Financial Risk Management Pricing Interest Rate Products Jan Annaert Ghent University Hull, Chapter 22

9-2 Why different? Stochastic process interest rate is different and more difficult Usually one rate is not sufficient, but the entire term structure Volatilities are different Interest rates are used to discount and to compute the pay-off

9-3 This chapter Simple analysis Standard model Dependence on one interest rate of price at one point in time Applications: –options on interest rate futures –implicit bond options –MBS –caps, floors, collars

9-4 Black’s Model for European Options See chapter 13: Distribution V T is lognormal Standard deviation ln(V T ) is  sqrt(T) Interest rates are non-stochastic No geometric Brownian motion

9-5 And for interest rates? V is an interest rate, a bond price, a difference between interest rates Let F = forward price for V r = spot rate T Approximations/errors –Forward price = futures price ?? –Discount using “constant” interest rates, pay-off with stochastic rates

9-6 European bond options Bond price at T is lognormally distributed Standard error ln(price) is  sqrt(T) Compute F Compute K (clean versus dirty) Compute  (is dependent on remaining term to maturity)

9-7 Yield volatilities In practice yield volatilities are commonly used Compute price volatility of

9-8 Interest Rate Caps Cap: assures that a variable rate on a loan does not rise above a maximum rate maximum rate = cap rate Implicit cost or payment of premium

9-9 Libor r cap rate time Caps

9-10 Caps: Example Nominal amount: 10 million USD rate: 3-month LIBOR revision (‘tenor’): quarterly cap rate: 8% Payment by FI:

9-11 Caps as options max(R-cap,0) caplets is the payoff for a call option with later payment cap = a portfolio of call options

9-12 Payment at time (k+1)  k : Caps: General Pricing using Black’s model: Discounted to time k  k  :

9-13 Caps à la Black Attention: Black assumes that the forward rate volatility is constant For short maturities this is not realistic Inconsistency with regard to the treatment of interest rates

9-14 Caps à la Black Approximation Use Black’s model to compute the implied volatility for Eurodollar futures options Use these to price caplets with similar maturities = forward forward volatilities

9-15 Swaptions A option to enter a swap (and pay a fixed interest rate) Swap = exchange of a fixed rate bond versus a floating rate bond At start swap: floating rate bond = face value swaption = exchange of a fixed rate bond versus the face value = put option on a fixed rate bond

9-16 Pricing Swap rate is lognormally distributed at maturity Payoff swaption: m times a year, n years Total: T i =T+i/m

9-17 Convexity Adjustments In Black’s model future values are replaced by their RN expectation (the forward price) The relation between yield and price is non- linear, so the (RN) expected rate is not necessarily the forward rate

9-18 Bond price Rate Y3Y3 B 1B 1 Y1Y1 Y2Y2 B 3B 3 B 2B 2 Convexity

9-19 When do we need a convexity adjustment? For most interest rate products there is a time difference between the determination of the cash flow and effective payment If the cash flow depends on the  -period rate, and the difference between determination and payment is also , NO adjustment is needed If not, it is