9.5 Interest Rate Options, Swaptions & the Black Model

9.5 Interest Rate Options, Swaptions & the Black Model

How this fits: section 9.3 built the binomial model for equity options and section 9.4 built Black–Scholes–Merton and the Greeks. This section covers the remaining contingent-claims learning outcomes, all of which are interest-rate specific: valuing an interest rate option on a two-period binomial tree, the Black model for futures options, and the Black model applied to European interest rate options and swaptions.

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1. Interest Rate Options on a Binomial Tree

An interest rate call option pays off when a reference rate rises above the exercise rate:

$\text{Call payoff} = \text{Notional} \times \max(0,\ r_{\text{reference}} - r_X) \times \frac{\text{Days}}{360}$

An interest rate put option pays when the reference rate falls below the exercise rate:

$\text{Put payoff} = \text{Notional} \times \max(0,\ r_X - r_{\text{reference}}) \times \frac{\text{Days}}{360}$

Three features distinguish these from equity options and are the source of most errors:

1. The underlying is a rate, not a price. A rise in rates is good for the call holder, whereas a rise in rates is bad for a bond-price call.
2. Payment is in arrears. The rate is observed at the start of the settlement period and paid at the end, so the payoff must be discounted one period from observation to payment.
3. Discounting uses the node-specific rate, not a single flat rate. Each node on the interest rate tree carries its own one-period rate, which is precisely the rate used to discount from that node backwards.

Two-period valuation procedure

Step 1. Take the calibrated binomial interest rate tree. Rates at time 2 are $r_{uu}$, $r_{ud}$, $r_{dd}$; at time 1 they are $r_u$ and $r_d$; at time 0 the rate is $r_0$.

Step 2. Compute the option payoff at every time-2 node using the payoff formula.

Step 3. Discount back to time 1 using the risk-neutral probability of 0.5 for each branch (the standard assumption in the curriculum's calibrated tree) and the time-1 node rate:

$V_u = \frac{0.5\,V_{uu} + 0.5\,V_{ud}}{1 + r_u}$

Step 4. Add any time-1 exercise payoff for an option with intermediate expiry, then discount from time 1 to time 0 at $r_0$.

Worked example. A two-period interest rate call on a notional of 10,000,000 with an exercise rate of 3.00%, on an annual reset. The calibrated tree gives $r_0 = 2.50%$, $r_u = 3.60%$, $r_d = 2.20%$.

Payoffs occur at the end of each period based on the rate observed at the start:

• At the up node (rate 3.60%): payoff $= 10{,}000{,}000 \times (0.0360 - 0.0300) = 60{,}000$, paid one year later, so its value at the up node is $60{,}000/1.0360 = 57{,}915$
• At the down node (rate 2.20%): the option is out of the money, so the value is 0
• Value at time 0: $\dfrac{0.5(57{,}915) + 0.5(0)}{1.0250} = \dfrac{28{,}957}{1.0250} = \textbf{28{,}251}$

The two errors this construction sets up are (a) forgetting to discount the payoff from payment date back to the observation node, and (b) discounting at the time-0 rate throughout instead of using each node's own rate.

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2. Caps, Floors, and the Swap Equivalence

A cap on a floating-rate liability is a series of interest rate call options — caplets — one for each reset date. A floor is a series of interest rate put options — floorlets.

$\text{Value of a cap} = \sum_{i=1}^{n} \text{Value of caplet}_i$

The exam-critical relationship:

Long cap + short floor at the same exercise rate = pay-fixed interest rate swap.

The intuition: with a cap you receive when rates exceed the strike; with a short floor you pay when rates fall below it. Combined, you pay the strike and receive the floating rate at every reset — which is exactly a pay-fixed swap. It follows that when the exercise rate equals the current swap fixed rate, the cap and the floor have equal value, because the swap itself has zero value at initiation.

Practical uses in vignettes:

• A borrower with floating-rate debt buys a cap to limit its interest cost while retaining the benefit of falling rates.
• The same borrower sells a floor to fund the cap premium, creating a collar that fixes the cost within a band. A zero-cost collar sets the two strikes so the premiums offset.
• A floating-rate lender buys a floor to protect its income.

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3. The Black Model

The Black model (Black-76) values European options where the underlying is a futures or forward price rather than a spot price. Its structure is Black–Scholes–Merton with the spot price replaced by the discounted futures price:

$c = e^{-rT}\,[F_0\,N(d_1) - X\,N(d_2)] \qquad p = e^{-rT}\,[X\,N(-d_2) - F_0\,N(-d_1)]$

$d_1 = \frac{\ln(F_0/X) + (\sigma^2/2)T}{\sigma\sqrt{T}} \qquad d_2 = d_1 - \sigma\sqrt{T}$

Two structural points:

1. There is no carry term in $d_1$. The futures price already embeds the cost of carry, which is the reason the model is used for commodities, currencies, and rates.
2. The whole payoff is discounted at $e^{-rT}$, because a futures position requires no initial outlay for the underlying.

Applications

4. Swaptions

A swaption is an option to enter an interest rate swap at a stated fixed rate on a stated future date.

The bond-equivalence line is the one that traps candidates: a payer swaption behaves like a put option on a bond, because rising rates that benefit the payer are exactly the rates that reduce bond prices.

Valuation with the Black model

$\text{Payer swaption} = (AP) \times \text{Notional} \times e^{-rT}\left[R_{FS}\,N(d_1) - R_X\,N(d_2)\right]$

where $R_{FS}$ is the current forward swap fixed rate for the swap that begins at the option's expiry, $R_X$ is the exercise rate, and $AP$ is the present value of an annuity matching the swap's payment schedule — the sum of the discount factors over the life of the underlying swap, which converts a rate difference into a value.

The receiver swaption reverses the terms inside the bracket, using $N(-d_2)$ and $N(-d_1)$.

Put–call parity for swaptions: a long payer swaption plus a short receiver swaption at the same exercise rate equals a forward pay-fixed swap. When the exercise rate equals the forward swap rate, the payer and receiver swaptions have equal value, exactly mirroring the cap–floor relationship.

Uses in vignettes

• A company planning to issue floating-rate debt in six months buys a payer swaption to lock a maximum fixed cost while retaining the benefit if rates fall.
• A holder of a callable bond is effectively short a receiver swaption; buying one back neutralises the call exposure.
• A pension fund expecting to receive fixed in future buys a receiver swaption to protect against a fall in rates before the swap is executed.

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5. Implied Volatility and Its Role

Implied volatility is the value of $\sigma$ that makes a model price equal the observed market price. It is not estimated from history; it is extracted from prices.

Four points the curriculum emphasises:

1. Volatility is the only unobservable input. Underlying price or rate, exercise price, time to expiry, and the risk-free rate are all observable, so the market price of an option is a statement about volatility and nothing else.
2. Options are quoted in volatility terms in professional interest rate and currency markets, precisely because the volatility is what is being traded. A dealer quotes a swaption at "78 vols", not at a currency price.
3. Implied volatility is forward-looking and typically exceeds realised volatility over the same period, a gap interpreted as a volatility risk premium.
4. A volatility surface — implied volatility varying by strike and by expiry — demonstrates that the constant-volatility assumption of Black–Scholes–Merton and the Black model does not hold. The skew in equity index options, where downside strikes carry higher implied volatility, reflects demand for crash protection and fat-tailed return distributions.

For interest rate options, implied volatility connects directly to fixed income: the volatility used to calibrate the binomial interest rate tree in section 8.2, and the volatility that determines the option-adjusted spread in section 8.3, is the same market quantity. A rise in swaption implied volatility raises the value of the embedded call in a callable bond and therefore widens its OAS at a constant price.

Level II traps in this material

1. Treating an interest rate call as benefiting from falling rates. It benefits from rising rates.
2. Failing to discount the interest rate option payoff from the payment date back to the observation node — payment is in arrears.
3. Discounting a binomial interest rate option at a single flat rate rather than at each node's own rate.
4. Reversing the swaption equivalence: a payer swaption is analogous to a put on a bond.
5. Omitting the annuity factor in a swaption valuation, which produces a rate difference rather than a value.
6. Using Black–Scholes–Merton with a carry term for a futures option; the Black model has no carry term because the futures price already embeds it.