European Options on Futures

To price European options on futures we can use the Black model.

c = e ^{-r T} [F₀(T) N(d₁) - X N(d₂)]
p = e ^{-r T} {X [N(-d₂)] - F₀(T) [N(-d₁)]}

where:

d₁ = [ln(F₀(T)/X) + (σ²/2)T] / (σ T^1/2)
d₂ = d₁ - σ T^1/2
F₀(T) = the futures price at Time 0 that expires at Time T

The volatility, σ, is the volatility of the continuously compounded change in the futures price.

This model can be obtained directly from the BSM formula. It can also be described in a similar way:

• The Black model has two components, a futures component and a bond component. A call option, for example, is simply the futures component minus the bond component.
• The value of a futures option is simply the present value of the difference between the futures price and the exercise price.

Example 1

A forward contract is priced at $65. European options on the forward contract have an exercise price of $70 and expire in 180 days. The continuously compounded risk-free rate is 5.25% and volatility is 0.17. Calculate the prices of a call option and a put option on the forward contract.

First calculate T: T = 180/365 = 0.4931.

Then calculate the values of d₁ and d₂.
d₁ = [ln(65/70) + (0.17)²/2 x 0.4931] / (0.17 x 0.4931^1/2) = -0.5612
d₂ = -0.5612 - 0.17 x 0.4931^1/2 = -0.6805

Using the normal distribution table:
N(d₁) = N(-0.5612) = 1 - N(0.5612) = 1 - 0.7123 = 0.2877
N(d₂) = N(-0.6805) = 1 - N(0.6805) = 1 - 0.7517 = 0.2483

c = e^{-0.0525 x 0.4931}(65 x 0.2877 - 70 x 0.2483) = 1.2858
p = e^{-0.0525 x 0.4931}[70 x (1 - 0.2483) - 65 x (1 - 0.2877)] = 6.1580

Interest Rate Options

Interest rate options can be valued based on a modified Black futures option model in which the underlying is a forward rate agreement (FRA), there is an accrual period adjustment as well as an underlying notional amount, and care must be given to day-count conventions.

Note that you are supposed to "describe how the Black model is used to value European interest rate options and European swaptions." There is no need to memorize the formulas to calculate the prices of interest rate call and put options.

Example 2

An interest rate call option based on a 90-day underlying rate has an exercise rate of 5.5% and expires in 90 days. The forward rate is 5.25% and the volatility is 0.08. The continuously compounded risk-free rate is 4%. Let's calculate the price of the interest rate call option using the Black model.

The time to maturity is T = 90/365 = 0.2466.
F₀(T) = 0.0525
X = 0.055
d₁ = [ln(f₀(T)/X) + (σ²/2)T] / (σ T^1/2) = [ln(0.0525/0.055) + 0.08²/2 x 0.2466] / (0.08 x 0.2466)^1/2 = -1.1512
d₂ = d₁ - σ T^1/2 = -1.1512 - 0.08 x 0.2466 ^1/2 = -1.1910
N(d₁) = N(-1.1512) = 1 - N(1.1512) = 0.1251
N(d₂) = N(-1.1910) = 1 - N(1.1910) = 0.117

c = e^{-0.04 x 0.2466} (0.0525 x 0.1251 - 0.055 x 0.117) = 0.000131

We need to make two adjustments:

• The answer is given under the assumption that the option payoff occurs at the option expiration. However, this interest rate option expires in 90 days and pays off 90 days after that (advanced set, settled in arrears). Therefore, we need to use the forward rate to discount the result back from day 180 to day 90: 0.000131 x e ^{-0.0525 x (90/365)} = 0.00013.

• As the underlying rate and exercise rate are expressed as annual rates, the answer is an annual rate. However, interest rate option prices are often quoted as periodic rates. We need to convert the result to a periodic rate based on a 90-day rate, using the customary 360-day year: 0.00013 x (90/360) = 0.0000325.

Swaptions

A swaption is an option on a swap. That is, it is an option to enter into a swap at a fixed rate. The most widely used swaption is the plain vanilla interest rate swaption. It allows the holder to establish a fixed rate on the underlying swap in advance and have the option of entering into the swap with that fixed rate or allowing the swaption to expire and entering into the swap at the fixed rate that prevails in the market.

• A payer swaption is an option on a swap to pay fixed and receive floating. The buyer hopes the fixed rate goes up before the swaption expires.
• A receiver swaption is an option on a swap to receive fixed and pay floating. The buyer hopes the fixed rate goes down before the swaption expires.

A swaption is based on a specific underlying swap. For example, MPK considers engaging in a $10 million three-year swap in two years. Worried about rising rates, it buys a payer swaption at an exercise rate of 11.5 percent. Swap payments will be annual. The underlying can be viewed as a five-year swap at the time the swaption is initiated and a three-year swap when the swaption expires.

The Black model can be modified to value a swaption. For example, the payer swaption valuation model is:

PAY_SWN = (AP) PVA[R_FIXN(d₁) - R_XN(d₂)]

Note that there are two adjustments, one for the accrual period and one for the present value of an annuity. Other important differences are:

• There is no discount factor. The payoff is a series of payments.
• The underlying is the fixed rate on a forward interest rate swap.
• The exercise price is an interest rate.
• Both the forward swap rate and the exercise rate should be expressed in decimal form and not as percentages.

Similar to other option models, the swaption model can be described as:

• the present value of the expected option payoff at expiration.
• a swap component +/- a bond component.

There are many useful equivalence relationships. For example, payer swaptions are equivalent to puts on bonds. Similarly, receiver swaptions are equivalent to calls on bonds. Another example: to long a receiver swaption and short a payer swaption with the same exercise rate is equivalent to entering a receive-fixed, pay-floating forward swap.