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The Standard Market Models

The Standard Market Models. Financial Innovation & Product Design II Dr. Helmut Elsinger « Options, Futures and Other Derivatives », John Hull, Chapter 22. BIART Sébastien. Introduction. What are IR derivatives ? Why are IR derivatives important ?. IR derivatives : valuation.

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The Standard Market Models

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

  2. Introduction • What are IR derivatives ? • Why are IR derivatives important ?

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

  4. IR derivatives : valuation 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

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

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

  7. Black’s Model • K : strike price • F0 : forward value of variable • T : option maturity • s : volatility

  8. The Black’s Model: Payoff Later Than Variable Being Observed • K : strike price • F0 : forward value of variable • s : volatility • T : time when variable is observed • T *: time of payoff

  9. 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

  10. 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

  11. 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, ST – 98) • The spot price of zero-coupon at the maturity of the option depend on the 3-month interest rate prevailing at that date. • ST = 100 / (1 + rT0.25) • Exercise option if: • ST > 98 • rT < 8.16%

  12. Example : Options on zero-coupons vs. Options on IR • 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

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

  14. 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

  15. Yield Volatilities vs Price Volatilities • This relationship implies the following approximation : where sy is the yield volatility and sis the price volatility, y0 is today’s forward yield • Often is quoted with the understanding that this relationship will be used to calculate

  16. 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

  17. Interest Rate Caps The cash flow for each caplet at time t is: Max[0, M (rt – R) ] • M is the principal amount of the cap • R is the cap rate • rt 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 rt < R : CF from borrowing = –Mrt • If rate rT > R: CF from borrowing = –M rT + M (rt – R)  = – M R

  18. Black’s Model for Caps • The value of a caplet, for period [tk, tk+1] is • L: principal • RK : cap rate • dk=tk+1-tk • Fk : forward interest rate • for (tk, tk+1) • sk: interest rate volatility

  19. Example 22.3 • 1-year cap on 3 month LIBOR • Cap rate = 8% (quarterly compounding) • Principal amount = $10,000 • Maturity 1 1.25 • Spot rate 6.39% 6.50% • Discount factors 0.9381 0.9220 • Yield volatility = 20% • Payoff at maturity (in 1 year) = • Max{0, [10,000  (r – 8%)0.25]/(1+r 0.25)}

  20. 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  0.9220 [7%  0.2851 – 8%  0.2213]  0.25 = 5.19 cash flow takes place in 1.25 year

  21. Example 22.3 The Cap as a portfolio of Bond Options : 1-year cap on 3 month LIBOR Cap rate = 8% Principal amount = 10,000 Maturity 1 1.25 Spot rate 6.39% 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 Using Black’s model with: F = 10,025K = 10,000r = 6.39%T = 1 = 0.35% Put (cap) = 4.607 Delta = - 0.239 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%

  22. When Applying Black’s ModelTo 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

  23. 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 sK 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 sT is market swap rate at time T

  24. European Swaptions The value of the swaption is s0 is the forward swap rate; s is the swap rate volatility; ti is the time from today until the i th swap payment; and

  25. 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

  26. 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

  27. … Thank you !

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