Interest rate derivatives are instruments whose payoffs are dependent in some way on the level of
interest rates. They are more difficult to value than equity and foreign exchange derivatives for the
1. The behavior of an individual interest rate is more complicated than that of a stock price or an
2. For the valuation of many products it is necessary to develop a model describing the behavior of
the entire zero coupon yield curve.
3. The volatilities of different points on the yield curve are different.
4. Interest rates are used for discounting the derivative as well as defining its payoff.
The three most popular OTC interest rate option products discussed are: Bond options, Interest rate
caps/floors and swap options.
Black model and its extensions provide a popular approach for valuing European style Interest rate
In the context of Interest rate derivatives, delta risk is the risk associated with a shift in a zero curve.
Because there are many ways in which the zero curve can shift, many deltas can be calculated.
European bond options do not provide a description of how interest rates evolve through time.
Consequently, they cannot be used for valuing interest rate derivatives such as American style swap
options, callable bonds and structured notes.
To overcome these limitations, term structure models are used for the evolution of all zero coupon
Sr. No Topic Page Nos.
1. Introduction to interest Rate derivatives 06
2. Valuation of Interest rate derivatives 10
3. Hedging of Interest Rate Derivatives 15
4. DerivaGem Software
Introduction to Interest Rate Derivatives
An interest rate derivative is a derivative where the underlying asset is the right to pay or receive
a notional amount of money at a given interest rate.
Interest-Rate Derivative Instruments
In finance, a bond option is an OTC-traded financial instrument that facilitates an option to buy
or sell a particular bond at a certain date for a particular price. It is similar to a stock option with
the difference that the underlying asset is a bond. Bond options can be valued using the Black
A European bond option is an option to buy or sell a bond at a certain date in future for a
An American Bond option is an option to buy or sell a bond on or before a certain date in
future for a predetermined price.
The term "bond option" is also used for option-like features of some bonds. These are an
inherent part of the bond, rather than a separately traded product. These options are not mutually
exclusive, so a bond may have lots of options embedded.
1. A callable bond allows the issuer to buy back the bond at a predetermined price at certain
time in future. The holder of such a bond has, in effect, sold a call option to the issuer.
Callable bonds cannot be called for the first few years of their life. This period is known
as the lock out period.
A puttable bond allows the holder to demand early redemption at a predetermined price at
certain time in future. The holder of such a bond has, in effect, purchased a put option on
A convertible bond allows the holder to demand conversion of bonds into the stock of the
issuer at a predetermined price at certain time period in future.
An exchangeable bond allows the holder to demand conversion of bonds into the stock of
a different company, usually a public subsidiary of the issuer, at a predetermined price at
certain time period in future.
Interest Rate Caps / Floors / Collars
Caps and floors are essential tools in managing floating rate liabilities while minimizing hedging
and opportunity costs. They protect against adverse rates risk, while allowing gains from
favorable rate movements. Caps and floors are forms of option contracts, conferring potential
benefits to the purchaser and potential obligations on the seller. When purchasing a cap or floor,
the buyer pays a premium- typically up-front. The premium amount depends on the specified cap
or floor rate and time period covered, which may range from a few months to several years. Caps
and floors greatly enhance a treasurer's flexibility in managing financial assets and liabilities.
Used together or in combination with other hedging instruments, caps and floors are efficient
tools for reconfiguring a company's financial risk profile.
Caps and floors are used to:
Hedge floating-rate liabilities
Reduce borrowing costs
Increase investment returns
Create synthetic investments
Neutralize options embedded in assets or liabilities
Interest Rate Caps:
A cap creates a ceiling on floating rate interest costs. When market rates move above the cap
rate, the seller pays the purchaser the difference. A company borrowing on a floating rate basis
when 3 month LIBOR is 6% might purchase a 7% cap, for example, to protect against a rate rise
above that level. If rates subsequently rise to 9%, the company receives a 2% cap payment to
compensate for the rise in market rates. The cap ensures that the borrower's interest rate costs
will never exceed the cap rate.
Interest Rate Floors:
A floor is the mirror image of a cap. When market rates fall below the floor rate, the seller pays
the difference. A 6% floor triggers a payment to the purchaser whenever market rates drop below
6%. Asset managers buy floors to guarantee a minimum return on floating rate assets. They sell
floors to generate incrementally higher returns. Debt managers buy floors to protect against
opportunity losses on fixed rate debt when rates fall. They may sell floors as a component of a
hedge strategy involving other derivative instruments.
Swaptions are options on swaps. Like swaps, they offer protection against adverse movements in
interest rates, and are frequently used to minimize financing or hedging costs. Combined with
other instruments, swaptions are often used to solve more complex risk management challenges.
Interest rate swaptions give the holder the right, but not the obligation, to enter into or cancel a
swap agreement at a future date. The buyer may purchase either the right to receive a fixed rate
in the underlying swap or to pay the fixed rate.
Valuation of Interest Rate Derivatives
Interest rate caps
The simplest and most common valuation of interest rate caplets is via the Black model. Under
this model we assume that the underlying rate is distributed log-normally with volatility σ. Under
this model, a caplet on a LIBOR expiring at t and paying at T has present value
P(0,T) is today's discount factor for T
F is the forward price of the rate. For LIBOR rates this is equal to
K is the strike
Notice that there is a one-to-one mapping between the volatility and the present value of the
option. Because all the other terms arising in the equation are indisputable, there is no ambiguity
in quoting the price of a caplet simply by quoting its volatility. This is what happens in the
market. The volatility is known as the "Black vol" or implied vol.
As a bond put
It can be shown that a cap on a LIBOR from t to T is equivalent to a multiple of a t-expiry put on
a T-maturity bond. Thus if we have an interest rate model in which we are able to value bond
puts, we can value interest rate caps. Similarly a floor is equivalent to a certain bond call. Several
popular short rate models, such as the Hull-White model have this degree of tractability. Thus we
can value caps and floors in those models.
The value of a swap is the net present value (NPV) of all estimated future cash flows. A swap is
worth zero when it is first initiated, however after this time its value may become positive or
negative. There are two ways to value swaps: in terms of bond prices, or as a portfolio of forward
The valuation of swaptions is complicated in that the result depends on several factors: the time
to expiration, the length of underlying swap, and the "moneyness" of the swaption. Here the "at
the money" level is the forward swap rate, the forward rate that would apply between the
maturity of the option (t) and the tenor of the underlying swap (T) such that the swap, at time t,
has an "NPV" of zero. For an at the money swaption, the strike rate equals the forward swap rate,
and "moneyness" therefore is determined based on whether the strike rate is higher, lower, or at
the same level as the forward swap rate.
Given these complications, quantitative analysts attempt to determine relative value between
different swaptions, in some cases by constructing complex term structure and short rate models
which describe the movement of interest rates over time.
However a standard practice - particularly amongst traders where speed of calculation is more
important - is to value European Swaptions using the Black model, where, for this purpose, the
underlier is treated as a forward contract on a swap. Here:
The forward price is the forward swap rate.
The volatility is typically "read-off" a two dimensional grid of at-the-money volatilities
as observed from prices in the Interbank swaption market. On this grid, one axis is the
time to expiration and the other is the length of the underlying swap. Adjustments may
then be made for moneyness; see Implied volatility surface under Volatility smile.
Short Rate Models
In the context of interest rate derivatives, a short rate model is a mathematical model that
describes the future evolution of interest rates by describing the future evolution of the short rate.
The short rate
The short rate, usually written rt is the (annualized) interest rate at which an entity can borrow
money for an infinitesimally short period of time from time t. Specifying the current short rate
does not specify the entire yield curve. However no-arbitrage arguments show that, under some
fairly relaxed technical conditions, if we model the evolution of rt as a stochastic process under a
risk-neutral measure Q then the price at time t of a zero-coupon bond maturing at time T is given
where is the natural filtration for the process. Thus specifying a model for the short rate
specifies future bond prices. This means that instantaneous forward rates are also specified by
the usual formula
And its third equivalent, the yields are given as well.
Particular short-rate models
1. The Rendleman-Bartter model models the short rate as
2. The Vasicek model models the short rate as
3. The Ho-Lee model models the short rate as
4. The Hull-White model (also called the extended Vasicek model sometimes) posits
. In many presentations one or more of the
parameters θ,α and σ are not time-dependent.
5. The Cox-Ingersoll-Ross model supposes
6. In the Black-Karasinski model a variable Xt is assumed to follow an Ornstein-Uhlenbeck
process and rt is assumed to follow rt = expXt
7. The Black-Derman-Toy model has for time-
dependent short rate volatility and otherwise.
Other interest rate models
The other major framework for interest rate modelling is the Heath-Jarrow-Morton framework
(HJM). Unlike the short rate models described above, this class of models is generally non-
Markovian. This makes general HJM models computationally intractable for most purposes. The
great advantage of HJM models is that they give an analytical description of the entire yield
curve, rather than just the short rate. For some purposes (e.g., valuation of mortgage backed
securities), this can be a big simplification. The Cox-Ingersoll-Ross and Hull-White models in
one or more dimensions can both be straightforwardly expressed in the HJM framework. Other
short rate models do not have any simple dual HJM representation.
Hedging of Interest Rate Derivatives
Protecting against income risk
Position to hedge Purpose of hedging Instruments
GAP>0 (Asset-sensitive) To immunize against + call on bonds
reductions in interest rates
+ call on bonds futures
whithout loosing the benefits
short cash (asset) from rate increases + IR floors (-cap)
GAP<0 (Liability-sensitive) To immunize against interest + put on bonds
rates rises without loosing the
+ put on bond futures
benefits of rate reductions
long cash (asset) + IR cap (-floor)
Protecting against capital risk
Position to hedge Purpose of hedging Instruments
Dgap <0 To immunize against + call on bonds
reductions in interest rates
+ call on bond futures
without loosing the benefits
short cash (asset) from rate increases + IR floor (-cap)
Dgap >0 To immunize against interest + put on bonds
rates rises without loosing the
benefits from rate reductions + put on bond futures
long cash (asset) +IR cap (-floor)
Hedging with options on bonds
The hedging with bond options is not very frequent. Much more popular is the hedging with
futures options. The latter are equivalent to those on bonds if both the options are European and
the futures and options expire simultaneously. Since futures and spot prices are identical when
the options can be exercised (only at expiration), the payoffs of the futures options and of the
bonds options are identical. Therefore the price of a European futures option and a European
option on bonds must always be identical.
Comments on bond options and futures option strategies:
GAP>0 and Dgap <0 (short asset): imply exposure to interest rate reductions: short
term assets financed with long term liabilities give rise to 1) reinvestment risk 2) capital risk.
To the same type of risks is exposed a fund manager who: holds a bond portfolio containing
bonds scheduled to mature in short times; or who knows that, in short times, he will have a
cash inflow to invest in bonds; or who holds too large short-term investments and too small
long-term ones. Also, it is exposed to the same risks a hedge fund manager who holds a high
proportion of medium-long term liabilities and a high proportion of short-term assets.
Problems of the same type have to be faced by an institution (f. i. a pension fund manager)
with a substantial portion of its funds invested in T-Bill or in money market instruments
(perhaps to avoid the risks of long-term investments) but willing to participate in significant
increases of long-term bonds without forgoing the advantages of shor-term investments (a
sensible strategy might be to buy call options on T-bond futures); etc.
GAP<0 and Dgap >0 (long asset): imply exposure to interest rate increases: capital
risk for assets held or income risk for long-term assets financed with short-term liabilities. To
the same type of risks is exposed a fund manager who forecasts a cash outflow and hence is
necessitated to sell assets in short times; or who holds a long-term bonds portfolio but wants
to protect the market value of the portfolio while retaining the opportunity to profit if bond
prices should increase (a possible strategy might be to buy put options on T-bond futures); a
company is indebted at a floating rate; etc.
Independently of the size of the GAP or of the Dgap, if a manager holds a portfolio of T-
bonds, thinks that the interest rates will stay stable and wants to increase the current portfolio
returns, then he might write a call option on T-bond futures against bonds held in portfolio
(it is a covered option writing). Similarly, if an institution plans to sell, in the near future,
bonds currently held in portfolio and desires to obtain an above-the-market net price when the
bonds are sold, it might write a call option on futures. There seems to be some evidence that
implementing covered call writings with futures options provides larger returns than using
Financial managers buy or sell swaptions to hedge future interest rate exposures or manage
Hedge Contingent Financing. A company's future financing needs may be uncertain, or
contingent upon other events. A swaption provides protection against rising rates without
obligating the purchaser in the event the financing doesn't materialize.
Lower Borrowing Costs. One way to reduce financing costs is to use swaptions to
monetize points of indifference. For example, an active issuer of debt in several
maturities may be indifferent at any point in time to issuing in one maturity versus
another. A company indifferent between three- and five-year debt can combine the issue
of three-year fixed- rate notes with the sale of a 3 x 2 receiver swaption (the right to
receive the fixed rate in a two year swap starting three years hence).
The premium received on the swaption provides immediate cash inflow and reduces the
company's net borrowing cost compared to both the straight three-year or five year debt issue.
Capture Excess Call Value. Swaptions can be used to translate the value of call options
embedded in debt securities into cash.
Investors in these instruments use swaptions and related derivatives to manage unwanted
Swaptions, alone or in combination with other hedging tools offer hedgers and portfolio
managers significantly increased risk management flexibility.
A little value for gamma indicates that by definition, the rate of change of delta is little.
This means rebalancing of the hedge-portfolio may be carried out in larger intervals of time.
Conversely, larger gamma values are an indication that delta is very sensitive with respect to
shifts in the underlying, resulting in the increase in risk inherent in a shift in portfolio value.
Because of the cost of frequent hedging, it is natural to try to minimize the need to rebalance the
portfolio too frequently. The corresponding hedging procedure is called a gamma-neutral
strategy. To achieve this objective, we have to buy and sell more swaptions, not just the swap.
By simple differentiation, you can check that a position in the underlying asset has
Thus, we cannot change the gamma of our position by adding the underlying. However, we can
add another swaption in quantity, which will make the portfolio gamma-neutral. By holding two
different swaptions we can make the portfolio both delta- and gamma-neutral. Note that a delta-
neutral portfolio ¢Port = 0 has gamma equal to ¡ and a traded swaption has gamma equal to ¡0. If
the number of traded swaptions added to the portfolio is w0, the gamma of the portfolio is
Hence, the portfolio becomes gamma-neutral, if our position in the traded swaption is equal to
w0 = −¡/¡0. Of course, as we add the traded swaption, the delta of the portfolio changes. So the
position in the underlying (swap) then has to be changed to maintain delta-neutrality. Due to
the quantity Δ Port of the underlying (swap) has to be added for hedging.
The so called delta-hedging is a dynamic hedging strategy. Here, it is sought, price changes of
the swap to be compensated with price changes of the swaption. This is achieved by setting up a
portfolio by holding (or shorting) the derivative (swaption) and shorting (or holding) a quantity ¢
of the underlying (swap); this is referred to as hedge portfolio.
In this way, within the portfolio, price increases of the swap are compensated by price drops of
the swaption and vice-versa. Risks caused by fluctuations of the underlying security are
practically eliminated. As can be verified, this portfolio has a delta of zero (let Δ Port be the price
of the portfolio):
Therefore, by way of delta-hedging, one can eliminate (at least theoretically and to a great extent
practically) the risk. The proportion of the underlying security in the portfolio must be
continuously changed since the quantity ¢ depends on both the price of the underlying and the
remaining period to maturity of the swaption. This process is called dynamic hedging (or
rebalancing) of the portfolio. Therefore (theoretically), one continuously has to buy and sell
swaps. However, in the case of a discrete model, rebalancing of delta is done at discrete time
Znterest rate collar
The simultaneous purchase of an interest rate cap and sale of an interest rate floor on the same
index for the same maturity and notional principal amount.
The cap rate is set above the floor rate.
The objective of the buyer of a collar is to protect against rising interest rates.
The purchase of the cap protects against rising rates while the sale of the floor generates
A collar creates a band within which the buyer’s effective interest rate fluctuates
It is the buying of an interest rate floor and simultaneously selling an interest rate cap.
The objective is to protect the bank from falling interest rates.
The buyer selects the index rate and matches the maturity and notional principal amounts
for the floor and cap.
Buyers can construct zero cost reverse collars when it is possible to find floor and cap
rates with the same premiums that provide an acceptable band.
Combining Caps and Floors to Create Collars. A collar is created by purchasing a cap or floor
and selling the other. The premium due for the cap (floor) is partially offset by the premium
received for the floor (cap), making the collar an effective way to hedge rate risk at low cost. In
return the hedger gives up the potential benefit of favourable rate movements outside the band
defined by the collar. A borrower who purchases an 8% cap and sells a 6% floor guarantees a 6-
8% base rate on a floating rate loan. An investor in floating rate CD's might do exactly the
opposite, buying a 6% floor and financing it with the sale of an 8% cap. A costless collar is
created when the cap and floor levels are set so that the premiums exactly offset each other.
Caps, floors and collars are a simple but very effective way to control risk and manage hedge
costs. The option characteristics of caps and floors offer unique opportunities to minimize
borrowing costs or achieve higher investment returns.
For floating rate borrowers, interest rate caps offer significant hedging advantages:
• Caps help limit exposure to rising rates yet allow the borrower to benefit when rates fall;
• Caps have no credit implications—once the hedger has paid for the cap his obligations
to the hedge provider are complete;
• Caps are easy to explain to executive managers and boards of directors;
• Caps come with no ―regret factor‖—no matter what happens to market rates after the
cap is in effect, the hedger wins.
For all their obvious benefits, caps tend to be used less than other hedging instruments. The
reason is the up-front premium. For some firms any hedging expense is too much hedging
expense. Hence caps are often passed over as a hedging alternative. This need not be the case. A
wide range of cost-effective cap-based hedging strategies can be put together by the thoughtful
treasurer or CFO.
Purchasing a cap at one strike and selling a cap with a higher strike creates a corridor of interest
rate protection. Selling the higher strike cap reduces the cost of the hedge, but it also sets a limit
on the benefits that the cap buyer can receive. While this makes corridors less interesting as long
term hedges, they are quite useful for shorter-term (1-2 years) hedges where the risk of extreme
movements in rates is reduced. For example, today a 5.75%/6.25% 18 month corridor carries the
same premium as a 6.00% cap. At the same time it protects completely (no deductible) against
the next 50 basis points of upward rate movement.
Adding a knockout feature to the corridor reduces hedging costs further. Because the hedge
knocks out (disappears) whenever the upper boundary of the corridor is breached, it carries a
lower net premium than the standard corridor – almost 50% lower for the example above. The
knockout feature also allows the hedger to widen the corridor band without changing the cost. In
the above example, the corridor band widens from 5.75%/6.25% to 5.75%/6.75% when done as a