Swaptions
Actually, no.
The second component needs to capture the payment structure.
When you barter with another counterparty to exchange goods or services, you're essentially swapping one good or service for another with the goal of getting the best price.
Now imagine another opportunity. Say you're given the chance to enter negotiations at certain points for a price. In some cases, you could take advantage of another person's negotiating skills, while at other times, you'd probably gain nothing.
In addition to the bond and swap components, swaptions can also be thought of as various swap structures. For example, a long receiver swaption and a short payer swaption with the same exercise rate is the same as entering into a receive-fixed, pay-floating forward swap.
What would the floating rate need to do for the swap holder to benefit?
Exactly!
Not quite.
The swap holder is short on the floating rate, meaning that if it rises, the long receiver swaption or short payer swaption will have to pay, just like a receive-fixed, pay-floating forward swap. The opposite is also true. A long payer swaption and a short receiver swaption with the same exercise rate is equivalent to entering a receive-floating, pay-fixed forward swap.
If the exercise rate is the same on both the receiver and payer swaptions, then the values are the same, so the exercise rate is equal to the at-market forward swap rate. That means the put-call parity relationship can provide additional valuation.
But swaptions and forward rates aren't the only similar instruments. Think about how a callable fixed-rate bond could involve a swaption.
So if you're long the callable bond, you're effectively long a straight fixed-rate bond and short a receiver swaption. That's because the receiver option gives the holder the right to receive a fixed rate. But in this case, you'd be short that right, so you're holding the right to get a fixed rate put to you. If the bond is called, that usually means that rates have decreased, and new bonds can be issued at lower rates, so you're essentially put a lower rate through the callable bond.
So the callable bond is just like a straight bond with a short receiver swaption.
To summarize:
[[summary]]
Bingo!
It's a swap because it captures the multiple cash flows that will be swapped over time.
For a payer swaption, the swap component is
$$\displaystyle (AP)PVA(R_{FIX})N(d_1)$$.
And the bond component is
$$\displaystyle (AP)PVA(R_X)N(d_2)$$.
So a payer swaption is the difference between the swap component less the bond component.
For a receiver swaption, the swap component is
$$\displaystyle (AP)PVA(R_{FIX})N(-d_1)$$.
And the bond component is
$$\displaystyle (AP)PVA(R_X)N(-d_2)$$ .
So a receiver swaption is the bond component less the swap component.
This outlines what a swaption basically is: an option on a swap. It gives its holder the right, but not the obligation, to enter a swap at the preagreed swap rate—the exercise rate.
Interest rate swaps can be either receive-fixed, pay-floating or receive-floating, pay-fixed, where a payer swaption pays fixed, and a receiver swaption receives fixed.
What do fixed interest rates need to do for a payer-swaption holder to benefit?
Actually, no
You got it!
This is a slightly tricky question because the swap adds another aspect. That's because when exercised, the payer-swaption holder is able to enter into a pay-fixed, receive-floating swap at the predetermined exercise rate, _R__X_. If rates rise, the holder can realize a gain by entering into an offsetting at-market receive-fixed, pay-floating swap at the current fixed rate.
In this case, both floating rates will offset, leaving the holder with a gain as the difference between the agreed-upon fixed rate at initiation and the current fixed rate.
This opportunity can be priced off as another type of financial contract. What is it?
No.
A zero-coupon bond wouldn't involve the consistent payment of interest.
Not quite.
That consistent payment is technically risk free, so an equity isn't the best choice.
You got it!
Swaptions can be valued as an annuity to capture the swap reset dates. The contracts are settled as advanced set, settled in arrears.
The valuation equation for a payer swaption is the following.
$$\displaystyle PAY_{SWN} = (AP)PVA[R_{FIX}N(d_1) - R_X N(d_2)]$$
And the receiver swaption valuation model is
$$\displaystyle REC_{SWN} = (AP)PVA[R_XN(-d_2) - R_{FIX}N(-d_1)]$$
where
$$\displaystyle d_1 = \frac{ln[\frac{R_{FIX})}{R_X}] + \frac{\sigma ^2}{2}T}{\sigma \sqrt{T}}$$
$$\displaystyle d_2 = d_1 - \sigma \sqrt {T}$$.
_R__FIX_ denotes the fixed swap rate starting when the swap expires, noted as before with _T_, quoted on an annual basis; _R__X_ denotes the exercise rate starting at time _T_, again on an annual basis; _AP_ notes the accrual period; and σ notes the volatility of the forward rate swap.
This is another adjustment to the Black model. The first adjusts for the fact that the swap is a series of payments, not a single payoff, based on an underlying that is the fixed rate of a forward interest rate swap.
Since the underlying is an interest rate, the exercise price is expressed as an interest rate, so the forward rate swap and the exercise rate should be expressed in decimal form and not as a percent.
But even with all these adjustments, the swaption model can still be described as the present value of the expected option payoff at expiration. It can be broken down into two components. The first is a bond. What's the second component?
No.
The underlying's not an equity, so that's not it.
Increase
Decrease
Fall
Rise
Equity
Annuity
Zero-coupon bond
Swap
Equity
Interest rate
Continue
Continue
Continue
Continue
Continue
Continue
