Financial Engineering
IC302
6. Interest Rate Options
David Oakes
–
Autumn Term 2020/1
Interest Rate Options
2
Interest Rate Derivatives
• Interest rate derivatives make payoffs that depend in some way on the level of interest rates.
• They are among the most widely traded derivatives.
• Valuing interest rate derivatives can be difficult because:
1. Behaviour of individual interest rates can be complicated
2. For some products we may need a model that describes the behaviour of the entire zero-coupon yield curve
3. Volatilities may differ at different points on the curve
4. Interest rates are used for discounting the derivative as well as for defining its payoff
3
Interest Rate Options
• Bond options
– Over-the-counter (OTC) options on individual bonds – Embedded options (e.g. callable or puttable bonds)
• Options on interest rate futures or bond futures
• Interest rate caps and floors • Swaptions
4
Martingales and Measures
5
Interest Rate Options
• Until now we have assumed that interest rates were constant when valuing options. But the whole point of interest rate options is that interest rates are not constant.
• The risk-neutral valuation principle allows us to value derivatives by calculating the expected payoff on the assumption that the expected return on the underlying asset equals the risk-free rate and discounting that expected payoff to the present at the risk-free rate.
• But how should we apply this principle when interest rates are stochastic? What interest rate should we use?
6
Other Risk-Neutral Worlds
• The answer to these questions lies in recognizing that there are many different risk-neutral worlds, each defined by a specific assumption about the market price of risk.
• In the traditional risk-neutral world of the last lecture, all market prices of risk are zero. This works well for valuing derivatives when we can assume that rates are constant.
• When interest rates are stochastic, it is more helpful to work in other risk-neutral worlds defined by different market prices of risk. That is the subject of this section.
7
Risky Variable
• Consider derivatives that depend on a single variable 𝜃 that follows a process:
𝑑𝜃 ≡ 𝑚 𝑑𝑡 + 𝑠 𝑑𝑧 𝜃
where 𝑑𝑧 is a Wiener process and the parameters 𝑚 and 𝑠 are the expected growth rate of 𝜃 and the volatility of 𝜃, respectively (see Hull, ch. 14).
• The variable 𝜃 need not be the price of an investment asset, or even a financial variable.
8
Derivatives
• Suppose that 𝑓) and 𝑓* are the prices of two derivatives
that depend only on 𝜃 and 𝑡 and follow the processes: 𝑑𝑓! = 𝜇! 𝑑𝑡 + 𝜎! 𝑑𝑧
𝑑𝑓” = 𝜇” 𝑑𝑡 + 𝜎” 𝑑𝑧 𝑓”
where 𝜇), 𝜇*, 𝜎), and 𝜎* are functions of 𝜃 and 𝑡.
• The 𝑑𝑧 here is the same as in the equation for 𝜃 because it is the only source of uncertainty in 𝑓) and 𝑓*.
𝑓!
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Instantaneously Riskless Portfolio
• The prices 𝑓) and 𝑓* can be related using an analysis similar to that in the Black-Scholes-Merton model.
• The discrete versions of the processes for 𝑓) and 𝑓* are: Δ𝑓! = 𝜇! 𝑓!Δ𝑡 + 𝜎!𝑓!Δ𝑧 (1)
Δ𝑓” = 𝜇” 𝑓”Δ𝑡 + 𝜎”𝑓”Δ𝑧 (2)
• Eliminate the Δ𝑧 by forming an instantaneously riskless portfolio with 𝜎*𝑓* of the first derivative and −𝜎)𝑓) of the second derivative. This portfolio has value Π given by:
Π= 𝜎”𝑓” 𝑓!− 𝜎!𝑓! 𝑓” (3)
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No-Arbitrage Condition
• The dynamics of this portfolio are (using (1) and (2)):
ΔΠ = 𝜎”𝑓” Δ𝑓! − 𝜎!𝑓! Δ𝑓” = 𝜇!𝜎”𝑓!𝑓” − 𝜇”𝜎!𝑓!𝑓” Δ𝑡 (4)
• Since the portfolio is instantaneously riskless, it must earn
the risk-free rate if there is to be no arbitrage. Hence:
ΔΠ = 𝑟ΠΔ𝑡
• Substituting from (3) and (4) gives:
𝜇!𝜎” − 𝜇”𝜎! = 𝑟𝜎” − 𝑟𝜎! → 𝜇! − 𝑟 = 𝜇” − 𝑟
• Define 𝜆 as the value of each side of this equality:
𝜇! − 𝑟 = 𝜇” − 𝑟 = 𝜆 𝜎! 𝜎”
𝜎! 𝜎”
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Market Price of Risk
• Dropping subscripts, if 𝑓 is the price of a derivative
dependent only on 𝜃 and 𝑡 with: 𝑑𝑓 = 𝜇 𝑑𝑡 + 𝜎 𝑑𝑧
then 𝜇−𝑟=𝜆 (5) 𝜎
• The parameter 𝜆 is the market price of risk of 𝜃.
• No-arbitrage requires that 𝜆 be the same for all derivatives that depend only on 𝜃 and 𝑡. Note that 𝜆 may itself depend on 𝜃 and 𝑡.
𝑓
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Market Price of Risk
• The market price of risk measures the tradeoff between
risk and return for securities that depend on 𝜃.
• We can interpret 𝜎 loosely as the quantity of 𝜃-risk in 𝑓.
• Rearranging the no-arbitrage condition just found gives: 𝜇 − 𝑟 = 𝜆𝜎 (6)
• The expected return on 𝑓 in excess of the risk-free rate must be equal to the quantity of 𝜃-risk it contains multiplied by the price of 𝜃-risk.
13
Alternative Risk-Neutral Worlds
• The process followed by the derivative price 𝑓 is:
𝑑𝑓 = 𝜇 𝑓 𝑑𝑡 + 𝜎 𝑓 𝑑𝑧 (7)
• The value of 𝜇 depends on the risk preferences of investors. In a world where the market price of risk is zero, we have from (6) that 𝜇 = 𝑟, so that:
𝑑𝑓 = 𝑟 𝑓 𝑑𝑡 + 𝜎 𝑓 𝑑𝑧
• This is the traditional risk-neutral world in which we valued options in the previous lecture.
• Other assumptions about the market price of risk lead to other internally consistent risk-neutral worlds.
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Girsanov’s Theorem
• Using (6) and (7) again, we see that in these worlds:
𝑑𝑓 = 𝑟+𝜆𝜎 𝑓𝑑𝑡+𝜎𝑓𝑑𝑧 (8)
• The market price of risk of a variable determines the
growth rates on all securities dependent on that variable.
• As we move from one market price of risk to another, the expected growth rates of security prices change but their volatilities remain the same.
• This is Girsanov’s theorem, which we saw in practice (but did not name) when implementing the binomial model.
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Change of Probability Measure
• Choosing a particular market price of risk is also referred
to as changing the probability measure.
• Girsanov’s theorem says that when we change probability measure (i.e. when we move from a world with one set of risk preferences to a world with another set of risk preferences) the expected growth rates of security prices change but their volatilities remain the same.
16
Martingales
• A martingale is a zero-drift stochastic process. A variable
𝜃 is a martingale if its process has the form: 𝑑𝜃 = 𝜎 𝑑𝑧
where 𝑑𝑧 is a Wiener process and 𝜎 may be stochastic.
• A martingale’s expected value at any future time is equal
to its value today:
𝐸𝜃# =𝜃$
• It follows that the expected change in 𝜃 between time 0 and time 𝑇 must also be zero.
17
Relative Price and Numéraire
• Suppose that 𝑓 and 𝑔 are the prices of traded securities dependent on a single source of risk. The securities pay pay no income in the time period under consideration.
• Define the relative price of 𝑓 with respect to 𝑔 as: 𝜙 = 𝑔𝑓
• We call 𝑔 the numéraire. It defines the units in which we measure the price of 𝑓.
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Equivalent Martingale Measure
• The equivalent martingale measure result shows that, when there are no arbitrage opportunities, 𝜙 is a martingale for some choice of the market price of risk.
• For a given numéraire 𝑔, the same choice of market price of risk makes 𝜙 a martingale for all securities 𝑓.
• This choice of market price of risk is the volatility of 𝑔.
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Equivalent Martingale Measure
• Suppose that 𝜎8 and 𝜎9 are the volatilities of 𝑓 and 𝑔 and
𝑟 is the instantaneous risk-free rate.
• From (8), in a world where the market price of risk is 𝜎9:
𝑑𝑓= 𝑟+𝜎%𝜎& 𝑓𝑑𝑡+𝜎&𝑓𝑑𝑧 𝑑𝑔= 𝑟+𝜎%” 𝑔𝑑𝑡+𝜎%𝑔𝑑𝑧
• Applying Itô’s lemma (see Hull, ch. 14) gives: 𝑑ln𝑓= 𝑟+𝜎%𝜎&−𝜎&”⁄2𝑑𝑡+𝜎&𝑑𝑧
𝑑 l n 𝑔 = 𝑟 + 𝜎 %” ⁄ 2 𝑑 𝑡 + 𝜎 % 𝑑 𝑧
𝑑ln𝑓−ln𝑔 = 𝜎%𝜎&−𝜎&”⁄2−𝜎%”⁄2𝑑𝑡+ 𝜎&−𝜎% 𝑑𝑧
so that:
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Equivalent Martingale Measure • This can be rewritten as:
𝑓𝜎−𝜎”
𝑑ln=−& %𝑑𝑡+𝜎&−𝜎%𝑑𝑧 𝑔2
• Apply Itô’s lemma again to find the process for 𝑓⁄𝑔 : 𝑑 𝑓 = 𝜎 −𝜎 𝑓𝑑𝑧
𝑔&%𝑔
• This shows that 𝑓⁄𝑔 is a martingale and proves the equivalent martingale measure result.
21
Application to Derivatives Pricing
• From our earlier definition of a martingale, it follows that:
𝑓$=𝐸 𝑓# 𝑔$ %𝑔#
𝑓#
or 𝑓$=𝑔$𝐸% 𝑔# (9)
where 𝐸9 denotes the expected value in a world defined by the numéraire 𝑔.
• This defines a general approach to risk-neutral valuation that can be applied for alternative choices of numéraire.
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Alternative Numéraires
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Money Market Account
• The money market account is a security that is worth $1
at time 0 and earns the instantaneous risk-free rate 𝑟 : 𝑑𝑔 = 𝑟𝑔 𝑑𝑡
• Note that 𝑟 may be stochastic.
• The drift of 𝑔 is stochastic but its volatility is zero. It follows from the last section that 𝑓⁄𝑔 is a martingale in a world where the market price of risk is zero.
• This is the traditional risk-neutral world that we used to price options in the last lecture.
Money Market Account
• From (9) we have that:
𝑓# 𝑔#
neutral world.
• For the money market account, 𝑔! = 1 and:
𝑓 $ = 𝑔 $ 𝐸>
where 𝐸5 denotes expectations in the traditional risk-
• so that
𝑔 # = 𝑒 ∫!” ( ) * 𝑓$=𝐸> 𝑒+∫!”()*𝑓#
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Money Market Account
• This can be written as:
𝑓 $ = 𝐸> 𝑒 + ( ̅ # 𝑓 # ( 1 0 ) where 𝑟̅ is the average value of 𝑟 between 0 and 𝑇.
• One way to value interest rate derivatives is therefore to simulate 𝑟 in the traditional risk-neutral world.
• If the short-term interest rate 𝑟 is constant, then: 𝑓 $ = 𝐸> 𝑒 + ( # 𝑓 #
• This is the special case that we saw in the last lecture.
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Zero-Coupon Bond Price
• Define 𝑃 𝑡, 𝑇 as the price at time 𝑡 of a risk-free zero-
coupon bond that pays $1 at time 𝑇.
• Because𝑔@ =𝑃 𝑇,𝑇 =1and𝑔A =𝑃 0,𝑇 wehave:
(11) where 𝐸@ is expectation in a world with this numéraire.
• Notice that, unlike in (10), the discounting takes place outside the expectations operator. This simplifies valuation for securities that make payoffs only at 𝑇.
𝑓$=𝑃0,𝑇 𝐸# 𝑓#
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Zero-Coupon Bond Price
• Consider any variable 𝜃 that is not an interest rate.
• A forward contract on 𝜃 with maturity 𝑇 pays 𝜃@ − 𝐾 at time 𝑇. Define 𝑓 as the value of this contract. From (11):
𝑓$=𝑃0,𝑇 𝐸# 𝜃# −𝐾
• Theforwardprice𝐹of𝜃isthevalueof𝐾forwhich𝑓A=
0. It follows that:
𝑃0,𝑇𝐸# 𝜃#−𝐹 =0 → 𝐹=𝐸# 𝜃# (12)
• The forward price of any variable (except an interest rate) is its expected future spot price in a world defined by the numéraire 𝑃 0, 𝑇 .
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Forward Interest Rates
• For variables that are interest rates, we need to modify slightly our choice of numéraire.
• Define 𝐹 𝑡 as the forward interest rate as seen at time 𝑡 for the period between 𝑇 and 𝑇∗ with compounding period 𝑇∗ − 𝑇 (e.g. 0.25 for quarterly compounding).
• Define 𝑅 as the realized interest rate for the same period expressed with the same compounding frequency.
• A forward rate agreement (FRA) paying 𝐹 𝑡 − 𝑅 at time 𝑇∗ is worth zero at time 𝑡.
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Forward Interest Rates
• It follows from (12) that:
𝑃0,𝑇∗ 𝐸#∗ 𝐹𝑡 −𝑅 =0 → 𝐹𝑡 =𝐸#∗ 𝑅 (13)
• Note that 𝐹 𝑡 and 𝑅 can be calculated from any yield curve and that this may be different than the risk-free zero curveusedtocalculate𝑃0,𝑇∗ .
• In particular, 𝐹 𝑡 and 𝑅 can be calculated from LIBOR or Euribor rates and 𝑃 0, 𝑇∗ calculated from OIS rates. As we saw in lectures 2 and 3, this is current market practice.
30
Forward Interest Rates
• Notice that each forward rate is a martingale with respect to its own numéraire and associated probability measure.
• These are sometimes called forward measures.
• These results are used in the standard market model for interest rate caps and floors.
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Annuity Factor
• Consider a fixed-for-floating swap that starts at future
time 𝑇 and has payment dates 𝑇), 𝑇*, … , 𝑇J.
• Define 𝑇A = 𝑇 and assume the notional principal is $1.
• Suppose that the forward swap rate (i.e. the fixed swap rate for which the swap has zero value) is 𝑠 𝑡 .
32
Annuity Factor
• The value of the fixed leg of the swap is:
𝑠𝑡𝐴𝑡
where
• 𝐴 𝑡 is the annuity factor for the fixed leg of the swap.
• Defining the floating leg of the swap as 𝑉 𝑡 . Equating the values of the fixed and floating legs, we have:
𝑠𝑡𝐴𝑡=𝑉𝑡 → 𝑠𝑡=𝑉𝑡 𝐴𝑡
0+!
𝐴 𝑡 = L 𝑇.1! − 𝑇. 𝑃 𝑡, 𝑇.1! ./$
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Annuity Factor
• We can apply the equivalent martingale measure result by
setting𝑓equalto𝑉 𝑡 and𝑔equalto𝐴 𝑡 . Thisgives: 𝑠 𝑡 = 𝐸2 𝑠 𝑇 (14)
where 𝐸N denotes expectations in a world defined by the numéraire 𝐴 𝑡 .
• Using the valuation result in (9) gives:
𝑉0=𝐴0𝐸 𝑉𝑇 (15) 2𝐴𝑇
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Annuity Factor
• The probability measure associated with the annuity
factor is sometimes called the swap measure.
• Note that 𝑉 𝑡 can be calculated from any yield curve
while 𝐴 𝑡 is calculated from the risk-free zero curve.
• In particular, the swap can be a fixed-for-floating swap that references LIBOR or Euribor rates while the risk-free rates are calculated from OIS rates.
• These results are used in the standard market model for swaptions.
35
Black’s Model
36
Black’s Model
• Black’s model (see Hull, ch. 18) can be used to price options in terms of the forward or futures price of the underlying asset when interest rates are constant.
• Traders often prefer to use Black’s model instead of the Black-Scholes-Merton model to value options on the spot price because it can be used for investment or consumption assets without the need to estimate the income (or convenience yield) on the underlying asset.
• The arguments of the last section can be used to show that Black’s model also works when rates are stochastic.
European Call
• Consider a European call option on an asset with strike
price 𝐾 that expires at time 𝑇.
• From (11), the price 𝑐 of the option is: 𝑐 = 𝑃 0, 𝑇 𝐸# max 𝑆# − 𝐾, 0
where 𝑆@ is the price of the asset at 𝑇 and 𝐸@ denotes expectations in a world defined by the numéraire 𝑃 0, 𝑇 .
38
Spot Price and Forward Price
• Define 𝐹A and 𝐹@ as the forward price of the asset at time
0 and time 𝑇 for a contract maturing at time 𝑇.
• Because𝑆@=𝐹@:
𝑐 = 𝑃 0, 𝑇 𝐸# max 𝐹# − 𝐾, 0
• Assume that 𝐹@ is lognormal in the world considered, with
standard deviation of ln 𝐹@ equal to 𝜎T 𝑇 (e.g. because the forward price follows a process with volatility 𝜎T).
39
Expected Payoff
• This implies (see Hull, Appendix to ch. 15) that: 𝐸#max𝐹#−𝐾,0 =𝐸#𝐹#𝑁𝑑! −𝐾𝑁𝑑”
where
𝑑 ! = l n 𝐸 # 𝐹 # ⁄ 𝐾 + 𝜎 3″ 𝑇 ⁄ 2 𝜎3 𝑇
𝑑 ” = l n 𝐸 # 𝐹 # ⁄ 𝐾 − 𝜎 3″ 𝑇 ⁄ 2 𝜎3 𝑇
is the cumulative distribution function for a • From(12),𝐸@ 𝐹@ =𝐸@ 𝑆@ =𝐹A.
and 𝑁 𝑥
variable with a standard normal distribution.
40
Black’s Model • Hence:
where
𝑐=𝑃0,𝑇 𝐹$𝑁𝑑! −𝐾𝑁𝑑” (16) 𝑑 ! = l n 𝐹 $ ⁄ 𝐾 + 𝜎 3″ 𝑇 ⁄ 2
𝜎3 𝑇
𝑑 ” = l n 𝐹 $ ⁄ 𝐾 − 𝜎 3″ 𝑇 ⁄ 2 𝜎3 𝑇
• Similarly:
𝑝=𝑃0,𝑇 𝐾𝑁−𝑑” −𝐹$𝑁−𝑑! (17)
for a European put. • This is Black’s model.
41
Applicability of Black’s Model
• Black’s model therefore can be used when interest rates are stochastic, provided that 𝐹A is the forward asset price for a contract with the same maturity as the option.
• We interpret 𝜎T as the volatility of the forward price.
42
Bond Options
Bond Options
• A bond option is an option to buy or sell a particular bond by a specified future date at an agreed price.
• Options may be traded OTC on individual bonds.
• More usually, bonds or loans may contain specific features
that are, in effect, embedded options.
• Common examples include callable bonds, puttable bonds, loan commitments, and prepayment privileges.
Callable Bonds
• A callable bond can be repurchased by the issuer from investors at a pre-specified price on certain future dates.
• The price that the issuer will pay for the bond if they choose to exercise the option is the call price.
• In financial engineering terms, a callable bond is equivalent to a non-callable bond combined with a short position in a call option on the bond:
=-
Callable Bond
Non-callable Bond
Call Option on Bond
45
Callable Bond
• A callable bond is less valuable to investors than an otherwise equivalent non-callable bond because of the call option that has been given to the issuer.
• Other things remaining equal, issuers will therefore have to pay a higher coupon to issue a callable bond.
• But the call provision allows issuers to buy back the bonds and refinance if interest rates fall during the bond’s life.
• Callable bonds usually cannot be called for the first few years of their life, and the call price may decline over time.
46
Puttable Bond
• Investors in a puttable bond can force early redemption of the bond at a pre-specified price on certain future dates.
• The price at which investors can sell the bond to the issuer if they choose to exercise the option is the put price.
• In financial engineering terms, a puttable bond is equivalent to a non-puttable bond combined with a long position in a put option on the bond:
=+
Puttable Bond
Non-puttable Bond
Put Option on Bond
47
Puttable Bond
• A puttable bond is more valuable to investors than an otherwise equivalent non-puttable bond because of the put option that allows them to force early redemption.
• Other things remaining equal, issuers will therefore pay a lower coupon to issue a puttable bond, and the bond will trade a lower yield than similar non-puttable bonds.
• The put provision offers investors some protection against a decline in the bond’s value prior to redemption.
48
Prepayment Rights
• In some countries (e.g. USA), borrowers in fixed-rate mortgages may choose to repay their mortgage early or to accelerate repayment without financial penalty.
• They have an incentive to do so if rates fall, since they can then refinance by taking out a new loan at a lower rate.
• Financially-motivated prepayment disadvantages lenders, since the value of the mortgage loan (like the price of a bond) would otherwise have increased as rates fell.
• Prepayment rights are like a call option on a bond.
49
Loan Commitments
• A lender may quote a fixed rate to a potential borrower which is good for a specified time (e.g. the next 3 months).
• This is common practice in mortgage lending.
• These loan commitments are like a put option on a bond.
• The borrower has an incentive to exercise the option if rates increase, since it gives them the right to sell the lender a loan on the agreed terms at its face value.
50
Valuing Bond Options
• European bond options can be valued using Black’s model.
• Set 𝐹A equal to 𝐹W (the forward bond price) and 𝜎T equal to 𝜎W (its volatility) in (16) and (17).
• The standard deviation of the logarithm of a bond’s price changes as we look further ahead, initially increasing (due to uncertainty about rates) and then falling to zero at maturity (because the price equals face value at maturity).
• The volatility of the forward bond price for a given bond typically declines as the option maturity increases.
51
Interest Rate Caps and Floors
Interest Rate Cap
• An interest rate cap protects the cap buyer from an increase in a reference interest rate (e.g. 3m USD LIBOR) above a specified strike level during the life of the cap.
• The cap is structured as a series of caplets, one for each settlement period for the reference interest rate.
• Each caplet is like a call option on the reference rate.
• Same strike rate for all the caplets in the cap.
Hedging with a Cap
• Company wants protection against rising interest costs on a 1-year, $100m loan which resets quarterly to 3m LIBOR.
• It buys a one-year cap (three caplets) with strike rate 2%.
Caplet 2 Caplet 1
Caplet 3
months 0 3 6 9 12
One-year cap
• Each caplet is an individually exercisable call option on the forward LIBOR rate for a specific reset date.
54
Caplet Payoffs and Interest Costs
• Caplet payoff offsets interest cost each period in excess of the cap strike plus the amortized cost of the cap.
LIBOR Capped Cost
LIBOR
Caplet P/L
Interest Cost
0
LIBOR
Amortized Annual Cost of Cap
2%
1% 2%
55
Caplet Exercise and Settlement
• Suppose that LIBOR for one of reset dates fixes at 3%.
• Payoff to caplet (assuming 91 days in the period) is: 𝑃𝑎𝑦𝑜𝑓𝑓 = 𝑁𝑜𝑡𝑖𝑜𝑛𝑎𝑙 𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙 × max 𝐿𝐼𝐵𝑂𝑅 − 𝑆𝑡𝑟𝑖𝑘𝑒, 0 × 𝐷𝑎𝑦𝑠
= $100 𝑚𝑖𝑙𝑙𝑖𝑜𝑛 × max 0.03 − 0.02, 0 × 4! = $252,777.78 56$
• This exactly offsets the additional interest cost on the loan due to LIBOR fixing 1% above the cap strike of 2%.
𝑌𝑒𝑎𝑟 𝐵𝑎𝑠𝑖𝑠
56
Caps vs. FRAs, Futures and Swaps
• Caplet is only exercised if LIBOR fixes above the cap strike.
• Buyer of cap can therefore still benefit from any fall in rates, unlike with hedges based on FRAs or futures.
• Like a swap, the cap covers multiple LIBOR resets, but with individually exercisable optionality for each reset date.
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Caps as Bond Options
• Caps are portfolios of call options on forward LIBOR rates.
• But a cap can also be interpreted as a portfolio of put options on zero-coupon bonds with payoffs on the puts occurring at the time they are calculated (see Hull, ch. 29).
58
Interest Rate Floor
• The buyer of an interest rate floor receives positive payoffs if reference interest rate fixes below agreed strike.
• Floor is structured as a series of floorlets, each of which is like a put option on the reference rate.
Floorlet P/L
K = 1%
• Buying a floor can protect a lender against a fall in rates.
59
LIBOR
Interest Rate Collar
• Interest rate caps can be expensive.
• Upfront cost can be reduced by having the cap buyer sell a floor at a lower strike to the bank that sells it the cap.
• The resulting position is an interest rate collar.
• For example, a company that buys a 1-year cap with strike price 2% may offset part of the cap cost by selling the arranging bank a 1-year floor with strike price 1%.
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Collars and Interest Costs
•
P/L
0
Caplet plus floorlet gives one component of collar.
Collar
1% 2%
Interest Cost
LIBOR
LIBOR
Capped Cost With Collar
0
1%
2%
LIBOR
•
Company is protected against LIBOR fixing above 2% but gives up benefit of paying lower rate if LIBOR is below 1%.
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Zero-Cost Collar
• Hedging cost with collar is cap value minus floor value.
• It may be possible to find a floor strike such that the floor value exactly offsets the value of a cap with a given strike.
• This creates a zero-cost collar (i.e. upfront cost is zero).
62
Cap and Floor Value Drivers
• Strike price
– Higher strike price reduces value of cap and increases value
of floor.
• Maturity
– Longer maturity increases the value of both caps and floors
(more options and longer dated options).
• Slope of yield curve
– Steeper curve means higher forward rates, which makes caps more expensive and floors cheaper.
63
Cap and Floor Value Drivers
• Volatility of forward rates
– Higher volatility increases the value of both caps and floors.
64
Interest Rate Cap
Refinitiv Eikon
65
Interest Rate Cap
5y Cap on 3m USD LIBOR
Refinitiv Eikon
66
Interest Rate Cap
5y Cap on 3m USD LIBOR
Cap Strike
Refinitiv Eikon
67
Interest Rate Cap
5y Cap on 3m USD LIBOR
Cap Strike Premium (bp)
Refinitiv Eikon
68
Interest Rate Cap
5y Cap on 3m USD LIBOR
Cap Strike
Premium (bp) Premium (USD)
Refinitiv Eikon
69
Interest Rate Cap
5y Cap on 3m USD LIBOR
Cap Strike
Premium (bp) Premium (USD)
Refinitiv Eikon
Premium
(implied volatility)
70
Interest Rate Floor
Refinitiv Eikon
71
Interest Rate Collar
Refinitiv Eikon
72
Black’s Model for Caplets
• Caplet values can be calculated using Black’s model:
𝐶𝑎𝑝𝑙𝑒𝑡 𝑉𝑎𝑙𝑢𝑒 = 𝐿 𝛿7𝑃 0, 𝑡71! 𝐹7𝑁 𝑑! − 𝑅8𝑁 𝑑” (18) 𝑑!=ln𝐹7⁄𝑅8 +𝜎7″𝑡7⁄2
𝜎7 𝑡7
𝑑”=ln𝐹7⁄𝑅8 −𝜎7″𝑡7⁄2 𝜎7 𝑡7
where
𝐿 = 𝑛𝑜𝑡𝑖𝑜𝑛𝑎𝑙 𝑝𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙
𝛿7 = 𝑡71! − 𝑡7
𝐹7 = 𝑓𝑜𝑟𝑤𝑎𝑟𝑑 𝑟𝑎𝑡𝑒 𝑎𝑡 𝑡𝑖𝑚𝑒 0 𝑓𝑜𝑟 𝑝𝑒𝑟𝑖𝑜𝑑 𝑏𝑒𝑡𝑤𝑒𝑒𝑛𝑡7 𝑎𝑛𝑑 𝑡71! 𝑅8 = 𝑐𝑎𝑝 𝑠𝑡𝑟𝑖𝑘𝑒 𝑟𝑎𝑡𝑒
𝜎7 = 𝑣𝑜𝑙𝑎𝑡𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑡h𝑒 𝑓𝑜𝑟𝑤𝑎𝑟𝑑 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒
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Theoretical Justification
• Extending Black’s model to value a caplet can be justified by considering a world defined by a numéraire equal to a risk-free zero-coupon bond maturing at time 𝑡tu).
• The current value of any security is its expected value at time 𝑡tu) in this world multiplied by the price of a zero- coupon bond maturing at time 𝑡tu) (see (11)).
• The expected value of a risk-free interest rate for the period 𝑡t to 𝑡tu) equals the forward interest rate in this world (see (13)).
74
Theoretical Justification
• The first of these results shows that the price of a caplet
that provides a payoff at time 𝑡tu) is: 𝐶𝑎𝑝𝑙𝑒𝑡𝑉𝑎𝑙𝑢𝑒=𝐿𝛿7𝑃 0,𝑡71! 𝐸71! max 𝑅7 −𝑅8,0
where 𝐸tu) denotes expectation in a world defined by a numéraire equal to a zero-coupon bond maturing at 𝑡tu).
• When the forward interest rate underlying the caplet has constant volatility, 𝑅t is lognormal in this world and the standard deviation of ln 𝑅t is equal to 𝜎t 𝑡t .
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Theoretical Justification
• This allows us to write the caplet value (see Hull, Appendix to ch. 15) as:
𝐶𝑎𝑝𝑙𝑒𝑡 𝑉𝑎𝑙𝑢𝑒 = 𝐿 𝛿7𝑃 0,𝑡71! 𝐸71! 𝑅7 𝑁 𝑑! − 𝑅8𝑁 𝑑”
𝑑 ! = l n 𝐸 7 1 ! 𝑅 7 ⁄ 𝑅 8 + 𝜎 7″ 𝑡 7 ⁄ 2 𝜎7 𝑡7
𝑑 ” = l n 𝐸 7 1 ! 𝑅 7 ⁄ 𝑅 8 − 𝜎 7″ 𝑡 7 ⁄ 2 𝜎7 𝑡7
• The second result implies that: 𝐸71!𝑅7 =𝐹7
which leads directly to Black’s model for a caplet in (18).
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Cap/Floor Volatilities
• Each caplet can be valued separately using (18).
• A different volatility may be used for each caplet; these are called spot volatilities.
• Caplet volatilities are often quoted, however, as flat volatilities, with the same volatility applied to each caplet.
• There is smile or skew in cap and floor implied volatilities, which may be modelled using more sophisticated approaches such as the SABR model.
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Cap/Floor Implied Volatilities
Refinitiv Eikon
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Swaptions
Swaptions
• A swaption is an option on an interest rate swap.
• The swaption holder has the right to enter into a pre- defined interest rate swap on a specified future date.
• Swaptions create exposure to forward swap rates, and therefore also to forward LIBOR or Euribor rates.
• Because they are options, they also create exposure to interest rate volatility in forward periods.
Payer Swaption
• Payer swaption: right but not obligation to pay fixed in a swap of specified maturity (‘tenor’) at an agreed rate (‘strike’), effective on the swaption exercise date
• 1y5y payer at strike 1.5%: right but not obligation to pay fixed at 1.5% in a 5-year swap that starts 1 year from now
• Under what conditions will it make sense for the swaption holder to exercise this swaption?
Payer Swaption
• Payoff at expiry to a payer swaption is like payoff to a long call option on the underlying swap rate.
Swaption Payoff
1.5% Swap Rate
Receiver Swaption
• Receiver swaption: right but not obligation to receive fixed in a swap of specified maturity (‘tenor’) at an agreed rate (‘strike’), effective on the swaption exercise date
• 5y10y receiver at strike 2%: right but not obligation to receive fixed at 2% in a 10-year swap starting 5 years from now
• Under what conditions will it make sense for the swaption holder to exercise this swaption?
Receiver Swaption
• Payoff at expiry to a receiver swaption is like payoff to a long put option on the underlying swap rate.
Swaption Payoff
2% Swap Rate
Swaption Exercise
• European swaption: can only be exercised at option expiry
• Bermudan swaption: can be exercised on a discrete set of predetermined dates
• American swaption: can be exercised at any time during the option’s life, up to and including the expiry date
• Most swaptions are European.
Swaption Settlement
• Physical settlement: long counterparty is delivered into a position in the underlying swap when the swaption is exercised. Swap position may be in a cleared swap.
• Physical settlement has been the market standard in USD swaption market, but cash settlement is also possible (e.g. based on collateralized cash price).
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Swaption Settlement
• Collateralized Cash Price: long counterparty receives cash settlement based on the collateralized cash price of swap
• Collateralized cash price is market value of the physical swap when the swaption is exercised, where the swap is traded under a market standard collateral agreement.
• Cash settlement based on collateralized cash price is the market standard in EUR swaptions.
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Describing Swaptions
• Swaptions are identified by: – Option expiry
– Swap tenor
– Swaption type
– Strike (i.e. fixed rate in underlying swap)
• 1y5y payer (read ‘1-into-5-year payer’) at strike 1.5%
Option expiry
Swap tenor
Swaption type
Strike price
Swaption Value Drivers
• Moneyness
• Underlying exposure of a swaption is to a forward-starting swap rate, so moneyness is about the current market forward swap rate relative to the swaption strike price.
• Time to expiry
• Volatility
• The relevant volatility is that of the underlying swap rate. 89
Payer Swaption
Refinitiv Eikon
90
Payer Swaption
2y5y European Payer Swaption
Refinitiv Eikon
91
Payer Swaption
2y5y European Payer Swaption
Refinitiv Eikon
Notional USD 10 million
92
Payer Swaption
2y5y European Payer Swaption
Refinitiv Eikon
Premium (bp)
Notional USD 10 million
93
Payer Swaption
2y5y European Payer Swaption
Premium (bp)
Notional USD 10 million Premium (USD)
Refinitiv Eikon
94
Payer Swaption
2y5y European Payer Swaption
Premium (bp)
Notional USD 10 million
Premium (USD)
Premium (implied volatility)
Refinitiv Eikon
95
Receiver Swaption
Refinitiv Eikon
96
Swaption Volatility Cube
• Analysis of the implied volatility surface can help identify relative value opportunities in option markets and help us choose the most cost-effective way to hedge with options.
• For swaptions, this surface has three dimensions: expiry, strike, and underlying swap tenor.
• We call this surface the implied volatility cube.
• As with cap/floor volatilities, smile or skew may be modelled using the SABR model.
97
Swaption Volatility Cube
Refinitiv Eikon
98
Black’s Model for Swaptions
• Swaption values can be calculated using Black’s model: 9: 𝐿
𝑃𝑎𝑦𝑒𝑟 𝑆𝑤𝑎𝑝𝑡𝑖𝑜𝑛 𝑉𝑎𝑙𝑢𝑒 = L./! 𝑚 𝑃 0, 𝑇. 𝑠3𝑁 𝑑! − 𝑠8𝑁 𝑑” (19) 𝑑!=ln𝑠3⁄𝑠8 +𝜎”𝑇⁄2
𝜎𝑇
𝑑”=ln𝑠3⁄𝑠8 −𝜎”𝑇⁄2 𝜎𝑇
where
𝐿 = 𝑛𝑜𝑡𝑖𝑜𝑛𝑎𝑙 𝑝𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙
𝑚 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑎𝑦𝑚𝑒𝑛𝑡𝑠 𝑝𝑒𝑟 𝑦𝑒𝑎𝑟 𝑖𝑛 𝑡h𝑒 𝑠𝑤𝑎𝑝
𝑇. = 𝑠𝑤𝑎𝑝 𝑝𝑎𝑦𝑚𝑒𝑛𝑡 𝑑𝑎𝑡𝑒𝑠
𝑠3 = 𝑓𝑜𝑟𝑤𝑎𝑟𝑑 𝑠𝑤𝑎𝑝 𝑟𝑎𝑡𝑒 𝑎𝑡 0 𝑓𝑜𝑟 𝑎𝑛 𝑛 𝑦𝑒𝑎𝑟 𝑠𝑤𝑎𝑝 𝑠𝑡𝑎𝑟𝑡𝑖𝑛𝑔 𝑎𝑡 𝑇 𝑠8 = 𝑠𝑡𝑟𝑖𝑘𝑒 𝑝𝑟𝑖𝑐𝑒 (𝑖. 𝑒. 𝑓𝑖𝑥𝑒𝑑 𝑟𝑎𝑡𝑒 𝑖𝑛 𝑢𝑛𝑑𝑒𝑟𝑙𝑦𝑖𝑛𝑔 𝑠𝑤𝑎𝑝)
𝜎 = 𝑣𝑜𝑙𝑎𝑡𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑡h𝑒 𝑓𝑜𝑟𝑤𝑎𝑟𝑑 𝑠𝑤𝑎𝑝 𝑟𝑎𝑡𝑒
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Black’s Model for Swaptions • Defining the annuity factor 𝐴 by:
9: 1
𝐴 = L. / ! 𝑚 𝑃 0 , 𝑇 .
the payer swaption value can be re-written as:
𝑃𝑎𝑦𝑒𝑟 𝑆𝑤𝑎𝑝𝑡𝑖𝑜𝑛 𝑉𝑎𝑙𝑢𝑒 = 𝐿 𝐴 𝑠3𝑁 𝑑! − 𝑠8𝑁 𝑑” (20)
100
Theoretical Justification
• Extending Black’s model to value a swaption can be justified by considering a world defined by a numéraire equal to the annuity factor 𝐴.
• The current value of any security is the current value of the annuity multiplied by the expected value of:
𝑆𝑒𝑐𝑢𝑟𝑖𝑡𝑦 𝑉𝑎𝑙𝑢𝑒 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑇 𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡h𝑒 𝑎𝑛𝑛𝑢𝑖𝑡𝑦 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑇
in this world (see (15)).
• The expected value of the swap rate at time 𝑇 in this world equals the forward swap rate (see (14)).
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Theoretical Justification
• The first result shows that the value of the swaption is:
𝐿𝐴𝐸2 max 𝑠# −𝑠8,0
• This allows us to write the swaption value (see Hull, Appendix to ch. 15) as:
𝑆𝑤𝑎𝑝𝑡𝑖𝑜𝑛𝑉𝑎𝑙𝑢𝑒=𝐿𝐴 𝐸2 𝑠# 𝑁 𝑑! −𝑠8𝑁 𝑑”
𝑑!=ln𝐸2𝑠#⁄𝑠8 +𝜎”𝑇⁄2 𝜎𝑇
𝑑”=ln𝐸2𝑠#⁄𝑠8 −𝜎”𝑇⁄2 𝜎𝑇
102
Theoretical Justification • The second result shows that:
𝐸2𝑆# =𝑠3
which leads directly to the extension of Black’s model to a payer swaption in (20).
103
References Main reading:
• Hull, ch. 28 and 29
• Kosowski and Neftci, ch. 17
Background reading:
• Hull, ch. 14, 15 and 18
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