The introduction of the new 6% notional coupon for T-bond and T-note futures, beginning with the March 2000 contracts, brought to the forefront an important concept that had long been disregarded: valuing the embedded quality option. This embedded option has important pricing implications not only for the T-bond and T-note futures contracts, but also for the options on these futures.
Chicago Board of Trade options on Treasury bond and note futures are a much more complicated derivative than simple OTC Treasury options. The CBOT options are a compound option, or an option on an underlying security with an embedded option. To understand the features of these options, one must first understand the underlying option embedded in the futures contract.
The short seller of a futures contract has the choice, or "quality option," of what bond in the deliverable basket he would like to deliver. As yields increase (decrease), the longer (shorter) duration securities in the basket tend to become cheapest to deliver (CTD). With the overall level of yields of all securities in the deliverable baskets relatively close to the 6% notional coupon, switches in the CTD are likely, and the embedded option has significant value.1
The problem with valuing options on futures using a lognormal price-based model is that prices of T-bond and T-note futures are not lognormally distributed when there are switches in the CTD. Because the futures contract 'tracks' the CTD, as yields increase (decrease), the duration of the contract begins to increase (decrease) as it tracks longer (shorter) duration bonds.2 When examining the modified duration of the December 10-year T-Note futures contract at December option expiration (see chart below), we can see that this relationship continues to hold for reasonably large moves in prices.3 The low/negative convexity characteristic of the contracts associated with the switches in the CTD and the resulting increase (decrease) in modified duration in a rising (declining) interest rate environment alter the futures price distribution. Options on futures are somewhat analogous to options on mortgage-backed securities pass-throughs, which also exhibit negative convexity characteristics.
One may initially think switching into longer-duration bonds in a selloff and shorter duration bonds in a rally would cause at-the-money puts to trade at a premium to ATM calls, thus violating the rules of put/call parity. But T-bond and T-note futures trade at a discount to the forward price of the CTD, divided by its conversion factor, due to the embedded quality option. As expiration nears, the futures price is expected to rise if rates stay stable, due to the declining time value of the embedded quality option. Therefore, when there is value to the quality option, ATM calls already are effectively in the money by the value of the quality option. This offsets the effects of the low/negative convexity, putting ATM calls and puts at parity.
To properly value out-of-the-money options on T-bond and T-note futures using a Black-Scholes model, we must adjust our implied price volatility for out-of-the-money options according to the changing duration of the futures contract, caused by switches in the CTD, as its price moves to the strike price. Assuming a flat normalized yield vol curve, the implied price volatility of different strike options should have a specific relationship to the modified durations of the contract at those strike price levels. Therefore, out-of-the-money puts should trade at a higher implied price vol than OTM calls, due to the higher modified duration of the futures contract at lower prices. The more likely it is that switches occur and the greater the change in the modified duration associated with these switches, the greater the price vol difference should be between OTM puts and calls.
The price of the futures contract initially will change at a rate based on the modified duration of the contract at current market levels and then at a rate based upon the modified duration of the contract at the strike price if the futures price moves to that level. Therefore, a simple rule of thumb is to use an average of these two durations as the approximate duration to use when valuing options at that particular strike. To illustrate the math (note that these are not current market levels), let us assume that a futures contract is trading at a price of 100 with a modified duration of 10, and the ATM calls trade at a price vol of 10%. Assuming a flat normalized yield vol curve, if the futures contract would have a modified duration of nine at a price of 110, then the 110 calls should trade at a price vol of approximately 9.5% (the ratio of the implied price vol of the OTM to ATM call option should approximately equal the ratio of the average of the modified duration at the current price and the strike price to the modified duration at the current price.)
Estimated Price Vol 110 calls / Price Vol 100 calls = Average Duration / Mod Dur at 100
X% / 10% = (10% + 9%) / 10%
X = 9.5% is the Estimated Price Vol of 110 calls
Using this same logic for pricing out-of-the-money puts in this example, if the futures contract would have a modified duration of 11 at a price of 90, then the 90 calls should trade at a price vol of approximately 10.5%. Therefore, we can see that the price vol curve should be skewed higher for out-of-the-money puts versus out-of-the-money calls.
By adjusting the implied vol of OTM options according to the above methodology, one can still use a Black-Scholes model for pricing options on T-bond and T-note futures, with good results.
* Prices of T-bond and T-note futures are not lognormally distributed when there are switches in the CTD. Because the futures contract 'tracks' the CTD, as yields increase (decrease), the duration of the contract begins to increase (decrease) as it tracks longer (shorter) duration bonds.
* To properly value out-of-the-money options on low/negatively convex T-bond and T-note futures, we must adjust our implied price volatility according to the changing duration of the futures contract.
* Therefore, out-of-the-money puts should trade at a higher implied price vol than out-of-the-money calls, due to the higher modified duration of the futures contract at lower prices.
1. If deliverable bond yields become significantly higher (lower) than the 6% notional coupon, the contract will trade as a mark-to-market forward on the longest (shortest) duration bond, with almost no chance of a switch in the CTD (as was the case with the old 8% notional T-bond and T-note contracts for many years.) The duration of the contract would then behave like a normal positively convex bond.
2. The modified duration of a futures contract is an average duration of deliverable securities, probability-weighted by their chance of being CTD.
3. In this example, the futures price would have to go down to about 89 or up to about 105 before the contract would start behaving like a normal positively convex bond.
This week's Learning Curve was written byJoe Shatz, v.p., futures and options and government strategies groups atMerrill Lynchin New York.