Chapter 29: Interest Rate Derivatives: The Standard Market Models

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Bond Options

An option to buy or sell a bond at a specified price. Exchange-traded bond options are American-style; OTC bond options are typically European.

Using Black's Model for Bond Options

Assume the bond price at the option's maturity is lognormal:

$c = P(0, T)[F_B N(d_1) - K N(d_2)]$

$d1=ln⁡(FB/K)+σB2T/2σBT,d2=d1−σBTd_1 = \frac{\ln(F_B/K) + \sigma_B^2 T/2}{\sigma_B \sqrt{T}}, \quad d_2 = d_1 - \sigma_B \sqrt{T}$

where $F_B$ is the forward bond price, $σB\sigma_B$ is the bond price volatility, and $P (0, T)$ is the discount factor to the option maturity.

Limitation: Bond prices cannot be exactly lognormal — as maturity approaches, bond prices converge to par value (pull to par). Bond price volatility is non-constant.

Yield-Based Pricing

An alternative: assume bond yields are lognormal. The bond price is a nonlinear function of yield, so the option payoff is computed from the yield distribution.

Interest Rate Caps and Floors

Cap

A portfolio (strip) of caplets. Each caplet provides a payoff on a specific reset date if the reference rate exceeds the cap rate:

$Payoff=L⋅τ⋅max⁡(R−RK,0)\text{Payoff} = L \cdot \tau \cdot \max(R - R_K, 0)$

where $L$ = principal, $τ\tau$ = accrual period, $R$ = realized rate, $R_K$ = cap rate.

The payoff is made at the end of the accrual period (in arrears).

Floor

A portfolio of floorlets. Each floorlet pays if the reference rate falls below the floor rate:

$Payoff=L⋅τ⋅max⁡(RK−R,0)\text{Payoff} = L \cdot \tau \cdot \max(R_K - R, 0)$

Pricing Caplets Using Black's Model

Each caplet is a call option on the reference rate. Under the forward measure for the payoff date:

$Caplet=L⋅τ⋅P(0,ti+1)⋅[FiN(d1)−RKN(d2)]\text{Caplet} = L \cdot \tau \cdot P(0, t_{i+1}) \cdot [F_i N(d_1) - R_K N(d_2)]$

$d1=ln⁡(Fi/RK)+σi2ti/2σiti,d2=d1−σitid_1 = \frac{\ln(F_i/R_K) + \sigma_i^2 t_i/2}{\sigma_i \sqrt{t_i}}, \quad d_2 = d_1 - \sigma_i \sqrt{t_i}$

where $F_i$ is the forward rate for period $i$ , and $σi\sigma_i$ is the caplet volatility.

Volatility Quoting Conventions

Caps are quoted using flat volatilities — a single volatility that, when applied to all caplets, gives the market price. In practice:

Spot volatilities: Each caplet has its own volatility
Implied flat volatility: Averages spot volatilities to match the cap price

Put–Call Parity for Caps and Floors

$Cap−Floor=Swap\text{Cap} - \text{Floor} = \text{Swap}$

A long cap + short floor = pay floating, receive fixed swap. This is because at each reset date:

$max(R - R_K, 0) - \max(R_K - R, 0) = R - R_K$

European Swap Options (Swaptions)

A swaption gives the right to enter into a swap at a future date. Two types:

Payer swaption: Right to pay fixed, receive floating
Receiver swaption: Right to receive fixed, pay floating

Pricing Using Black's Model

Under the swap measure (numeraire = annuity factor), Black's model gives:

$Swaption=A⋅[S0N(d1)−SKN(d2)]\text{Swaption} = A \cdot [S_0 N(d_1) - S_K N(d_2)]$

where $A$ = annuity factor (PV of $1 p ai d o n e a c h s w a pp a y m e n t d a t e),$ S_0 $= f or w a r d s w a p r a t e,$ S_K$ = strike swap rate.

This is analogous to Black's model with the forward swap rate as the lognormal underlying.

Swaptions and Bond Options

A receiver swaption can be viewed as a call option on a coupon-bearing bond with strike equal to par. A payer swaption is a put option on a coupon-bearing bond. However, the lognormal assumption differs: swaptions assume lognormal swap rates; bond options assume lognormal bond prices.

Hedging Interest Rate Derivatives

Interest rate derivatives are typically hedged using:

Delta hedging with interest rate futures or bonds
Vega hedging with other options (to manage volatility exposure)
Bucket hedging: Hedging exposure to different maturity segments (buckets) of the yield curve individually

The yield curve exposure is decomposed into exposures to different parts of the curve (short end, belly, long end), and each bucket is hedged separately.