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

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

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

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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

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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

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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 :

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

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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

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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

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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

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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 = 95.60 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%

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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

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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 )]

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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

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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

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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

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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

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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

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1-year cap on 3 month LIBOR Cap rate = 8% (quarterly compounding) Principal amount = $10,000 Maturity11.25 Spot rate6.39%6.50% Discount factors0.93810.9220 Yield volatility = 20% Payoff at maturity (in 1 year) = Max{0, [10,000 (r – 8%)0.25]/(1+r 0.25)} Example 22.3

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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 0.9220 [7% 0.2851 – 8% 0.2213] 0.25 = 5.19 cash flow takes place in 1.25 year Example 22.3

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1-year cap on 3 month LIBOR Cap rate = 8% Principal amount = 10,000 Maturity 1 1.25 Spot rate6.39% 6.50% Discount factors 0.938 0.9220 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 / 0.9381 = 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 = - 0.239 The Cap as a portfolio of Bond Options :

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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

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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

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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

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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

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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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