Bond Economics

Cutting Live Meetings Is Silly

2026-10-30T09:32:59.531-04:00

Fed Chair Kevin Warsh made an unusual announcement this week — one Fed Open Market Committee (FOMC) meeting in 2027, and two meetings in 2026 will not be “live”: they will sit around and yack, but will not change rates. And as a reminder to those who are unfamiliar with the FOMC, they have 8 scheduled meetings per year (roughly every six weeks), although they can make emergency rate moves in between scheduled meetings (usually panicked cuts).

Losing one live meeting is not that big a deal, but cutting it to six meetings is starting to change the complexion of things. The problem is straightforward: central bankers generally like to have smooth rate change regimes. Until the last cycle, the preference evolved towards 25 basis point hikes per meeting (“gradualism”). The last cycle featured more aggressive rate hikes (which surprised me), but they felt that they got behind the curve. That cycle was unusual, and if we have more “normal” inflation behaviour, the Fed is going to be in the position of pre-emptively tightening before inflation gets out of hand (which is what the Bank of Canada is doing now).

The problem with six meetings per year is that if gradualism is followed, that only allows for 150 basis points in hikes a year. That is not going to be enough if the Fed follows its previous track record and doing hefty rate cuts when it panics as a recession hits, reversion speeds need to be faster than that. (The panicked nature of rate cuts in practice means that the meeting schedule is not an issue for rate cut campaigns.) This means that the Fed would likely have to intersperse fatter rate hikes — 50 or 75 basis points. This in turn will cause markets to panic, as they will likely feel that the Fed is panicking about inflation (like in the early 2020s). The end result is the bond market having to price in a greater risk (term) premium, which is an outcome that all financial commentary wrings its hands about. (“Oh no, the fiscal risk premium will go up, raising costs for borrowers!” is standard fodder for the financial press, particularly in the United Kingdom, where the financial press are fiscal risk nutters.)

This does not matter if policymakers think the rate settings are roughly correct, but it is somewhat stupid to premise counter-cyclical policy settings on the basis that we will never get another recession.

The only justification I see for this is that the Fed Chair is tired of being hassled by the President about cutting rates, and this move gets him peace and quiet for a couple months during the year. This is an unusual situation, and will not persist.

Does the Fed Need Time To Discuss Strategy?

The argument for declaring meetings not to be “live” is to allow more strategy discussions. Believe it or not, there exists modern telecommunications tools that allow groups of people to yack from locations all over the world, they just need to connect to the “internet.” Since such meetings have very limited coordination time costs, one could have dozens of such meetings in a year without greatly affecting other duties.

If you are doing macro strategy in the private financial sector, there is an argument for fairly regular strategy meetings. The reason for this is straightforward: you are buying and selling securities (and derivatives) that reprice continuously during the business day (and after hours), and timing buys and sells matters.

This is not true for central bankers. They are adjusting policy variables in response to macroeconomic conditions (and very occasionally, responding to financial crises). Useful economic data for the real economy is generally collected at a monthly frequency, and the data are noisy. Outside of a crisis, central banks are concerned with “supply constraint” measures (like the output gap) that move slowly. If you are a believer in some variant of an output gap, if the output gap indicates that there is a lot of spare capacity right now, in the absence of a crisis, there will spare capacity three months in the future. Unless you subscribe to extremely wacky theories about monetary policy (e.g., Market Monetarism), deviations in the policy rate by 25-50 basis points will have almost no measurable effect on the economy. Outside of a crisis, monetary policy is about adjusting policy variables slowly based on forecasts one- to two-years out. Unless your models stink, those forecasts should not move by much month-to-month, so we should expect policy rate changes to follow sweeping up/flat/down trajectories.

The reason why central bankers need periodic meetings is to discuss whether new data require revising their forecasts, and to make those slow changes that their long-term beliefs require.

Cutting the number of live meetings forces policy away from gradualism, it implies a more hyperactive monetary policy. Policymakers need to jump immediately to the “correct” value. In his “gradualism” speech, Bernanke called “bang-bang control”, which is niche control systems jargon. If you ever spent afternoons as a teaching assistant in a control systems laboratory, you will discover that the description of “bang-bang control” is a literal description of how the systems operate. “Bang-bang control” was a child of 1960s optimal control theory, which was based on the typically bad assumption that you had an extremely good model of the system and know its state exactly. This worked for path-planning problems (getting to the moon was the apex of optimal control), but it breaks down in the presence of uncertainty. Central banking is shrouded in uncertainty, which is why more gradual policy changes are prudent.

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Primer: Dynamic Option Hedging

2026-10-26T09:42:37.533-04:00

This article attempts to provide a non-mathematical explanation of dynamic option hedging. Although the original reasoning to dive into this topic was the hedging of options on interest rate products, for simplicity, it will discuss an equity option on the basis that more people are somewhat familiar with stocks and stock options. It may even be of use to readers who are in similar position the author was in when he entered into finance: mathematical background, but little knowledge of finance. It is possible to plunge into the mathematics of financial derivatives1 without having an intuition for why they behave the way that they do, but that intuition is needed to grasp how they fit into the financial system.

Options

An option can either be traded on an exchange or as an “over-the-counter” legal contract traded between two counterparties. The most basic class of options are the ones considered here, which give the counterparty that “owns the option” the option to buy or sell a specific underlying security at a fixed price on a fixed date in the future (with the counter-party being the party that is selling/buying, it has “sold the option”). Since it is an option, the owner can just let the option expire (presumably because the transaction is not profitable).

There are therefore a few key parameters that determine the value of the option, that are fixed in the contract definition.

Is it an option to buy or sell the security? In equities, the options to buy are termed “calls,” while the option to sell is a “put.” The same terminology holds for fixed income exchange-traded options, but the terminology is wackier for options on swaps.
At what price is the transaction to buy/sell taking place at? This is known as the “strike.”
At what date does the option expire, which is the last (or only) date the owner of the option can make the transaction. Since the contract has no enforceable terms after that date, the option is then worthless. (Technical note: there is an extra parameter that I attempted to absorb into a single mangled sentence — is it possible to exercise the option ahead of expiry? We are going to ignore that possibility, since we are not pricing the option.)

In order to avoid using too much jargon, I will stick with one option configuration in this text. The parameters are as follows. The owner of the option has the right to buy 1 share of a company2 at a date in the future (e.g., 3 months from now) at the price of $100. This is therefore a call option. The “date in the future” is vague since I will not be discussing the value of the option, and all that matters for our purposes is that it is not about to immediately expire. We will assume that the current price of the share is around $100. We will also assume that the stock will not pay any dividends ahead of expiry.

(Background comment. This call option is ultimately more valuable if the price goes up. It matches the stock options given as incentives to employees. It also aligns with the option to prepay a mortgage. The borrower in a mortgage is “short” the debt (has a negative position). The option to prepay allows them to buy back the mortgage debt that owe at a fixed price. Going the other way, a put option benefits from the price falling. They correspond to owning insurance on an equity position.)

Jargon note: It was very difficult to write this article without using the jargon terms “long” and “short.” If you are long something — an equity, or an option — you own it. If you are short it, you have sold it and have a “negative position.” E.g., if you are short one share of the company, you are in the financial equivalent of owning -1 shares. I am also just going to write that we “buy” or “sell” the option, which sort-of works for exchange-traded options, but many options are contracts between two counterparties, so that terminology is stretched.

Valuation of the option is easy at expiry (last time the option can be exercised):

If the price of the equity > $100, then the value of the option is equal to (stock price) - $100. For example, if the stock price is $101, the option is worth $1. We can buy the option for $1, exercise the option by paying $100, and end up with an equity worth $101 (equal to the outlay). If the stock price is $102, the option is worth $2, etc.
If the price is less than or equal $100, the option is worthless — you can buy the equity cheaper (or at the same price) in the market instead of exercising.

The figure below shows this value at payoff graphically. The top panel shows the profits on owning the option, and the profit function consists of two line segments.

The bottom panel shows the profits on either purchasing the stock at $100 or entering into a forward contract to buy one share at $100. If the price ends up at $100, the position breaks even ($0 profit). Otherwise, the profits are a linear function (around the $100 point), with profits increasing as the price goes up, and losses if the price drops below $100.

The distinction between the two instruments is driven by the shape of the profit/loss line. The simple “buy the equity”/”buy the equity forward” strategy has a payoff that is a line, with both profits and losses. The option payoff is piecewise linear3 (which is a subclass of nonlinear functions), and is such that the option cannot end up with a negative value. (You can lose money buying options, but the loss comes from buying them at a higher price than what they expire at.)

The linear/nonlinear payoff distinction made its way into fixed income jargon, where instruments with options were “nonlinear” (such as mortgages with prepayment options) and those without are “linear” (most bonds, swaps, bond futures, etc.) Using this jargon, we are going to be discussing hedging nonlinear instruments with linear ones (which are more widely traded).

Option Valuation

Option-like securities (warrants) had been traded for a long time, even though straight options were ruled out regulations. Based on discussions with senior professors back when I was in control systems academia, people were leaving mathematical fields like control systems to make money trading warrants even in the mid-1960s. At that time, it was a bit of a black art, without “textbook” solutions. However, the Black-Scholes pricing formula arrived in 1973, along with deregulation that allowed trading in options.

The key insight to the Black-Scholes formula — which was not really clear to me when I first picked up The Mathematics of Financial Derivatives by Wilmot, Howison, and Dewynne — is that the value of an option is equal to the expected profits of hedging the option until expiry (where “expected profit” is the mathematical meaning). (The previously-mentioned textbook is the most straightforward option-pricing text that I am familiar with that goes into the mathematics.) Going the other way, the value of selling the option is equal to the expected hedging losses on that position. Although my intuition was that the value would be driven by the probability distribution at expiry, the hedging value is of more importance to financial market participants.

Why Hedge?

Before discussing how hedging works, let us look at why one hedges in the first place. In our example call option, the most natural story is the considerations of a market maker.

A market maker is a market participant that puts themselves in between others. They “make a market” by posting bid and ask prices — prices at which they are willing to buy (or sell) the instrument. If you want to survive as a market maker, you need to make your bid price less than your ask. Your profits are supposed to be made by earning the bid/ask spread. (You also hear bid/offer instead of bid/ask, but I believe bid/ask is preferred in equities and so is heard more often.) For example, say your bid/ask on a security is $99.50/$100.50 (which is pretty honking wide). If someone sells you the security and you can resell it immediately, you will net a $1 round-trip profit (you buy at $99.50, sell at $100.50) over a short time interval. Everyone else faces “transaction costs” associated with the bid/ask spread, that is how the market maker is supposed to make money (not by betting on the prices of the securities).

The problem is that you cannot ensure that you have clients that magically pop up to make trades that instantly close your position, you will end up stuck with inventory (or a “book”). The skill of market making is adjusting your bid/ask prices so that you keep your book balanced.

Returning to our example, imagine you are a market maker where some client sold you the call option just ahead of the market close. Although you liked the business, you are unlikely to close out the position with someone else buying it from you before the close. And you do not want to be stuck with the position before you leave for the weekend. (Yes, a $100 option position is too small to care about, but just add zeros to the position size so that the risk is more plausible.) Since you are a market maker, you do not want to go to another market maker to close the position (since you are eating the transaction cost). Instead, you dig out your options textbook and re-read the chapter on dynamic hedging.

Dynamic Hedging Without Maths

The secret to dynamic hedging options with “linear” instruments is the hedging ratio (often called the “delta”), or the price sensitivity of the instrument. In fixed income, we often to refer to the “DV01” — “the dollar value of one basis point.” That is, how much does the value of an instrument or portfolio change if we have a one basis point (0.01%) shock to the interest rate curve? Since this is an equity option, we are not interested in the interest rate sensitivity, we are interested in how much the value of the option changes if the price of the underlying equity changes (either in percentage terms, or absolute)?

Although we consult the Black-Scholes formula to get this sensitivity, we live in 2026 and use digital computers. We instead consult our portfolio analysis software, which spits out the hedging ratio.

In our case, we will assume that the hedging ratio is exactly 0.5 (which is what happens when the stock price is very near the strike price of $100). That is, the option value will have a price shock that is equal to 50% of the price shock change of the equity for small price changes. For example, if the equity rises by $0.02 (e.g., from $100 to $100.02), then the call option rises in value by $0.01 “all else equal”. But if the equity drops by $0.02, then the value of the option drops by $0.01.

We need to immediately address the “all else equal” part of the statement. There are multiple parameters that are input into the Black-Scholes formula to give the option value. We are assuming that all the other parameters are unchanged in this exercise. (Certain parameters — like the expiry date and the strike — are fixed in the contract, so they cannot change for this option.) The most important “other variable” is the so-called implied volatility — which I will return to below. Raising (lowering) the implied volatility will raise (lower) the price of the option, even if the price of the equity is unchanged. However, we cannot hedge the implied volatility with linear instruments — all we can hedge is the effect of price changes in the underlying instrument.

So what we ultimately do to hedge our long call equity option position (which is on one share) is to sell short 0.5 shares of the equity and call it a day. (To repeat the definition of short sales, this means that you end up with the equivalent of owning -0.5 shares.) The interpretation is straightforward: the call option benefits from the price rising, but the short sale loses money in that case. Since the call option has half the price sensitivity of the short sale, the short sale has half the notional position size.

What If Prices Move?

The price sensitivity measure we calculated were for very small movements. What happens if the price moves appreciably?

Let us say when we get back to the office, and the price of the equity jumped to $101. Our call option position will have risen in value (assuming unchanged implied volatility), but we lost money on the short equity position. If implied volatility was indeed stable, the two positions should have had largely cancelling profits and losses. The dynamic hedging eliminates (most of) the effect of the stock price change.

However, when we re-run our portfolio sensitivity analysis, we discover something troubling. The sensitivity of the short position is roughly constant, but the price sensitivity of the option rose. (I am not going to supply a number because I am lazy.) The sensitivity of option and short position no longer cancel out, so the position now gains (loses) money if the underlying stock price rises (falls). Since we are market makers and not stock speculators, this is unwelcome. So we need to sell short more equity to balance out the option. (The maximum sensitivity of the call option is 1, so the most we could end up shorting is just under 1 share.)

Now that we are hedged again, we go back to doing whatever market makers do all day. Then imagine we are stuck with the option position overnight again.

We discover the next day that the stock round-tripped back to close to $100. Once again, the sensitivity of the option is 0.5. But since we sold short some of the equity to bring back the net exposure to zero when it was at $101, we are now over-hedged (too large a short position). So we need to buy back the amount we bought earlier at $100 to bring the short position to being 0.5 shares.

Although the hedge is back in place, we now can note something. We were forced to sell short a small amount of the equity at $101 to hedge our option position. We then had to buy back that short sale at $100. Since we sold at a higher price than we bought, we made money on that adding and subtracting that extra hedging position.

More generally, we will see this pattern of behaviour if the price is jumping up and down, even if it does not conveniently return to its exact starting point. The hedging transactions generated by hedging a long option position are “counter-trend” moves that end up being profitable if price moves revert back-and-forth.

In an idealised probability setting, the expected profits from hedging will be driven by the “random noise” implied by the second moment of the probability distribution (which the standard deviation (or “sigma”) measures over a sample).4

Hedging a short position in an option is the reverse of the above. If we were short the call option, we then would have had to use a long position in the equity hedge the short option position. When the price went to $101, we would have had to buy to rebalance our hedging ratio. We would then have had to sell at $100 when we rebalanced again — eating a loss.

“Long (Short) Volatility”

The “directional hedge” discussed above (largely) removes the risk associated with the underlying equity changing price. However, the long position in an option is not risk-free: we will instantly gain (lose) money on the option if the implied volatility rises (falls). Our portfolio analysis software will spit out an explanation: there are more risk factors than just the underlying price change. By owning an option, we have direct exposure to the implied volatility (which is a measure of how expensive options are).

Since we make money if implied volatility goes up, we are “long volatility.” (We would be short volatility if the opposite were the case.) Linear instruments have no exposure to the implied volatility, so they cannot be used to hedge volatility risk. The only way to hedge that risk is trading other options, or wacky derivatives like volatility swaps.

There are other risk factors as well, but they are usually second order effects. The value of options decay as they get closer to expiry, so a long option position has a steady “time value” (“theta”) that eats away at its value. Interest rate derivatives with long expiries (like a prepay option 10 years in the future) are harder to model because the value of the option on exercise needs to be discounted by the interest rate — while interest rates are also the underlying instrument. (This is why you do not want to start with fixed income options if you are learning about options.)

Macro Effect of Hedging

Although hedging by itself is not that exciting, if everyone is doing it in the same direction, we can have problems. In particular, if we have a lot of entities with short option positions attempting to hedge them at the same time, they are forced to act in a pro-cyclic fashion: if the price rises, they need to buy more, while they are forced to sell more if prices fall.

In the equity world, problems would occur if a whole bunch of people were stupid enough to sell “portfolio insurance” and try to dynamically hedge their positions at the same time. (Whoops, what happened in 1987?) In the fixed income world, the structure of conventional mortgages means that mortgage issuers are selling prepayment options in incredibly large size to homeowners, and some poor devils attempt to hedge a portion of that position.

Concluding Remark

Option mathematics is complicated and annoying. But if we are managing a portfolio and not a quant developing pricing algorithms, we can just use portfolio analysis software to see the numeric sensitivity of our portfolio to risk factors (like stock prices rising). We can hedge out the “directional risk” of options using instruments without options embedded in them, but we will have to continuously adjust our hedging ratio. We will lose money from those adjustments if we are short options (but we hoped to have covered that from the money we got from selling the options).

The term financial derivative has nothing to do with the use of derivative in calculus. The term refers to instruments whose value derives from having contractual claims on an underlying instrument. For example, a bond future derives its value from bond futures ultimately being a claim on some underlying “cash” bond based on the futures specification. Options are the most interesting (and complex) member of the class of derivatives.

Exchange-traded options are normally board-lot sized (100 shares), but I am using one share to keep numbers simple.

My academic work ended up specialising in the analysis of piecewise linear systems, so this is of sentimental value to me.

There’s a certain amount of internet experts who sneer at the standard deviation and cackle at “sigma.” Yes, the Black-Scholes is not a perfect description of option pricing, but it is close enough that it (and its variants) are used as the option quotation benchmark. The main effect of imperfections is that implied volatility is not flat across strikes and expiries.

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Comments On U.S. Mortgage Hedging

2026-10-09T08:54:09.996-04:00

I ran into some discussions about “doom loops” involving mortgage hedging in the USD rates market. I am quite distant from current market conditions, so I cannot give market advice, nor am I completely on top of the current dynamics. Instead, this article gives a background explanation why mortgage hedging causes difficulties in USD rates markets.

One point to note that this is largely a phenomenon in the U.S. rates markets, the other major developed markets do not have as consumer-friendly mortgages as U.S. conventional 30-year mortgages. The U.S. conventional mortgage market was developed with the help of the government-sponsored enterprises (GSEs), that bought standardised mortgages. As will be discussed in this article, hedging the risk of the prepayment option in conventional mortgages is difficult, and securitisation (placing the mortgages in mortgage-backed securities — MBS) was the way to offload the risk of mortgage hedging for banks.

A conventional 30-year mortgage is amortising — the principal is paid down during the lifetime of the mortgage, which involves a fixed periodic payment. (As opposed to a bullet structure where the principal is only paid off at maturity. The standard bond structure is a bullet.) The interest rate is fixed, which means that we can guarantee that the principal is paid off by maturity at the latest. However, the borrower has the option to prepay the mortgage (in whole or in part), which is extremely convenient for consumers. If mortgage rates drop enough, they can refinance the mortgage at a lower rate. (There are fees associated with doing this, so the interest rate drop has to be enough to cover the fees.) Other countries either do not issue mortgages with fixed interest rates for 30 years (e.g., the effective maximum fixed rate available in Canada is (was?) 5 years, and so the rate had to be reset during a typical 25-year amortisation) and/or there are financial penalties associated with prepayment.

One small aside, which I have noted in the past. The amortising structure means that 30-year mortgages align closer to 7- to 10-year Treasury bullet securities. A 30-year Treasury has much longer duration than a 30-year mortgage, so it is not the proper comparison for mortgage/Treasury spreads.

Why Is Prepayment Bad (For Banks)?

A standard story is that banks “borrow short and lend long” — that is, they (allegedly) borrow at the overnight rate via deposit liabilities to finance long-dated assets like mortgages. The usual story is then the banks get wrecked if the deposit rates rise above the rate of interest on their assets.

This story is based on folk memories of the situation in the 1970s. Interest rates were regulated, and so banks could ensure that they had positive interest rate spreads without doing a whole lot of analytical work. (This led to the classic “3-6-3 rule” for small American savings and loans — pay 3% on deposits, get 6% on mortgages, hit the golf course at 3. It was a system that made so easy to run a bank that pretty much anyone could run one (and pretty much anyone did).

Once interest rates were deregulated, banks could no longer guarantee that short-term funding rates were below asset returns. In the 1970s, the beginning of the Savings and Loan crisis was driven by the rise in short-term rates squeezing small banks. (Fraud and financial speculation in the deregulated industry created far greater financial damage than the initial interest rate shock; the interest rate shock set up the psychology for risk-seeking behaviour that led to the fraud.)

How Do Banks Deal With the Risk?

I have written about this before, but the “borrow short/lend long” story misses three key strategies used to manage duration risk.

Dump the long-dated assets off the balance sheet via securitisations. The MBS market is very large for a very good reason.
Issue longer-dated liabilities (bank bonds, term deposits).
Use derivatives (interest rate swaps, or options on swaps — swaptions). Liability-matching investors are structurally short duration (not enough long-dated bonds), so they are willing to take the other side of derivatives trades where banks offload duration.

This means that dealing with duration/prepayment risk at banks is optional — they can either offload most of the risk, or they can let their treasury desk go wild with hedging strategies if they feel they are sophisticated enough.

How Do (Most) Bond Managers Deal With the Risk?

As an immediate disclaimer, I was involved in Canadian rates management, so my comments here are based on vibes. The easy way to deal with the prepayment risk on MBS (that were offloaded by banks) is to have MBS in your benchmark. If you are a bond fund manager, you are managing against an implicit liability — your benchmark. (Fixed income strategies and analytics are portable because everyone is managing against implicit/explicit liabilities.) So to deal with MBS, you just have MBS in your benchmark. If you have a market weighting in MBS, your aggregate prepayment risk is the same as the index — i.e., not a source of risk.

Since everyone other than index managers attempt to outperform the index courtesy of taking risks versus the benchmark weighting, it is up to the manager to decide how much mortgage risk to take. From what I saw, a fairly common strategies for smaller bond managers was to allocate the risk budget to corporate bonds (or duration decisions), or outsource their MBS allocation to a large institutional fund (if they are something like a pension fund and not competing directly with other fund managers).

How Do Asset Allocators Deal With This?

The standard way large investors manage their portfolios is to have a hierarchical system with an asset mix layer sitting on top of asset-class specific portfolio managers. The distinction between these two levels can be blurred, as senior portfolio managers possibly also have an input into the asset mix decisions, but the two roles are tracked independently. Nevertheless, it is probably a mistake to say that “bond portfolio managers make bond allocation decisions,” rather that is an asset allocator decision (although the asset allocators may be those portfolio managers wearing different hats).

At the asset mix level at firms that have actuarial liabilities, we tend to see that actuarial liabilities typically have a longer duration than their fixed income portfolios on two grounds. Firstly, it is “expensive” to have too high a bond portfolio weight, since expected returns are lower. Secondly, there is not enough long duration supply to allow many large investors to duration match (as was run into when the U.K. mandated pension fund liability matching).

Amortising mortgages are a bad fit to match against the cash flows needed to immunise required payments to pensioners whose life spans get distressingly long courtesy of improvements in medical science. As such, it is unlikely that anyone at the asset mix level is going to say “we should specifically load up on mortgages.” This means that decisions about mortgages get left at the level of portfolio managers.

Who’s Left?

If we look through the previous categories of investors who invest in fixed income, we have covered most of them in terms of the number of management firms. However, we are missing a small number of fixed income investors who do care about mortgage prepayment risk — and those investors are quite large. Given that influence on the market is weighted by portfolio size, the point about it being a small group may seem pointless. This is just me being pedantic about writing style — mortgage hedging is not really a major concern for “bond investors,” rather the subset that is “mortgage hedgers” (which is tautologically true).

Historically, the GSE’s were the largest mortgage hedgers, but there are also the big banks, investment banks, and mortgage (hedge) funds that have a mismatch between mortgage assets and their (implicit) liabilities. My understanding was that this activity was dialled down after the Financial Crisis (which saw a couple of the GSE’s put into receivership), but I am not on top of recent developments.

The Hedging Problem

Interest rate risk — in the absence of optionality — is the easiest/second easiest risk to hedge (foreign exchange hedging might be easier). All fixed income assets have a pricing sensitivity to the interest rate curve. The first order hedge to interest rate risk is to determine each portfolio asset’s sensitivity to a 1 basis point (0.01%) rise in rates across the curve (“DV01” — dollar value of .01%) and then add each DV01 value up, giving the DV01 of the portfolio.1 For example, if your have a portfolio of bonds with a DV01 of $-10,000, you expect to lose about $100,000 if there is a parallel rise of interest rates of 10 basis points across the curve. You then pick some hedging instruments (bond or short rate futures, swaps, or benchmark bonds) and then just do a trade to bring your DV01 back to its liability benchmark (e.g., the DV01 of an index portfolio that has the same market value as your portfolio).

If you want to be fancier, you decompose your interest rate exposure across the curve into maturity buckets, and you handle each independently — “key rate duration.”

DV01 and “duration” are related concepts, but in general, people refer to “duration risk” even when their concern is DV01. The advantage of duration is that it is return sensitivity for the portfolio (measured in percent) that is invariant to the size of the portfolio. However, the people managing portfolios care more about the dollar value of risks, and DV01 is a better fit. Duration is also not well defined for anything other than bonds, while DV01 can be used for any asset.

The ease of hedging results from the fact that market value changes are very close to linear with respect to shocks in interest rates. You will lose roughly twice as much money if rates rise 2 basis points instead of one, and you will get a profit roughly equal to the magnitude of the DV01 if rates fall one basis point instead of rising.

Fixed income options — and bonds with embedded options, such as prepayable mortgages — are not linear with respect to interest rate shocks. In the case of mortgages, they have the ugly property that the duration extends (the magnitude of the DV01 goes up) if interest rates rise. This means that if interest rates change by more than insignificant amounts, the actual gain/loss is no longer well approximated by (initial DV01)×(yield change).

(Vanilla option-free bond durations also change as yields change, but the duration change is beneficial. In the case of rising rates, this means that the duration drops very slightly as the yield goes up, so you lose less money than predicted by multiplying the starting DV01 times the yield change in basis point. Going the other way, the duration extends as yields fall, so you make more money than predicted. In practice, you need a rate move of 50 basis points to really notice this effect. This means that bond returns outperform what is predicted by the linear sensitivity — there is a positive second order return sensitivity that is called convexity. Since bonds with embedded options always underperform what is predicted by linear extrapolation, practitioners will say that they have negative convexity.)

Hedging Embedded Options

There are two strategies for hedging the risks of embedded options.

Buy interest rate options that roughly align with the embedded options in your mortgage portfolio.
Dynamically hedge the risk.

Buying options to cancel out the risk from mortgage prepayments is generally expensive, but it does reduce other risks. However, options trading is zero sum: for every buyer, there needs to be a seller. Although there is some structural demand for selling volatility due to dubious academic finance theories,2 there is not enough to cover the demand created by large mortgage portfolios. So, in aggregate, the hedgers are forced to dynamically hedge at least part of their aggregate option risk.

Dynamic hedging requires periodically adjusting your hedging ratio (DV01) so that it risk is neutralised. (This is done at a relatively high frequency, but typically not intraday.)3 To avoid excessive transactions, there will be a “dead zone” (yay for control systems terminology) where small hedging mismatches are ignored. However, this implies that if rates rise, the hedgers need to take actions that are equivalent to selling bonds (sell bonds, sell futures, pay fixed in swaps). Which means that they are doing a transaction that reinforces the current trend in yields.

Note: This description is probably too brief, so I expect that I will do a “dynamic hedging” primer as a follow up.

We therefore end up with the situation if the U.S. bond market tanks, the mortgage hedgers are piling on sell orders. Which reinforces volatility — the U.S. market has the tendency to adjust faster than other developed markets as a result.

Although that sounds like a plot right out of Monster Chiller Horror Theatre (now available on Netflix), we typically do not see interest rates march off to infinity. Sooner or later, valuation matters, and people step in to halt the rout.

As such, I do not see this as a “doom loop,” rather the reality that interest rate adjustments occur in on accelerated time span. We might lose a hedge fund or two, but that’s what happens when markets move.

Concluding Remarks

The structural need to hedge prepayment risks means that the USD rate market is a bit livelier than other markets when rates break out of a trading range. Although this is entertaining when it happens (if you are not on the wrong side of the trade), the effect typically burns out, so do not expect this effect to drive interest rates to infinity.

Remember that for bonds, “yield up” -> “price down.” So the DV01 is a negative number if the shock is a 1 basis point rise in rates. However, everyone orally gives DV01 as a positive number, which is the magnitude of the DV01. Some shops might give up and define DV01 as the dollar value of a 1 basis point drop in rates. However, the DV01 that people care about in practice is the DV01 gap between their portfolio and the index, which swings back and forth between positive and negative depend on duration views.

Modern portfolio theory says that the optimal portfolio manager generates steady outperformance. Do you know what generates steady outperformance? Strategies that effectively “sell volatility”: selling options, buying short-term credit products, carry trades. These strategies make money until they blow up, typically because everyone herds into the because everyone is judged on the basis of wanting steady outperformance.

If rates go in one direction then return to their starting point, dynamically hedging an option generates losses. In our example, if rates first rise, the hedger needs to sell to get back to neutral DV01. But if rates go back to where they were, they will need to undo that sale to return to neutral — which implies that they need to buy at a higher price than they sold. The reason why the Black-Scholes model was a big deal is that it related the price of an option to the cost of hedging the option — the more rates jump back and forth randomly, the greater the hedging losses. So, option prices depend upon the volatility of prices.

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