Master Chapter Six
Interest Rate Options Demystified
Interest rate Options – are options on interest rates that at it’s very basic construct can be described in three types a) caps b) Floors c) Swaptions (option on swaps). Most other structures and strategies can be understood by breaking them down into these basic types. Before taking a plunge into rate options let’s first understand the key variations made to the original Black Scholes Merton or Black Scholes (equity options) pricing model to arrive at the Black Model (also known as Black 76 based on Fischer Black’s research on commodity futures pricing in 1976) used to price interest rate options.
Black Scholes Merton (BSM) vs Black 76:
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- BSM pricing model used spot value of the underlying as a key parameter for arriving at option values, while Black 76 replaced it with the future/forward value of the underlying. But this isn’t anything more than a mathematical substitution as , hence it shouldn’t be seen as a fundamental difference.
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- Both models assume the underlying to have a lognormal distribution i.e. for the underlying to attain only positive values (Black 76 wasn’t built for negative interest rates). A key difference between the models however was on their assumptions on the nature of the movement of the underlying – BSM assumed a Geometric Brownian motion with a constant drift and volatility through time (following generalised weiner) while Black 76 assumed a driftless stochastic movement with the underlying’s volatility being a function of its absolute level and time (akin to an Ito process).
- By way of notation this can be expressed as: for an underlying with spot value S and future value F,
- Both models assume zero-arbitrage market conditions and use risk-neutral valuation (probability measure) i.e. risk preferences (degree of risk aversion on the underlying) do not play any role in pricing the option and all future expected values are discounted at the risk free rate. This implies that the value of the forward today (at time 0) is equal to the expected future value (spot earns the risk-free rate) discounted at the risk free rate.
- BSM however assumes a constant risk free rate at all times which is inappropriate for pricing interest rate options where the underlying would have a volatility of its own. Since the Black model prices options on the future values of the underlying any divergence between the expected future value and the realised forward is offset by margin calls in the run-up to the maturity of the futures contract (this is why the movement of the underlying is drift-less). Hence even if we considered a non-risk neutral world (where return on investment was higher or lower than the risk free rate) lending variability to the underlying rate of return r, this would make no difference to the present value/pay-off of the derivative investment if it’s future cashflows are discounted at the same rate of return, assuming of course that pay-off is only a function of r for the holding period t.
- Intuitively then, a stock that pays no dividend in the BSM world (a key assumption) can be seen as paying a dividend yield equal to the risk free rate in the Black 76 world.
- Pay off on an interest rate option is a non-linear function of interest rates. While vols get quoted in terms of the (basis point) interest rate vol, it’s conversion to a premium amount in cents needs to take into account the duration and convexity of the underlying swap across same notionals (more on this later).
Black model Equations
Types of Rate options
Let’s begin with Embedded Bond Options:
Call/Put Options embedded in Bonds
- Bonds are frequently structured with optionality in them for either the issuer or investor. Subject to relative demand supply conditions options are embedded to sweeten the deal for the investor and or cheapen the issuance for the issuer.
- Bond issuance that allows for an early redemption (issuer buying back the bond) has an embedded call option for the issuer i.e. holders of these bonds sell this call option (for an expiry shorter than the bond’s tenor) to the issuer at an enhanced yield versus a vanilla bond of the same tenor.
- While bonds that allow the investors to demand an early redemption from the issuer (i.e. put back the bond before maturity) is equivalent to the bond holders buying a put option (for an expiry shorter than the bond’s tenor) on the bond in addition to the bond itself.
- To illustrate – a corporate issuing a 10 year bond with a right to call it back at the end of 3 years is long a call option for a 3y expiry on a 7 year bond. First principles suggest that the coupon on this bond should be higher than that on a vanilla 10 year bond to compensate the investor for the option premium on the sale of call to the issuer. Additionally, the pricing would be a function of the current yield levels and the shape of the yield curve. A steeper yield curve would imply lesser premium for an issuer’s call and hence a lower kicker for the investor as option strike tends to be farther away from ‘at the money’ forward levels. Or seen another way a flatter yield curve which brings implied forwards closer to ATM increases the moneyness of the issuer’s call option enabling a large yield enhancement for the same implied yield volatility.
- Consider the issuer’s current yield curve to be: 1y: 3%, 2y: 3.5%, 3y: 4%, 5y: 4.5%, 7y: 4.75% 10y: 5%. Vanilla 10y bond can be issued at 5% but the bond with an embedded call after 3 years would be issued at a higher coupon to accommodate the call premium – say 5.25%. This premium would of course be a function of the yield volatility of this corporate’s bond curve. The issuer is effectively embedding a Call option on its 7 year bond with an expiry of 3 years. While the decision to exercise the call would depend on whether the 7y yield in 3 years’ time is below 5.25%, the pricing of it would depend on the 3y forward 7 year rate that’s at ~5.43% (i.e. OTM by 18bps).
- Impact of duration on changing option expiry, for the same bond – a key nuance of embedded bond options is how the changing embedded option expiry also changes the underlying (even if it’s the same issuer’s liability) itself as it gains/loses duration. To understand better – in our example above if the call was at the end of 7 years we would be pricing the option premium for a longer 7y expiry but on a shorter duration of the underlying (now a 3y bond). Coupon/strike on the bond would also vary (or sometimes not) based on the changing duration dynamics and shape of the yield curve. It’s possible that premium on a 7y expiry 3y tail is similar to the premium on a 3y embedded call in case the impact of 3y onward implied vols flattens out and intuitively the duration gain on the option expiry somewhat offsets the duration loss for the underlying tenor.
- Strike price for the issuer’s call – the strike price of the options we have discussed so far is determined purely by the coupon on the bond (that includes the premium in it). Issuers could also tailor a different premium arrangement for the investor to optically lower their own funding cost. Coupon on the callable bonds (while higher than plain vanilla) can be priced much lower i.e. more out of the money versus the implied forwards by pushing the option premium payment at the time (and in case) of the exercise of the call – this is typically achieved by redeeming the bond at a premium. This would result in a lower coupon premium for the full tenor of the bond. From an investor’s standpoint a redemption premium arrangement would mean a higher holding period return till the option expiry in case of an exercise (versus a vanilla bond of the same tenor as expiry). At the time of purchase the redemption premium is attractive enough to trade it off with a lower overall coupon in case of non-exercise.
- Instead of a strike/coupon at 5.25% in our example above – the issuer could choose to pay a premium in case of early redemption of let’s say 3% (also referred to as a contingent premium product) and reduce the coupon on the embedded bond to 5.05% which would optically reduce their overall cost of funding. In case the call is exercised at 103 (roughly 50bps lower on the 7 year rate after 3 years i.e. strike at 4.55%) the investor’s holding period return would be 5.05% + ~1% = ~6.05% for 3 years.
- Bonds with embedded put options – switching the right (but not obligation) to early redemption from the issuer to the investor would flip the long option from a call to the issuer to a put to the investor. Taking a cue from our example above – a 10y bond issued with a 3y put to the bond investor would bear a coupon lower than 4% (3y vanilla rate), as a way to pay for the option to extend duration in case rates go lower. Conversely a coupon on a puttable bond at or above 4% would mean the investor is either able to buy the option for free or is being paid for it, which would create a visible arbitrage to buy these bonds and short the 3y vanilla bond. Strike for the investor’s put would be the coupon on it, and the exercise price would logically always be par (an investor would usually not reinvest their money at a premium!).
- Trivia
Offshore Investment in India’s Corporate Bonds with both put and call options
As per the Foreign Portfolio Investment rules for India’s Fixed Income Markets (before the emergence of Voluntary Retention Route, VRR in March 2019), FPIs were only allowed to buy bonds with a minimum 3yr maturity. But for those who did not wish to take credit risk on the bond for as long as 3 years, buying a 3 year bond with say an embedded 1y put/call option was another way to structure it. By way of economics the pricing of this bond would be no different from a 1y vanilla bond as the investor would exercise the put if rates went up and the issuers would exercise the call if rates went down. This was also an attractive avenue for the issuers to tap capital market funding/offshore money especially when the front end of the yield curve was relatively flat and markets were expected to be range bound. Owing to higher re-issuance costs the issuer would typically not exercise the call unless yields collapsed. Offshore investors on the other hand would look to strike a balance between not taking longer than 1y credit risk (as much as possible) and a fast rallying long carry market.
Interest Rate Caps
- Interest Rate Caps – are a certain type of option on interest rates that cap the exposure to higher rates i.e. they pay off when market rate (or the relevant reference rate) is above the cap strike. As intuition would suggest they are frequently used by borrowers to hedge their funding costs. As an example a cap on an interest rate swap is the specified strike above which if the reference rate trades in the market, the buyer of the cap would be compensated the excess (Market rate – Cap Strike) yield on the contract notional.
- From a borrower’s perspective, you can think of a certain floating rate loan of say SOFR + 50bps for a 1 year tenor with quarterly payments. The interest cost on the liability can be hedged (especially in a rising rates environment) by buying a cap for a notional of USD 25 mio on SOFR at let’s say 5% (cap rate) i.e. the borrower would get compensated the excess (above the cap rate) by the dealer when SOFR fixes above 5%.
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- Snapshot of the cash flows below suggests that the effective cost of funding for the borrower in a rising rates environment would be lower given the gains from the purchase of the cap.
Table 1. Cash Flows of a 5% Cap Strike
Payment dates | Cap Rate | Market Rate (SOFR fix) | Day Count | Cap Payment (USD) |
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10 January 2010 | 5.0% | 5.0% | - | |
10 April 2010 | 5.0% | 6.5% | 90 | |
10 July 2010 | 5.0% | 7.0% | 91 | 94,792 |
10 October 2010 | 5.0% | 7.5% | 92 | 127,778 |
10 January 2011 | 5.0% | 92 | 159,722 |
Source: Pandemonium.
- Cap payment is made by the dealer in the following quarter using an actual/360 day count convention. For instance the gain of 150bps as per the April 10 fix would be realised on July 10 as:
- An interest cap therefore is a series of payments within the option contract, the payments decided as per the previous period floating rate reset i.e. if the reset rate is below the cap strike no payment would be made. This contingent payment period is referred to as the caplet. Also note that the first reset is excluded from payment considerations as the fixing is already known at the time of entering the contract.
- So a cap on a 5 year swap that resets quarterly, would comprise of 19 payment periods or caplets with each caplet (think of single period interest rate swap) being a call option on the underlying interest rate with the cap being the strike.
- Connecting the Dots
Caplets (or call option for a single period swap) on interest rates can also be understood as a put option on a coupon bearing bond with a same tenor as that of the caplet, strike price at par and cap rate as its coupon.
Consider a 5y cap with a strike of 5% and a quarterly reset. The caplet that starts in 12 months can be understood as a European put option on a 3 month bond exercisable at par 12m forward with face value at par (100 as per general references) and a coupon of 5% payable quarterly (i.e. single coupon for a 3m bond). If the 3m rate at the end of 12m is at 4% then the optionalised bond will be trading at a price of:
is the discount factor for the 3m period.
i.e. well below 100 and the option would be exercised for a payoff 100 (Put Strike) –
- Notice that this is the same as the present value of the excess (relative to cap strike) payable yield at the end of the 3m period i.e. the option payoff:
- In terms of cash flows therefore, a cap is a series of put options with expiries progressively extending with every reset but the same underlying bond tenor for each option; tenor equal to the reset period. In our example of a 5yr cap the different option tenors/expiries are 3m, 6m, 9, 12m…57m (total 19) with strike at par on a 3m underlying bond having a coupon rate of 5% p.a. payable quarterly.
- Floor on interest rates – are an option expression to protect oneself from a drop in interest rates by flooring them at a specific strike. A seller of a floor would pay the buyer the difference between the floor strike and market rate (as percentage of the notional) during a reset period for which the market rate was set below the floor strike. In the example above, if we set a floor on a fixed rate SOFR loan at 2% – the lending bank would earn a floor rate of 2% irrespective of how low interest rates go. For the buyer, a floor is a put option on interest rates or (as explained for caps above) a call option on floating rate bonds with tenors same as the floorlets.
- Trivia
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- Loan documents/Term sheets of Banks sometimes do not explicitly mention a floor on the benchmark lending index but may define interest payments as unidirectional i.e. to be made only by the borrower to the lender. Floating rate loans – benchmark index + a fixed margin (akin to a credit spread) – are exposed to risks of negative interest rates but the one sided obligation to pay interest implies a long exposure to a 0% (or sometimes higher) floor on the floating index for the lending bank, sold to them by the corporate borrower. This also keeps the floor market perpetually bid by corporates as they rush to hedge their liabilities especially in a low rates backdrop, creating significant downside skews towards 0 and lower strikes. While in a high interest rate backdrop when it’s cheaper to buy ~0% floors, borrowers wouldn’t anticipate a collapse in rates as big as to warrant these hedges. For banks however, the value of their long floor positions appreciates as rates collapse/flirt with negative values (observed post GFC and Covid rates cuts + liquidity stimulus) which have been monetized using trading of these derivatives separately from the loans itself.
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- Swap hedges for Loans – In case the borrower converts the floating loan to a fixed rate one through a vanilla swap, they need to be mindful of replicating the swap terms to match those of the loan that’s being hedged. If the swap isn’t structured to have the floor on the floating rate as that on the loan it could dangerously build up large payment obligations for the borrower in case rates plunge below the loan floor strike. For instance, in case of 0% loan floors and a paid fixed swap position (to hedge the floating rate loan) of the borrower, negative floating benchmark rate would imply cash outgo for the borrower on both the swap legs if the 0% floor isn’t embedded in the swap. This would render the supposed swap hedge ineffective and from an accounting standpoint the mark to market losses on the swap would need to be passed through to the borrower’s PnL.
- Connecting the dots
Caps and Floor Parity – akin to a put call parity this too follows the same logic of being long a cap (call) and short a floor (put) at the same strike and the difference between the two equating to a short interest rate swap (outright forward in case of put-call parity) position. To see how this is true just simply think of the same strike cash (K) flows of a long cap to be complementary of a short floor (only one of them materialise), and the difference between the two would amount to Market rate – K being paid at all times, whether strike is above or below the market rate. This is equal to being paid a fixed rate swap at a coupon of K, alongside receiving the market floating rate.
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- Swaptions – are options on interest rate swaps either with a right (but not an obligation) to enter a received or a paid position at a specific strike at the end of the option expiry, hence common reference is to a European swaption. Market lingo for a 3y forward 5y swaption addresses 3y as the expiry and 5y as the tail. Depending on the use and direction of hedge taken on interest rates there are two types of swaptions – receiver and payers; related strategies are a mix of these vanilla exposures in different variations (to be covered later).
- As an example a corporate borrower keen to tap capital markets for fund raising 6 months down the road can hedge their interest rate exposure by buying a 6-month expiry payer swaption for a duration equivalent to the funding tenor at a strike of K%. This would cap their deferred borrowing cost to K% as a worst case outcome but with no obligation to exercise if funding rates are below K% at the end of 6 months.
- Swaption vs a cap for a borrower’s hedge – key difference between the two option types is that a cap is a series of call options on the underlying caplets while a swaption is a series of annuity cash flows but still within a single option. Hence it’s important to think about whether a borrower wants to hedge a fixed or a floating rate liability via rate options. As per our example above a future fixed rate liability has been effectively hedged by a payer swaption, but if it were a current floating rate liability a single payer swaption wouldn’t protect against all the floating resets risks of the borrower’s interest obligation. To effectively hedge a floating rate obligation then, a cap on the reference rate can be purchased by the borrower for the full tenor of the loan hedging each of the reset risk.
- Connecting the Dots
A corporate which issues a 10y bond with an embedded call option at the end of 3 years (issuer’s right to call back the bond) is long a 3y call option on a 7y residual life bond or long a 3y receiver swaption on 7y swap rate.
- Trivia
- Valuation of a swaption – much like any other option on an underlying asset (interest rate in this case), valuation of a swaption is made up of both its intrinsic (moneyness) and extrinsic (time-dependent) values.
- As an example consider a payer swaption where,
P is the USD notional of the swaption,
T is the option’s time to maturity
K is the strike on the underlying swap
RT is the current (implied) forward rate
n is the tenor of the underlying swap
And lastly, m is the compounding frequency of the swap i.e. it exchanges cash flows ‘m’ times per year.
- As an example consider a payer swaption where,
at each interest payment date of the swap which resembles the cash flow of a long call option on RT (interest rate at option expiry) at strike K.
Let’s simplify the above and for the time being let’s avoid multi-period compounding i.e. assume annual compounding (m = 1). Think of a Bank that would like to hedge its locked-in fixed rate lending commitment to a borrower 1 year later for a period of 3 years by entering into a 1y forward 3y payer swaption (option expiry T = 1 year) at a strike of 10.50% for a notional of USD 25 mio.
Payoff of this swaption (at maturity) would be the sum of the discounted annuities over a 3 year period from when the option expires (residual maturity of the option).
If at inception below are the zero coupon forward rates:
1y forward 1y = 9.25%
1y forward 2y = 10.50%
1y forward 3y = 12%
Recall the valuation of an interest swap is same as that for a fixed rate bond with coupon equal to the par swap rate. Bootstrapping the zero rates above to arrive at the 1y forward 3y par swap rate RT we get 11.78%. This would be the observed implied forward swap rate at the time of initiating the swaption, but referencing the zero rates here is important to connect the dots later.
ATMF 1y forward 3y swap at 11.78% amounts to a 128bps per annum gain on a USD 25 mio payer swaption at a strike of 10.50%. That’s an annuity cash flow of USD 320,000 (call it annuity A) to be discounted by the projected rates above:
Swaption payoff or the PV of the cashflows received at expiry = USD 320,000 x (0.9153 + 0.8190 + 0.7118) = USD 782,752
Note that the vol of the underlying as the forward swap ages needs to be considered for valuing the swaption and the discounted cash flows above are only a sanity check/approximation for it. Time value of a swaption that’s determined by its time to expiry, volatility of the underlying in that horizon and the distance between the implied forward and strike would be the difference between the swaption value before and at expiry.
Standard market model expression (this is a common academic reference) of a single period payoff (extending from Black’s model) would be:
where Df (Ti) is the discount factor for cash flows for the period (0 to Ti). Note that the period Ti is longer than the option expiry T as ‘i’ signifies the swap cash flow period that appears m times per year. Hence for an n-year swap, i ranges from 1 to mn.
Summing up for all periods then, price of the payer swaption would be:
where
T is time to expiration of the option, expressed in years
𝜎2 is the variance of the underlying Swap rate
N is the cumulative standard normal distribution function
- Connecting the Dots
PT = K* Df1 + K* Df2 + (1+K)* Df3
PT = K* (Df1 + Df2 + Df3) + Df3
1-PT = K* (1-Df3) – k*(Df1 + Df2 + Df3)
(Notional) x Max { 1-PT, 0 } = Max {(1 – Df3) – K x (Df1 + Df2 + Df3), 0}
Taking the discount factor summation out, we arrive at:
I hope this also helps reaffirm/clarify the intuition earlier on the pay-off for caplets that are call option on a single period swap to be the same as for a put option for a same tenor coupon bearing bond (coupon same as the cap rate) with exercise price at par.
Receiver/Payers Parity – the parity equation makes a loyal appearance yet again for the swaptions world too. Assuming European options of course – receivers and payers on the same strike, with the same expiries and underlying swap tenors:
An In-the-money receiver and hence an out of the money payer would net off on their respective time value and the remaining intrinsic value would be the same as that of a received forward starting swap at the same strike. Conversely, an out of the money receiver/in the money payer would similarly reflect the negative PV of the received forward starting swap.
- The series of swap cash flows we have discussed so far are of the nature of an annuity – this is true because valuing the swaption or the underlying swap can be understood as an exchange of the current (implied) forward rate that’s an average fixed rate (effectively the same single period swap for every reset period) with a fixed strike rate that leads to fixed cash flows. Value of the swaption is thus the sum of the probability adjusted PV of each of these annuities, while that of the swap is simply the sum of the PV of all annuities.
- Connecting the Dots
The impact of duration of the underlying shows up in the premium paid for a long option on it through the number of periods of summation of a fixed annuity. The frequency with which the annuity is compounded and paid out is further weighted by the discount rate term structure for the respective forward cash flows which would determine the magnitude of the premium to be paid. Hence the premium is proportional to the tenor of the underlying swap and additionally gets impacted by an elevated and or steep discount rate term structure for longer expiries.
Table 2. Table on normalised vol across expiries/tenors
Tenor of the underlying swap | ||||||
Expiry | 1y | 2y | 3y | 5y | 10y | 15y |
3m | 117.5 | 148.4 | 143.6 | 130.2 | 103.7 | 94.7 |
6m | 135.7 | 148.2 | 143.0 | 127.6 | 104.3 | 95.9 |
12m | 151.1 | 142.0 | 134.0 | 120.5 | 102.4 | 94.5 |
2y | 133.6 | 127.2 | 122.3 | 112.6 | 98.5 | 91.2 |
3y | 121.1 | 116.0 | 113.2 | 106.0 | 93.8 | 87.1 |
5y | 106.6 | 103.2 | 101.0 | 95.9 | 86.8 | 80.7 |
10y | 84.5 | 82.5 | 80.5 | 77.2 | 72.0 | 66.9 |
Source: Pandemonium.
*all figures are annualised vol expressed in basis points
*normal vol or an adjusted Black vol accommodates for negative interest rates i.e. distribution of the underlying isn’t lognormal
Table 3. Table on Spot premium across expiries/tenors
Tenor of the underlying swap | ||||||
Expiry | 1y | 2y | 3y | 5y | 10y | 15y |
3m | 44.9 | 111.0 | 159.0 | 233.0 | 342.0 | 434.0 |
6m | 72.5 | 156.0 | 221.0 | 319.0 | 483.0 | 616.0 |
12m | 112.0 | 207.0 | 288.0 | 419.0 | 658.0 | 843.0 |
2y | 135.0 | 254.0 | 361.0 | 537.0 | 868.0 | 1114.0 |
3y | 145.0 | 274.0 | 395.0 | 598.0 | 982.0 | 1262.0 |
5y | 155.0 | 296.0 | 428.0 | 655.0 | 1095.0 | 1411.0 |
10y | 147.0 | 283.0 | 407.0 | 628.0 | 1082.0 | 1404.0 |
Source: Pandemonium.
*all figures are annualised premium expressed in cents
Table 4. Table on Forward premium across expiries/tenors
Tenor of the underlying swap | ||||||
Expiry | 1y | 2y | 3y | 5y | 10y | 15y |
3m | 45.5 | 113.0 | 161.0 | 236.0 | 347.0 | 440.0 |
6m | 74.5 | 160.0 | 227.0 | 328.0 | 496.0 | 633.0 |
12m | 118.0 | 218.0 | 303.0 | 441.0 | 694.0 | 888.0 |
2y | 148.0 | 278.0 | 396.0 | 588.0 | 952.0 | 1221.0 |
3y | 164.0 | 311.0 | 448.0 | 677.0 | 1112.0 | 1430.0 |
5y | 187.0 | 357.0 | 515.0 | 790.0 | 1320.0 | 1701.0 |
10y | 209.0 | 401.0 | 577.0 | 890.0 | 1534.0 | 1990.0 |
Source: Pandemonium.
*all figures are annualised premium expressed in cents
- Tables above are a regular business day (truncated) snapshot of annualised basis point (normal) volatility of ATM USD OIS swaps across option expiries and tenor of the underlying swaps and their corresponding spot and forward premia as on July 18, 2023. Just a few observations below:
- The crucial vol input and its forward term structure importantly determines the premium amount in the forward space. The option expiry period and how it accommodates the degree of the uncertainty around the path of interest rates in the forward horizon would among the key factors governing the vol term structure. Up to 1 year swaption expiries above have fairly elevated implied vols as that’s the horizon which carries the uncertainty of flipping from a hard landing to a soft landing to a no landing/arguably over-heating narrative for the US economy, affording the central bank to bring forward rate cuts/keep rates elevated with a long pause/keep up with the hikes to sustainably achieve the 2% inflation target. It is also the area under on the vol surface where demand for optionality from hedgers and speculators overwhelms the supply. The marked decline in vols on 2y and onward expiries implies a lower degree of uncertainty in that horizon along with an overhang from vol supply (from callables, range accruals and other such structured products).
- Spot premium for a specific option expiry across tenors of the underlying swap is fairly proportional to the respective duration of these swaps. This is of course true if we compare the premium on the same swaption notional for different tenors of the underlying swap which would mean different DV01s; a conditional spread strategy would usually be done DV01-neutral.
- Large divergence between spot and forward premium for longer expiry options is a function of the discounting curve used to compute the present value of the premium (spot premium) from its forward value. From the table above, the spot premium on 10y forward 1y is lower than the 5y forward 1y swaption owing to the high discount rate in the current environment.
- Annuity Discount factor A – for ease of notation the payer option price equation above can be re-written as:
Convexity adjustment factor
- As a recap from the Time Value section – Convexity is a second order derivative with respect to yield change to signify the change in the rate of change of price or the change in duration, while the duration itself depicts the first order change in price with respect to change in yield. Interestingly when we bring together products with linear and nonlinear (a non-zero second order move) payoffs in them it’s important to think about the convexity adjustment in the overall trade.
- First principles as already discussed earlier prove that bond prices are convex and any payoff based on a bond price (or the like i.e. swaps) movement therefore would be convex too. To illustrate better let’s consider a box trade of a Eurodollar Futures contract and a 3 month cash bond with the same start date as that of the future; it doesn’t matter whether or not the bond has a coupon as even for a zero coupon bond duration would change as time passes. Both the futures and the bond are priced on the same underlying interest rate. The movement in pnl on the futures position is linear to the move in underlying interest rate i.e. every 1bp move results in USD 25 move in its value hence has a duration of ~0.25. The same 1bp move however would move the bond position pnl non-linearly as its duration would change depending on its coupon/yield. A long position in a bond vs a short position in futures would create a net long convexity play and would make more money/lose less money as interest rates go down/up. The dollar amount by which the futures pnl needs to be adjusted to equate to the bond pnl is the convexity adjustment factor. You can also think of it as the notional adjustment required in the futures position to create the same per basis point move in value as that for the bond position.
Graph 1 - Zoomed in view of a linear payoff for Eurodollar future and convex payoff for a 3 month cash bond

Source: Pandemonium.
- Convexity adjustment notation – to complete the argument let’s also discuss the academic reference to this adjustment factor (uses Taylor series expansion that isn’t intended to be explained here) usually written as the adjustment to forward bond yield to equate to the expected bond yield. In a forward risk neutral world (with respect to a zero coupon bond with the same tenor as that of a forward contract) the forward bond price would equal the expected bond price. But because of the non-linear (convex) relationship between a bond price and its yield, the forward bond yield would need to be adjusted for this convexity to equal the expected bond yield. This can be denoted as (described in the appendix of John C Hull’s Options, Futures and Other Derivatives):
Where,
t is the time to maturity of the forward contract or derivative
Bt – Bond price at time t
Et (Bt) – expected value of the bond price at time t
y0 – forward bond yield observed today i.e. time 0
f (y0) – this is the same as forward bond price F0, denoted as a function of forward yield as of today i.e. time 0
yt – bond yield at time t
F0 – forward bond price corresponding to the forward bond yield as of today
i.e. at time 0
This above can be re-written as,
And when it’s not, we can think of hedging the forward/derivative position in a manner such that the forward bond price is realised at time t. The amount by which the position needs to be hedged is described by the RHS expression.
- Connecting the Dots
- Further expansion (skipping the math here) of the notation above yields the following as the convexity adjustment factor i.e. the magnitude by which the forward bond yield y0 needs to be adjusted to arrive at the expected bond yield:
- From the explanation above then, being long an option that amounts to being long gamma is often used interchangeably with being long convexity. But when we consider convexity more specifically with regards to the change in the price of an option (w.r.t. a basis point change in the underlying rate) that creates non-linear pnl swings we are effectively basing it off an underlying with changing duration. So while an interest rate cap and or floor that works like a simple call or a put option on interest rates has a linear pay-off at maturity, the change in the option valuation still exhibits convexity as the risk imitates being long calls/puts on a series of forward starting zero coupon bonds.
As for a swaption – back to first principles, the pay-off by definition would be convex (as it is based on the valuation of an in the money swap). Also recall being long a payer or a receiver at strike K is like being long a put or a call option on a bond with a coupon of K. Depending on the moneyness of the option the impact of convexity would imply -> 100bps in the money 5y receiver would pay more than double of 50bps in the money upon unwind.
An unwind/option valuation before expiry would also have a running time component (as the option ages) that would impact the option’s time value in addition to moneyness (true for all options) – for the same notional, a 100bps in the money 6m forward 5y receiver would have a higher option duration vs a 3m forward 5y receiver at the same strike, assuming 6m5y and 3m5y forwards are the same.
Critique of Black Models for Rate option valuation
- Rate options are generally priced using Black model as against using Black Scholes Merton model. The primary difference between the 2 approaches is that of estimating the value of the underlying at expiry. BSM uses spot + drift + a weiner/brownian process vs Black uses just a distribution of the forward price / interest rate and assumes that the underlying is lognormally distributed.
- We also apply different versions of the black model depending upon the rate option being traded. For bond options the assumption underlying is that bond forward price is lognormally distributed. For caps and floors the underlying assumption is that the interest rate for each caplet/ floorlet is log normally distributed and could have vols independent of each other. For swaptions the assumptions in the black model is that the forward swap rates are lognormally distributed and the probability distribution of the swap rate determines the option price.
- While the above 3 models are correct based on their respective assumptions – they aren’t consistent with each other. A lognormal distribution of a bond price is not consistent with a lognormal distribution of forward rates thus using the bond option valuation model may not give values consistent with the swaption model.
Interest rate strategies
Since I’ve already discussed all basic strategies as part of the FX/Equities options earlier, drawing a parallel with interest rate options won’t need as much detail unless we speak of pure rates curvature-driven option strategies like conditional spreads. Directionally lower rates strategies are usually structured with receivers in formats like – outright receivers, receiver spreads, receiver flies, receiver ladders, conditional bull steepeners or flatteners among others. Exposure to volatility regardless of direction is generally taken via longer expiry straddles/strangles, calendar spreads (sell short expiry to buy longer expiry on the underlying swap).
- We would go through lower rates strategies here so as a corollary it becomes easier to understand the higher rates ones.
Receiver Spreads
- Can be thought of either as a bear (put) spread on interest rates or a bull (call) spread on bonds. This is typically structured as a purchase of an at the money forward receiver and selling an out of the money forward (lower rate strike) receiver to cheapen the structure, for the same notional and expiry. Intuitively this is a bet on lower rates versus what’s currently implied by the forward yield, with the pay-off capped once we hit the out of the money strike.
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- Both strikes may well be out of the money in case of a stronger conviction for a directional move lower in rates.
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- Play on the Greeks – It’s a positive net delta exposure at inception with gamma and vega values the maximum and positive if the long strike is ATMF. Delta trajectory peaks in between the strikes while gamma and vega start coming off as the underlying rate moves away from ATMF and flips to negative values closer to the short side of the option. Option Theta is negative (value decays with time) for underlying interest rate values at or close to the long strike and has positive values as the underlying trades closer to the short strike.
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- Receiver flies – are the same as structuring a butterfly with receivers with the same expiry and underlying swap tenor, i.e. buying a lower and higher strike as the wings and selling twice the notional on the intermediate strike (typically mid-point of the range). The higher strike wing would imply being long a receiver spread while the lower strike would be like being short the spread, the intermediate strike being the short leg both times. As discussed earlier, the strategy doesn’t account for skews in the vol smile (unlike an outright non-ATMF receiver and or risk reversal) and is a conservative view on the underlying remaining within the long receivers range. Greeks would play out in the same manner as for a standard butterfly covered in Option strategies.
- Receiver ladders – are a series of receiver options structured to cheapen the overall structure, a long receiver ladder position is a purchase of an OTM receiver and sale of two respective more OTM (lower strikes) receivers to finance it, all for the same expiry. Receiver swaption being the same as a put option on an interest rates, this position is the same as a long put ladder on rates. A short position would be just the reverse i.e. buying the two lower strike receivers and selling the least OTM (or ATM even) to finance it. Strikes can be chosen to either make the structure zero cost or delta neutral or minimal net vega but it’s viability – strike placement – would depend on any skew on the vol smile (implied vols across strikes for an expiry).
Refer to the pay-offs of a long/short call and put ladders before we make further observations. K1 is the long/short strike while K2 and K3 are the more OTM short/long strikes for the respective long/short Call and Put ladders:
Graph 2-5: Pay off graphs on Long/Short Call/Put Ladders

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- Referencing the pay-offs above, a long ladder position has limited profit potential and can suffer unlimited loss (if the underlying goes far above/below the most OTM strike for a call/put ladder. Conviction to put on this position would come from a range-bound but directionally higher (in case of a call)/lower (in case of a put) view for the underlying. This would appear more tenable in a low vol backdrop.
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- A short ladder exposure on the other hand is an exchange of limited risk for unlimited profit potential if the underlying were to move significantly higher or lower typically in a high vol backdrop. You can think of this position as a tail hedge for a portfolio that wants to protect against unforeseen blow-up events.
- A long receiver ladder exposure into say the Silicon Valley Bank blow-up (completely unforeseen in an otherwise strong inflation/growth, almost goldilocks, backdrop) would have inflicted major portfolio losses. Short receiver ladder would have been a great hedge.
- Greeks of the strategy – A long ladder exposure would be short gamma, short vega and long theta while a short ladder exposure would be long gamma, long vega and short theta. Note that the rationale for the strategy shows up in the direction of gamma and vega exposure respectively.
- Trivia – Swaptions for Pension LDI
Coming back to the use of rate options for this investor segment, interest rate hedges are frequently timed with vanilla strategies but broader usage of swaptions and strikes on them would be a function of a) basis risk i.e. different discount factors for liabilities and the swaps meant to hedge interest rate risks the liabilities are exposed to b) gap risks on the underlying i.e. knee-jerk sell-offs that could create a hard trade-off between immediate rates hedging (eg. buying bonds if rates have sold off beyond target levels) or retaining the optionality c) widening bond-swap spreads d) vol term structure and payer/receiver skews, among others.
- Selling OTM Payer swaptions – where strike is placed at specific target levels high enough to trade off the improving funding ratio (lower present value of liabilities as discount rates increase) with going long interest rates. The premium earned by selling the swaptions also helps finance the purchase of fixed income assets once target levels are hit, but a meaningful sell-off beyond target levels would bleed funding (remember on the exercised swaption the LDI fund is received fixed at the strike).
- To prepare for a rates sell-off that’s bigger than the LDI manager expected, multiple payer swaptions can be sold at different strikes for better use of funding to hedge rates risks and attain better average entry levels on received positions.
- Swaption collars – the sale of an OTM payer can finance the purchase of an OTM receiver when the LDI manager is also conscious to hedge against a sudden collapse in rates. These are typically zero cost structures where in a rising rates backdrop, higher implied vols for higher strikes (topside skew) can finance OTM receiver strikes that are closer to ATMF, affording better downside protection.
- Conditional Spread strategies using rate options – now let’s look at option strategies on two different underlying i.e. curvature expressions where the payoff is conditional on a bullish or bearish interest rate environment. In other words these are conditional spread options that allow a more nuanced expression of a view as opposed to a plain steepener or flattener traded on an underlying swap spread. The strategy is usually DV01 neutral on both legs i.e. same as the underlying swap spread exposure in that regard.
- These spreads can have 4 different expressions: bull/bear flatteners and bull/bear steepeners. Much like bull/bear spreads that use calls/puts to structure them, a conditional bull spread uses receivers to express itself while a conditional bear spread uses payers. Leg-wise exposure to calls and puts can be understood from the table below:
Strategy | Option Type | Front Leg | Back Leg |
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Bull Flattener | Receiver | Short | Long |
Bull Steepener | Receiver | Long | Short |
Bear Flattener | Payer | Long | Short |
Bear Steepener | Payer | Short | Long |
Source: Pandemonium.
Motivation for initiating these strategies:
- Prime motivation as is true for any strategy is to be able to initiate it in the most cost effective manner. Big macro factors like changing central bank monetary and or QE policies, government fiscal dynamics, market-sensitive regulatory changes (BASEL/Accounting standard changes, IFRS implications for lifers’ et al), structured issuance dynamics (dealer gamma hedging) often generate more specific bull/bear views on curvature. Conditional spread strategies would be the way to express those views.
- The conditional aspect of the trade which reduces the probability of a payoff versus a vanilla swap spread trade and or a spread option (to be explained later) also makes it cheaper for instance entry levels are better for a zero PV conditional curve option.
- Or seen another way – a bull spread strategy structured with receivers limits the profit/loss scenarios to a rallying rates environment since in a sell-off the receivers are worthless. Similarly for a bear strategy structured with payers the profit/loss scenarios are limited to a rates-sell off environment.
- Relative implied volatility swings across tenors of the interest rate curve would typically help to cheapen the option spread. Vol of a specific tenor that’s (adequately) rich versus the others and its own history can help finance the long option leg of the spread. Furthermore, a strong bias to either receive or pay select tenors has the ability to create skews for OTM strikes for those tenors in the forward space, which if capitalised (i.e. selling these strikes) can enhance the spread entry levels.
- As an example let’s consider a recently popular bear strategy – 1y forward 2s10s USD SOFR conditional bear steepener i.e. going short a 1y forward 2y payer and long a 1y forward 10y payer. When we think about the rationale for this (or any other) trade, following template would generally come in handy (I’ll attempt to tie up every aspect with today’s market – Sep 6, 2023):
- Broad macro framework within which the trade is being structured – Consider the current US growth backdrop which has been surprisingly resilient after over 500bps of tightening over a span of barely 1.5 years, with a soft landing scenario a lot more likely than a hard landing/recession. This has also bought time for the US Fed to keep rates elevated (not rush into cuts), nudging the term premia in general higher. There’s also been an uptick in both government and corporate bond issuance amidst a bearish Global Fixed Income outlook (reinforced by BoJ raising its YCC target), that’s met with a buyers’ strike -> bear steepening on rates.
- Kinks and relative cheapness on spreads across the curve – Relentless flattening of the curve for most part of the hiking cycle has kept vols elevated for a 1y2y payer versus the back end (1y10y in this case). Trajectory of the ATMF and or ATMF + 25bps implied vol ratios for the two legs can be compared with its history to understand the relative cheapness/attractiveness of the skew of one leg versus the other.
- A zero cost structure is a better way to visualise the relative valuations – in the current market, premium on a 1y2y ATMF +25bps payer equals the premium on a 1y10y ATMF payer; strategy is DV01 neutral, hence a premium of 44cents for each leg would apply on different notionals). We would not get into the pricing methodology here as the intent is to intuitively make sense of the trade structure which offers a 25bps better entry level versus the ATMF spread.
- Compute the net carry on the trade – in other words what would be the change in premium over 3 months using the respective implied vols of the strikes, curve roll-down and time decay (that’s a function of vol term and curve carry among other things) after 3 months.
For starters this swap spread has a large negative roll if you notice the levels below, rolls 16bps negative over 3 months.
Spot 2s10s spread: -92bps
1y forward 2s10s ATMF spread: -24bps
9m forward 2s10s ATMF spread: -40bps
Table below gives a snapshot of the changing premium and implied vols for the respective legs over a 3 month period. We are short the implied vol on 1y2y that’s gone down a bit (is in our favor) and long implied vol on the 1y10y leg that’s barely changed much. As for the premium – the 9m2y has gained a bit given a 4.305 payer is less OTM/ closer to ATMF (at 4.25, versus 1y2y ATMF at 4.055) and 9m10y has suffered a time decay. Net premium is down to ~ -5cents (from flat earlier), mainly dragged down by the negative curve roll.
Table 6. Trade Snapshot
Tenor | Levels | Spot Premium | Implied vol |
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Spot 2y | 4.9 | - | - |
Spot 10y | 4.0 | - | - |
1y2y ATMF +25bps payer | 4.3 | 44.4 | 140.6 |
1y10y ATMF payer | 3.8 | 44.3 | 111.3 |
9m2y 4.305 payer | - | 45.3 | 138.4 |
9m10y 3.8 payer | - | 40.5 | 111.7 |
Source: Pandemonium.
- This trade expresses the recent bear steepening bias in the swaptions space and attempts to reduce the negative curve roll impact to the extent of the conditional curve delta (less than half of the underlying swap spread delta). But we need a strong bear steepening momentum for it to perform and more than offset the negative curve roll.