The Standard Market Models

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

• What are IR derivatives ?
• Why are IR derivatives important ?

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IR derivatives valuation

• Black-Scholes collapses
• Volatility of underlying asset constant
• Interest rate constant

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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

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Main Approaches to PricingInterest Rate Options

• 3 approaches
• Stick to Black-Scholes
• Model term structure Use a variant of Blacks
  model
• Start from current term structure Use a
  no-arbitrage (yield curve based) model

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Blacks 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 Blacks Model for pricing options
with
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Blacks Model

• K strike price
• F0 forward value of variable

• T option maturity
• s volatility

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The Blacks 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

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Validity of Blacks Model

• Blacks 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 Blacks 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, 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 rT 0.25)
• Exercise option if
• ST 98
• rT

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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

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European options on interest rates

• Options on zero-coupons
• Face value M(1R?)
• 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
• Max0, M (R-rT)? / (1rT?)
• A call option
• Payoff
• Max0, M (rT -R)? / (1rT?)

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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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Yield Volatilities vs Price Volatilities

• This relationship implies the following
  approximation
• where sy is the yield volatility and s is the
  price volatility, y0 is todays forward yield
• Often is quoted with the understanding that
  this relationship will be used to calculate

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Interest Rate Caps

• A cap is a collection of call options on interest
  rates (caplets).
• When using Blacks model we assume that the
  interest rate underlying each caplet is lognormal

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Interest Rate Caps

• The cash flow for each caplet at time t is
  Max0, 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
• If rate rT R CF from borrowing M rT ?
  M (rt R) ? M R ?

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Blacks Model for Caps

• The value of a caplet, for period tk, tk1 is

• L principal
• RK cap rate
• dktk1-tk

• Fk forward interest rate
• for (tk, tk1)
• sk interest rate volatility

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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)
• Max0, 10,000 ? (r 8)?0.25/(1r ? 0.25)

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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
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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 (18 0.25) Striking price
10,000
Using Blacks model with F 10,025K 10,000r
6.39T 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
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When Applying Blacks 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

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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
  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

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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

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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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Thank you !