Interest Rate Options
When you combine a long interest rate cap and a short interest rate floor with the same exercise rate, you're essentially initiating a receive-floating, pay-fixed interest rate swap. When the cap is in the money, the receive floating receives a net payment, but when the floor is in the money, the receive floating will make a payment.
If that's the case, then the opposite is also true: if you initiate a long interest rate floor with a short interest rate cap with the same exercise rate, you'd have a receive-fixed, pay-floating interest rate swap.
If the cap was in the money for the receive-fixed, pay-floating, who would make the payment?
"It's just the basics" or "that's pretty easy" are phrases you often hear to describe the building block or foundation pieces of a complex process. But when it comes to interest rate options, even the building block phase can be complex.
To start, what is the initial point of the underlying?
Yes!
If the call is in the money, the receive floating is paying on a receive fixed, pay floating. That's because you'd be short an interest rate call, so as the rate increases, the call goes in the money, and you'd have to pay.
It's crucial to remember that a long call option is tied to the receive-floating counterparty receiving the benefit of the interest rate increase.
Also, note that when the exercise rate is set equal on both the cap and the floor, that means that the value at initiation is zero, so it's called an at-market swap. That means that the initial cost of being long a cap and short the floor is also zero.
To sum it up:
[[summary]]
No.
Yes!
No.
If that's the case, how should the FRA rate compare to the spot rate in 12 months?
No.
That's a clear arbitrage opportunity.
That's not it.
That's a clear arbitrage opportunity.
You got it!
The FRA rate and spot rate on a three-month MRR deposit should be equal. That's a crucial point because it helps the Black model value the call option.
For interest rate payments, it's crucial to remember that payments are made in arrears, so the settlement is advanced set, settled in arrears. And don't forget that interest rates will be quoted on an annual basis, so an adjustment might be required.
With that information, you can value a call option on an FRA by using a variation of the Black model called the standard market model, which is expressed as the following.
$$\displaystyle c = (AP)e^{-r(t_{j-1}+t_m)}[FRA(0,t_{j-1},t_m)N(d_1) - R_XN(d_2)]$$
_AP_ denotes the accrual period in years, _R__X_ denotes the exercise rate, _t__j_-1 denotes the time to option expiration, and _t__m_ denotes the time to maturity.
$$\displaystyle FRA(0,t_{j - 1},t_m)$$
indicates that an FRA at Time 0 expires at time _t__j_ - 1 where the underlying expires at time _t__j_, meaning that _t__j_-1 + _t__m_ = _t__j_.
The equation is similar for a put option.
$$\displaystyle p = (AP)e^{-r(t_{j-1} + t_m)}[R_XN(-d_2) - FRA(0,t_{j-1},t_m,)N(-d_1)]$$
For both the call and the put,
$$\displaystyle d_1 = \frac{ln \left[ \frac{FRA(0,t_{j-1},t_m)}{R_X} \right] + \frac{\sigma ^2}{2}t_{j-1}}{\sigma \sqrt{t_{j-1}}}$$
$$\displaystyle d_2 = d_1 - \sigma \sqrt {t_{j-1}}$$.
Note that for both the call and the put formula, the assumed notional value is 1, so you'd take the result times the notional to find the total value.
It's pretty clear to see that those are complex equations, so start by thinking about the option itself. It gives the option buyer the right to certain cash payments based on observed interest rates. A call option gives the right to payments if the underlying interest rate exceeds the exercise rate, and a put option gives the right to payments if the underlying rate is below the exercise rate.
Now, try to view the interest option from the perspective of other FRAs. If you initiated a call option with the exercise rate equal to the FRA rate, then you'd receive payments if the rate increased above the FRA rate.
What type of swap does that sound like?
No.
There are no equities involved in an interest rate swap.
Not quite.
You're already receiving fixed through the deposit to be made in the future, so that's not it.
Bingo!
It's similar to a receive-floating, pay-fixed because you're already going to receive fixed during the deposit time period. If the interest rate "floats" higher during the FRA period, you'd receive payments on a long call or a short put position.
It's the exact opposite for a short call or long put position. In this case, the option can be thought of as receive-fixed, pay-floating.
So FRAs are the ingredients for interest rate swaps.
And interest rate options are the ingredients for interest rate caps. An interest rate cap is a portfolio or strip of interest rate call options (caplets) in which the expiration of the first underlying corresponds to the expiration of the second option, etc. That means that you can hedge a set of floating-rate loan payments with a long position in an interest rate cap, which is a series of interest rate call options.
If that's an interest rate cap, what would an interest rate floor be?
Yes!
No. That's an interest rate cap.
An interest rate floor is a portfolio or strip of interest rate put options (floorlets) in which the expiration of the first underlying corresponds with the expiration of the second option, etc. You can hedge a floating-rate bond or floating-rate lending situation with an interest rate floor, which is a series of interest rate put options.
An interest rate is the initial point of the underlying. Call options benefit when rates rise, while puts benefit when rates decline.
But now for the tricky part.
Take an interest rate call option on a three-month MRR with one year to expiration. The underlying interest rate for this call is a forward rate agreement (FRA) that expires in one year. The FRA is initiated today, and the underlying rate of the FRA is a three-month MRR deposit that will be investable in 12 months and mature in 15 months.
Receive-fixed
Receive-floating
Interest rate
Dividend rate
Higher
Lower
Equal
Equity swap
Receive-fixed, pay-floating swap
Receive-floating, pay-fixed swap
A strip of long put options
A strip of long call options
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