Mini Chapter Five
Annuity and Convexity
- 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 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.
