Financial Innovation & Product Design II Dr. Helmut Elsinger « Options, Futures and Other Derivatives », John Hull, Chapter 22 BIART Sébastien The Standard Market Models

Introduction  What are IR derivatives ?  Why are IR derivatives important ?

IR derivatives : valuation  Black-Scholes collapses 1. Volatility of underlying asset constant 2. Interest rate constant

Why is it difficult ?  Dealing with the whole term structure  Complicated probabilistic behavior of individual interest rates  Volatilities not constant in time  Interest rates are used for discounting as well as for defining the payoff IR derivatives : valuation

Main Approaches to Pricing Interest Rate Options  3 approaches: 1. Stick to Black-Scholes 2. Model term structure : Use a variant of Black’s model 3. Start from current term structure: Use a no-arbitrage (yield curve based) model

Black’s Model But Se -qT e rT is the forward price F of the underlying asset (variable)  This is Black’s Model for pricing options : The Black-Scholes formula for a European call on a stock providing a continuous dividend yield can be written as: with :

K : strike price F 0 : forward value of variable T : option maturity  : volatility Black’s Model

The Black’s Model: Payoff Later Than Variable Being Observed K : strike price F 0 : forward value of variable  : volatility T : time when variable is observed T * : time of payoff

Validity of Black’s Model Black’s model appears to make two approximations:  The expected value of the underlying variable is assumed to be its forward price  Interest rates are assumed to be constant for discounting

European Bond Options  When valuing European bond options it is usual to assume that the future bond price is lognormal  We can then use Black’s model

Example : Options on zero- coupons vs. Options on IR  Let us consider a 6-month call option on a 9- month zero-coupon with face value 100  Current spot price of zero-coupon =  Exercise price of call option = 98  Payoff at maturity: Max(0, S T – 98)  The spot price of zero-coupon at the maturity of the option depend on the 3-month interest rate prevailing at that date.  S T = 100 / (1 + r T 0.25)  Exercise option if:  S T > 98  r T < 8.16%

 The exercise rate of the call option is R = 8.16%  With a little bit of algebra, the payoff of the option can be written as:  Interpretation: the payoff of an interest rate put option  The owner of an IR put option:  Receives the difference (if positive) between a fixed rate and a variable rate  Calculated on a notional amount  For an fixed length of time  At the beginning of the IR period Example : Options on zero- coupons vs. Options on IR

European options on interest rates Options on zero-coupons  Face value: M(1+R)  Exercise price K A call option  Payoff: Max(0, S T – K) A put option  Payoff: Max(0, K – S T ) Option on interest rate  Exercise rate R A put option  Payoff: Max[0, M (R-r T ) / (1+r T )] A call option  Payoff: Max[0, M (r T -R) / (1+r T )]

Yield Volatilities vs Price Volatilities The change in forward bond price is related to the change in forward bond yield by where D is the (modified) duration of the forward bond at option maturity

 This relationship implies the following approximation : where s y is the yield volatility and s is the price volatility, y 0 is today’s forward yield  Often is quoted with the understanding that this relationship will be used to calculate Yield Volatilities vs Price Volatilities

Interest Rate Caps  A cap is a collection of call options on interest rates (caplets).  When using Black’s model we assume that the interest rate underlying each caplet is lognormal

The cash flow for each caplet at time t is: Max[0, M (r t – R) ]  M is the principal amount of the cap  R is the cap rate  r t is the reference variable interest rate   is the tenor of the cap (the time period between payments) Used for hedging purpose by companies borrowing at variable rate  If rate r t < R : CF from borrowing = – M r t   If rate r T > R: CF from borrowing = – M r T  + M (r t – R)  = – M R  Interest Rate Caps

Black’s Model for Caps  The value of a caplet, for period [t k, t k+1 ] is F k : forward interest rate for ( t k, t k+1 )  k : interest rate volatility L : principal R K : cap rate  k =t k+1 -t k

1-year cap on 3 month LIBOR Cap rate = 8% (quarterly compounding) Principal amount = $10,000 Maturity11.25 Spot rate6.39%6.50% Discount factors Yield volatility = 20% Payoff at maturity (in 1 year) = Max{0, [10,000  (r – 8%)0.25]/(1+r  0.25)} Example 22.3

The Cap as a portfolio of IR Options :  Step 1 : Calculate 3-month forward in 1 year :  F = [(0.9381/0.9220)-1]  4 = 7% (with simple compounding)  Step 2 : Use Black Value of cap = 10,000   [7%  – 8%  ]  0.25 = 5.19 cash flow takes place in 1.25 year Example 22.3

1-year cap on 3 month LIBOR Cap rate = 8% Principal amount = 10,000 Maturity Spot rate6.39% 6.50% Discount factors Yield volatility = 20% 1-year put on a 1.25 year zero-coupon Face value = 10,200 [10,000 (1+8% * 0.25)] Striking price = 10,000 Spot price of zero-coupon = 10,200 *.9220 = 9,404 1-year forward price = 9,404 / = 10,025 Price volatility = (20%) * (6.94%) * (0.25) = 0.35% Using Black’s model with: F = 10,025 K = 10,000 r = 6.39% T = 1  = 0.35% Put (cap) = 4.607Delta = The Cap as a portfolio of Bond Options :

When Applying Black’s Model To Caps We Must...  EITHER Use forward volatilities Volatility different for each caplet  OR Use flat volatilities Volatility same for each caplet within a particular cap but varies according to life of cap

European Swaptions  When valuing European swap options it is usual to assume that the swap rate is lognormal  Consider a swaption which gives the right to pay s K on an n -year swap starting at time T. The payoff on each swap payment date is where L is principal, m is payment frequency and s T is market swap rate at time T

European Swaptions The value of the swaption is s 0 is the forward swap rate; s is the swap rate volatility; t i is the time from today until the i th swap payment; and

Relationship Between Swaptions and Bond Options 1. Interest rate swap = the exchange of a fixed-rate bond for a floating-rate bond 2. A swaption = option to exchange a fixed-rate bond for a floating-rate bond 3. At the start of the swap the floating-rate bond is worth par so that the swaption can be viewed as an option to exchange a fixed-rate bond for par

Relationship Between Swaptions and Bond Options 4. An option on a swap where fixed is paid and floating is received is a put option on the bond with a strike price of par 5. When floating is paid and fixed is received, it is a call option on the bond with a strike price of par

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