Fixed Income derivatives, based on the binomial tree framework.
Interest Rate Swaps, Caps, and Floors
Floating Note
The payer of of a floating note will pay at where is not known until time . At the maturity date, the payer will pay the face value plus the last floating coupon
The price of a floating note should equal to the final face value .
Replication Method 1: from initial capital
Imagine that the buyer of this floating note give the payer at time 0. The payer just use to buy a ZCB maturity at 0.5, then at 0.5, the payer receive . Then the payer use the to buy another ZCB with maturity date on 1.0, and pay the left to the receiver as a “floating coupon”. At t=1.0, the payer obtain . Similarly, it use to buy another ZCB and pay the left. The process keep going until . At , the payer obtain , it pay as the face value and as the last floating coupon to the receiver.
Therefore, the price (time-0) value of the floating note is equal to .
Replication Method 2: replicate each floating coupon
Consider to replicate a single floating coupon, for example, the coupon at , which is .
At time 0, we can long a ZCB having face value = and maturity date = . Besides, short a ZCB having face value = and maturity date = . Then, at time , we obtain . Invest this into a new ZCB having maturity on . At , we obtain . Subtracting the we need to pay for the inital short selling ZCB, we gain .
Therefore, the time-0 value of the floating coupon payment at is equal to
The result is quite intuitive, we can observe the floating coupon as “fixed”, which is the fave value times the forward rate. But the forward rate is unknown until .
Interest rate swap
The holder (receiver) of a swap will receive fixed coupons and pay floating coupons. The payer will receive floating coupons and pay fixed coupons.
An m-period interest rate swap with fixed rate is a contract that pays at each of the times . The m-period swap rate is the value of that makes the time 0 price of this contract equal to zero.
It can be proved that the m-period swap rate is simply the m-period par coupon rate
Note that, holding a swap is equivalent to longing a coupon bond (receiving fixed coupon and face value in the end), together with shorting a floating note (paying floating coupons and face value in the end). Assume the face value of the floating note is . Then the face value of the coupon bond being longed should also equal to . Only in this way, the final received and payed face values are cancelled off, making the replication of the swap.
Since the time-0 price of a floating note with face value is . This means the time-0 value of the coupon bond should also be . Only in this way, the time-0 net cash flow is zero, making the replication work. Therefore, the coupon bond in the replication is a par-coupon bond having maturity and face value . The corresponding coupon rate is shown as above .
The payer of a swap (receive floaing rates, paying fixed rates) is equal to buying a floating note and shorting a coupon bond. Therefore, a swap has a DV01 < 0.
Caps & Floors
Interest rate cap (”call option” on the interest rate)
The coupon payment rate at is , and corresponding coupon payment is
where is the fixed rate.
Interest rate floor (”put option” on the interest rate)
The coupon payment rate at is , and corresponding coupon payment is
Floored Floater
The coupon rate at is
Inverse Floater with Floor at Zero
The coupon rate at is
Rule Of Thumb
In practice, it is difficult to calculate the price of above such caps and floors. However, we construct replication using these caps and floors into some easier calculated ones. Then we can derive the part of the portfolio based on some known part of the portfolio.
- longing a cap and shorting a floor is equivalent to shorting a swap (payer)
- longing a floored floater and shorting an inverse floater with floor of zero is equivalent to long an floating note.
Time-0 Prices
The time 0 prices of m-period swaps, caps, and floors with fixed rate are given by
Floor-Cap Parity
Since
we deduce that
This is similar to put-call parity.
Options
European Options
Can only be exercised on the expiration date.
Example
Standard procedure to calculate the deritatives:
- Fill in the binomial tree
- Calculate the underlying prices backward
- Calculate the deritatives values backward
American Options
An American option is characterized by an expiration data and an adapted process called the intrinsic value process. At each data , assuming that the option has not been exercised yet, the holder decides, based on the information available at time , whether to exercise now and collect or to wait. The option cannot be exercised more than once.
Backward Induction
At time , the value is its intrinsic value
At earlier time
It is optimal to exercise the option at the smallest time such that .
Note that, it is well known that if interest rates are positive, then it is not optimal to exercise an American call on a stock that does not pay dividends early. However, for a dividend-paying stock or a coupon bond, it may well be optimal to exercise an American call early.
Example
Bermudan Options
A Bermudan option has a (nonempty) set of possible exercise dates and an intrinsic value process , and we assume .
The option can be exercised once, at any date . A simple modification of the American backward induction algo works for Bermudan options. Let be the largest element of . We set . For earlier dates , if is not a possible exercise date, then
If is an exercise date, then
Forwards & Futures
Forward contracts and futures contracts are both designed to lock in a price now for purchase of an asset in the future.
A forward contract is an agreement between two parties concerning the sale of an asset at a future date , called the delivery date:
- The party taking the short position agrees to sell the asset at time at a set price , called the delivery price.
- The party taking the long position agrees to buy the asset at the price at time .
The delivery price is chosen so that at the time contract is made, the value of both positions is zero.
For , we define to be the value of that makes the price of both positions on the contract zero at time .
The amount of the asset to be sold on the delivery date must be specified as part of the contract.
Remark: Although the delivery price is chosen so that the value of both positions on forward contract is zero initially, as time evolves, the values of the long and short positions will generally both be nonzero (but always sum to zero).
Forwards v.s. Futures
There are practical difficulties with forward contracts. It may be difficult to find a party wanting to take the counterposition for the same quantity of the security on the same delivery date. Moreover, there is serious risk of default.
Futures contracts correct these difficulities. Investors do not make contracts with on another, but with a central exchange. Before taking a futures position, an investor must open a margin account. All investors with contracts for delivery of a given asset at date have the same delivery price. Futures price is adjusted every day (mark to market).
Forward Prices
Suppose that at time , two party of the contract decide a delivery price for time . Then for the longing party, its payoff at time is
the corresponding discounted payoff at time is
which should equal to zero. Therefore
Note that if the security pays no dividends or coupons, then where is the price of the security at time , and thus
Future Prices
An m-futures process is an adapted process such that
and
for .
An investor with a long position receives on day
There is exactly one adapted process satisfying above two conditions
Remark: Longing a Futures position means
- Contract is initiated at time
- Nothing is paid to enter the contract
- Long position receives at time for each
- Position can be closed out at no cost at any time
Forward v.s. Futures Prices
If and are uncorrelated under then
If and are positively correlated under , then
If and are negatively correlated under , then
For coupon bond, and are postively correlated.
Example
Callable and Putable Bonds
Consider a general N-period binomial rate model with interest rate process . Let and be given. Let be a set of possible “exercise dates”, and be a set of possible call or put prices.
We consider a bond with maturity , face value , and coupont rate .
The bond is callable with call dates and call prices provided that each time (assuming that the bond has not already been called), the issuer has the right to pay the bond holder and is then relieved of the obligation to make any further payments. ( is the coupon payment due at time , and is a payment that is made in place of paying at maturity.)
As for the putable bonds, similar to callable bonds, but the holder has the right to decide whether or not to sell back the bond to the issuer at the price afther the coupon has been paid (i.e., the holder receives the amount at time and no further payments.)
Note that, for both of the callable bonds and the putable bonds, if the optionality feature of the bond is exercised at time , the holder of the bond receives at time and no future payments. The difference is that the right subjects to the issuer for callable bonds while the right subjects to the holder for putable bonds.
Using backward induction,
- If the bond is callable, and we are at a possible call date , then
- If the bond is putable, and we are at a possible put date , then
Otherwise, if :
Remark:
When there is only one callable date, we can view a callable bond as a bond minus a call option, and the payoff is which is equal to . The result is coincident with above backward induction. Similarly, for a putable bond having a single putable date, it can be viewed as a bond plus a put option, having the payoff .
Example
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