## Tuesday, December 17, 2013

### The Yield Curve is a Call Option

At the zero bound, the forward yield curve should be understood as a Call Option on Interest Rates.

Here are the current forward rates in the Eurodollar market:

The main tweak that we need to make, in terms of understanding the yield curve this way, is to reconsider forward rates in terms of time instead of a rate.  When we do that, we have the three basic elements of the Black-Scholes formula:

For a call option, the determinants are:

S = the current price of the equity
r = the risk free interest rate
sigma = implied volatility of the equity price
K = the strike price of the option contract
T = time to expiration

In the context of forward interest rates, first, we need to assume that if short term rates rise from the current level near zero, they will rise at a relatively constant rate for some period of time.  This assumption is not substantially different from the experience of recent decades.

Then, we need to redefine the determinants:

S = the date of the first rate increase
r = the rate at which interest rates will rise after the first increase
sigma = the implied volatility of the potential movement of the date of the first rate increase
K = the expiration date of a position in Eurodollar futures

Here, the time factor, T, does not effect r, but it would be an implied part of sigma, with the typical decay pattern as the date of the first rate hike approaches.  (Note that, where time in a typical call option would normally be a known quantity, here it would fluctuate with changes in S.)

So, looking at the chart above, we can look at the current set of Eurodollar futures contracts (blue line), and break it apart into the elements of an option contract.  The red line would be the expected payouts at expiration.  The kink in the curve is the date of the first rate hike, the slope of the curve is the rate of the rate increases, and sigma is the level of uncertainty about changes in the expected date of the first increase.  Sigma determines the curvature of the yield curve.

In a normal option contract, premiums would reflect a combination of a time premium and a volatility premium.  As I model this, since our new r variable is not time dependent, there is no separate time premium.  There is only a volatility premium.  So, sigma is what bridges the difference between the expected values at expiration and the current contract prices.

We could imagine a different way to consider the model.  For instance, if the Fed announced credibly that they were going to start raising rates in September 2015 - no sooner and no later - then sigma would go to zero, S would be stationary, and the payout would be determined solely by r.  In that context, changes in market sentiment regarding future rates would be expressed through changing expectations in the rate of rate changes coming out of the zero bound, and the slope of the yield curve would be the only tradable factor.

Now that we have formulated yields in this way, we can more clearly see the appropriate methods for taking a range of positions.

For instance, going forward in time is like going into the money.  The farther into the money you go, the more an option behaves like the underlying security and the less it acts as an insurance policy.  So, if you want to hedge a portfolio that will benefit if interest rates increase but will suffer with low rates, a hedge against low rates might be most effective if part of the hedge was positioned as a long Eurodollar contract (a short position, in terms of the interest rate) in the March to September 2015 range.  This would have a payout similar to a short option, with a limited premium payoff potential and a large potential downside if it moves in-the-money (if the Fed starts raising rates earlier).  You would receive a premium for this hedge as sigma deteriorates over time.

On the other hand, a speculative position for rising interest rates should be positioned farther out on the curve, into late 2016 or later.  A speculative position taken on contracts expiring earlier than that will pay a premium, as it were, as sigma deteriorates.  Right now, the market expects the first rate hike around September 2015.  If you believe the first rate hike will instead happen around June 2015, and you sell September 2015 Eurodollars (which is a long interest rate position), you will have a loss even if your forecast is correct and the rate hike happens in June.  This is because you will have been paying for implied insurance due to the sigma related premiums for those contracts embedded in the current yield curve.

It should be theoretically possible to take several positions along the curve which would net out to give you a pure position with exposure to only one of these factors.  To speculate on a rate hike happening before the current market expectation, A position of 2 short June 2016 contracts and 1 long June 2017 contract should provide no net exposure to changes in slope (r), but this combination would be equivalent to holding a single short position at June 2015.  The advantage of this position is that it would not be exposed to the sigma premium that the raw June 2015 contract would be.  In fact, again in theory, this should be an arbitrage opportunity, since that synthetic position could be hedged against a long June 2015 contract.  The long June 2015 contract would "earn" a premium, as the yield curve slowly meets the expected payoff curve over time.  These positions should roughly net to zero over time, and the speculator would earn a premium of about 35bp over that time.  At current contract margin requirements, that would provide a return of about 25% in less than 2 years.