-Year Capped FRN

© K. Cuthbertson, D. Nitzsche Pricing a Cap (K=5%) See table 18.2 for the values of r i in the no-arbitrage lattice To price a cap, simply price the T=1, T=2, etc, caplets and sum the caplet premia

© K. Cuthbertson, D. Nitzsche Expected Payoff: 1x2 FRA Delayed Settlement FRA Pricing FRA’s

© K. Cuthbertson, D. Nitzsche MCS: Pricing a Caplet

© K. Cuthbertson, D. Nitzsche Vasicek’s mean reverting model: [18.53]r t – r t-1 = a ( b – r t-1 )  t +     t Long run rate is b= 0.08 (8%) Rate of convergence, ‘a’ = 0.20 Current short rate is r 0 = 0.10 (10%) Volatility,  = Caplet Strike rate K= 0.10 (10%) T = 1 year Divide 1 year into n=100 time units Take  t = T/n = 0.01 MCS: Pricing a Caplet (Excel T18.10)

© K. Cuthbertson, D. Nitzsche Generate n= 100 observations on r and calculate the average value of r, that is r av, over the horizon t=1 to t=T. Calculate the value of the caplet at expiration = max{r 100 – K, 0} The value of the caplet for the first Monte Carlo run is V (1) = exp(-r av T) x max{r 100 – K, 0} Repeat above steps for m=20,000 runs. The MCS value for the caplet is then C = (1/m)  1 m, V (i) MCS: Pricing a Caplet ( Excel T18.10).

© K. Cuthbertson, D. Nitzsche LECTURE ENDS HERE