A swaption is an option that gives the holder the right, but not the obligation, to enter into an interest rate swap at a future date. There are two types of swaptions:
A Payer Swaption allows the holder to pay a fixed rate and receive a floating rate. This is similar to a call option on
interest rates, benefiting from rising rates. A Receiver Swaption allows the holder to receive a fixed rate and pay a floating rate. This is akin to a put option on interest rates, advantageous
when rates fall.
Swaptions are used to hedge interest rate risk or for speculative purposes. For example, a firm might buy a payer
swaption to protect against rising borrowing costs, while a mortgage holder might buy a receiver swaption to protect against falling rates.
The value of a swaption is derived from the expected cash flows of the underlying swap, discounted to the present using
the zero-coupon curve. The forward swap rate and implied volatility are key inputs for valuation.
The forward swap rate is the fixed rate agreed upon today for a swap that will start at a specific future date. It
represents the rate at which the market expects the fixed payments of a swap to be exchanged for floating payments in the future, equivalent to pricing an IRS that starts in the
future.
The zero-coupon curve, the yields of zero-coupon bonds derived through bootstrapping from the prices of various
risk-free securities across different maturities is used to discount future cash flows to their present value.
It also provides forward rates essential for swap valuation in a no-arbitrage environment.
Implied volatility is also a used for swaption pricing as it captures the market’s expectations of future interest rate
movements and directly influences the premium of the swaption.
One method to price swaptions is Black's model, which uses the following formulas:
PV_payer(t) = NA[S * Φ(d1) - K * Φ(d2)]
PV_receiver(t) = NA[K * Φ(-d2) - S * Φ(-d1)]
With:
- N = Notional principal amount
- A = Annuity factor, the present value of the swap's fixed leg
- S = Forward swap rate
- K = Strike rate of the swaption
- Φ (Phi)= Cumulative standard normal distribution function
- d1 = [ln(S/K) + (σ^2 / 2) * T] / (σ * sqrt(T))
- d2 = d1 - σ * sqrt(T)
Let's consider the following example to price a payer swaption:
- Notional (N): $10,000,000
- Swaption Maturity (T): 1 year
- Underlying Swap Tenor: 5 years
- Strike Rate (K): 3%
- Forward Swap Rate (S): 3.5%
- Implied Volatility (σ): 20%
- Annuity Factor (A): Approximately 4.25 years
- Discount Factors (D1 = 0.95, D5 = 0.80)
- ln(3.5% / 3%) = ln(1.1667) ≈ 0.154
- σ^2 / 2 = 0.02
- d1 = (0.154 + 0.02) / 0.2 = 0.87
- d2 = 0.87 - 0.2 = 0.67
- Φ(0.87) ≈ 0.8078
- Φ(0.67) ≈ 0.7486
PV_payer = NA[S * Φ(d1) - K * Φ(d2)] = 10,000,000 * 4.25 * [(0.035 * 0.8078) - (0.03 * 0.7486)] = $247,362.50, premium for this payer swaption.
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