# Extracting the OAS out of an MBS?

I was reading about OAS and I'm wonder how one could "extract" (or "capture") the OAS out of a product by hedging out all the other risks.

One of the explanations I got from OAS from a book was this: "Essentially a positive OAS implies that once hedged against the forward LIBOR rates, a security will have positive returns".

This seems to jive with an explanation given by a member here: Interpretation of OAS on MBS

My question is, how does one actually go about hedging? Assume we live in a perfect model world, and I own an MBS product with 100 OAS. Which instruments would I use to hedge my product in order to try and extract that 100 basis points of OAS on average?

When calculating OAS, all the equations I've seen use the 1 period forward rate to discount paths forward, so would you hedge with the 1 period discount rate? In other words, at the start of each month would you sell short a 1 month libor contract?

• You could hedge USD interest rates risk by shorting cash U.S. treasuries, or exchange-listed treasury futures, or ED futures, or interest rate swaps (you pay fixed and receive floating). Your model should tell you how sensitive your risky instrument is to interest rates and therefore the notionals of your hedges. – Dimitri Vulis Mar 29 at 17:29
• @DimitriVulis so if each day, I came in calculated the effective duration of my bond holding OAS constant, and just picked say cash treasuries to hedge and rebalanced each day, I should in the long run approximately earn the OAS? Do I have that right? – James Riddle Mar 29 at 17:39
• The reason I'm trying to be precise, is that the other answer I linked in my question called it "free money" at least in a model/theoertical sense. And I'm just trying to understand how exactly one would get that "free money" assuming the world behaved like the model – James Riddle Mar 29 at 17:45
• Using one duration figure (effectively assuming that all interest rates move in parallel, i.e. you can hedge your sensitivity to 3mo libor with 30y treasuries) may leave you with P&L from rates moving not in parallel (e.g. curve's slope changing). You can use key rate durations to figure out the hedge notionals, or dollar sensitivities to these key rates. (easier to understand in my humble opinion). – Dimitri Vulis Mar 29 at 18:04
• As a general principle, spread never earns "free money", rather it compensates for some kind of risk. For examlple, maybe you definitely earn it if you hold to maturity, but if you must unwind early, then you get only a "fire sale" price, and not what you consider fair price (think retail CD). And there are many other kinds of obcure risks. – Dimitri Vulis Mar 29 at 18:12

how does one actually go about hedging?

Your OAS model determines your hedging instruments, hedge ratios, and hedging (model) risk. If your OAS model calibrates to the LIBOR swap curve and swaption vols (which is common), then you can generate hedge ratios by computing key rate durations of the underlying swap and swaption instruments.

For example:

Hedging Instruments/Calibrated model Instrument Duration MBS Duration Hedge Ratio
Zero Coupon LIBOR Swap 5.0 2.5 (key rate) 1/2 x 5yr Swap
Zero Coupon LIBOR Swap 10.0 1.0 (key rate) 1/10 x 10 yr Swap
LIBOR Swaption (5yr expiry into 10yr Swap) -1.0 -3.0 (vega) 3x Swaption

The table above provides a stylized example of a duration neutral hedging program. It employs simplifying assumptions like ignoring key rate exposure from swaptions.

The hedge ratios inherit model risk from the OAS model which embeds both interest rate model and prepayment model risk. Such a simplified approach would expose the hedger to historical biases in the prepayment model and distributional assumptions of the interest rate model.

Hedge ratios become stale after several days due to (possibly large) movements in the market prices of the underlying calibrated instruments and the specific MBS instrument we are hedging, which forces us to recalibrate the OAS model to current market prices, recompute hedge ratios and rebalance the (synthetic mortgage) hedging portfolio accordingly.

"Essentially a positive OAS implies that once hedged against the forward LIBOR rates, a security will have positive returns": I don't believe this statement is correct. Essentially, the OAS is a statement about expected excess returns (after hedging out duration and volatility risk as described in the comments above) and says little to nothing about what return can be realized on any one realized path of interest rates.

Reviewing the construction of OAS: we generate a number of interest rate paths using a term structure model, project cash flows for the relevant bond along each of these paths, present value these cash flows using discount factors extracted from the underlying yield curve plus a spread to get a price for each path, and then average all these prices to see if we match the market price. If not, we iterate again until we find the spread that achieves a match.

Now, if we pick a path for which the price (after discounting with the OAS) is below the market price and it somehow happens that rates (and volatility) evolve exactly along this path then in fact we're guaranteed an excess return less than predicted by the OAS.

While I think my critique above of being able to extract the OAS is spiritually correct, it wasn't completely satisfying and here's an attempt to make it more rigorous. We'll follow Tuckman's Fixed Income Securities (3rd edition, pp. 222-224). Denote the market price of the security at time $$t$$ by $$P_t(x, OAS)$$, where $$x$$ is a risk-factor (say, interest rates). Then, under the assumption of risk neutrality we have ($$r$$ is a short-term rate):

\begin{align} E[\frac{dP}{P}] &= (r + OAS)dt \\ dP &= (r + OAS)Pdt + \frac{\partial P}{\partial x} (dx - E(dx)) + \frac{\partial P}{\partial OAS} d(OAS) \end{align}

The first equation says that the expected return under risk-neutral probabilities is the short-term rate plus the OAS. The second equation says that the return of the security along a realized interest-rate path is equal to a carry-roll-down component, a component due to changes in interest rates, and a component due to OAS changes.

Now, if we hedge out $$x$$ and finance the position at $$r$$ then the return is equal to the OAS plus a contribution equal to change in OAS times the spread duration.

To summarize, while the expected return of the hedged position under the risk-neutral process is equal to the OAS (because the expected OAS change is 0), the specific return along different realizations of $$x$$ depends on changes in the market pricing of the OAS risk premium. This makes sense because even though a hedge may synthetically replicate the interest rate and volatility exposures of an MBS, it will not account for changes in OAS. In short, hedging does not allow you to extract the OAS every single time, just "on average."

• wgajate is correct below. You are forgetting the options that are supposed to hedge the discrete path that you mention that generates less than the expected oas spread. – Edward Watson Apr 3 at 16:12
• By construction, the hedge ratios obtained from the OAS model correspond to some sort of average over multiple paths. Using these hedge ratios doesn't guarantee you the OAS along a specific path. From what I understand, wgajate is speaking about how to use the OAS model to hedge, he's not saying anything about realizing this OAS along a specific interest rate path. – Sharad Apr 3 at 16:25
• by hedging you are attempting to realize the oas. The option is capturing the move to that sub-optimal path and partially offsetting it's effect on the return. OAS is an adjusted Zspread where the Zspread minus the option cost is the option adjusted spread after hedging your key rate durations and negative convexity with options. Also receiving swaps and selling options is synthetic mortgage. – Edward Watson Apr 3 at 16:37