MR 12 – Arbitrage Pricing with Term Structure Models

When yield curve moves are broken down with principal components analysis, a handful of factors explain almost all of the variation across maturities. That empirical fact is what makes term structure modelling practical. If a few random factors drive every rate, then a modeller can write down how those factors behave over time, add the requirement that no riskless profit is available, and from those two ingredients obtain prices and hedge ratios for bonds of every maturity and for any security whose cash flows depend on rates.

Securities whose payoffs are tied to interest rates, such as bond options, caps or swaps, are called interest rate contingent claims. A term structure model is the combination of three jobs: making an assumption about how the rate factors evolve, pricing fixed income instruments that are consistent with that assumption, and computing how sensitive those prices are to the factors so that positions can be hedged.

This lesson works in the simplest setting that still shows every moving part. There is one factor, the six-month interest rate, and it can only move up or down over each six-month step. Two kinds of security appear throughout. The first kind is a set of bonds whose market prices are taken as given; these are the underlying securities. The second kind is the contingent claim we want to value, and its price must come out of the given bond prices through an arbitrage argument.

Where the lesson is heading

A tree fitted to market prices can value claims the market has not priced directly. It also yields the option-adjusted spread (OAS), which measures how far a market price sits from the model price and, if the model is right, the extra return a holder can expect.

Start with two market spot rates, both semiannually compounded: 2% for six months and 2.15% for one year. Fixing these two rates is the same thing as fixing the prices of a six-month zero-coupon bond and a one-year zero-coupon bond, so these two bonds are our underlying securities.

Next comes the modelling assumption. Six months from today the six-month rate will be either 2.50% or 1.50%, each with probability one half. A diagram with exactly two branches out of every node is a binomial tree. Each column of the tree is a date: today is date 0 and six months from now is date 1. Each possible outcome on a date is a state of the world. The higher rate, 2.50%, is called the upstate, and the lower rate, 1.50%, is called the downstate.

Price trees for the two zeros

Every price tree is built from right to left, starting at maturity. Take $1,000 face value throughout and leave currency symbols off the tree nodes. The six-month zero matures on date 1 and pays 1,000 in both states, so today it is worth that amount discounted for one half-year at 2%. Notice that as you move right along a price tree, the remaining life of the security shrinks: the six-month zero on date 0 is simply a maturing bond on date 1.

The one-year zero pays 1,000 on date 2 whatever happens. On date 1 it has six months left, so it is discounted at whichever six-month rate prevails in that state. Its date-0 price, by contrast, comes straight from the 2.15% one-year spot rate.

The last row matters for what follows. The six-month zero is riskless over the first step, while the one-year zero is risky because its date-1 value differs by 4.902 across the two states. That risky bond is the only source of rate exposure available for building replicating portfolios in this first step.

Using the probabilities of one half, the average value the one-year zero will have on date 1 is 0.5 × 987.654 + 0.5 × 992.556 = 990.105. Bringing that average back to today at the date-0 six-month rate of 2% produces what is called the expected discounted value.

The market price of that same bond is 978.842, which is 1.460 less than 980.302. The gap is expected, because investors do not set prices by a measure that ignores risk.

Why the gap is a risk penalty

Over the coming six months the one-year zero will be worth either 987.654 or 992.556. Compare it with a hypothetical claim that pays 990.105 on date 1 with certainty. Such a riskless claim would be worth 990.105 / 1.01 = 980.302 today. An investor who dislikes uncertainty prefers the sure claim, so the risky bond with the same average must trade below 980.302. The market expresses that dislike by pricing the bond at 978.842. In other words, the one-year zero carries a risk penalty, and investor risk aversion is what creates it.

This gap is why options cannot be valued by simple averaging, and why a different set of probabilities, the risk-neutral ones, is needed.

Consider a European call that expires on date 1 and gives the right to buy 1,000 face value of what will then be a six-month zero for a strike of 990. The bond in question is the one-year zero today. If the rate rises to 2.50%, that bond is worth 987.654 and nobody pays 990 for it, so the call is worthless. If the rate falls to 1.50%, the bond is worth 992.556 and the call pays 992.556 less 990 = 2.556. The option is a contingent claim in the plainest sense: its payoff is a function of the bond value, and the bond value is a function of the rate.

Matching the payoff in both states

Pricing by arbitrage means finding a portfolio of the underlying bonds that reproduces the option payoff in every state and charging what that portfolio costs. Because the payoff depends on the rate, the portfolio has to work in both states at once.

Let F^0.5 be the face amount of six-month zeros and F¹ the face amount of one-year zeros held on date 0. On date 1 the six-month zeros mature and are worth their face, while each dollar of face in the one-year zeros is worth 0.987654 in the upstate and 0.992556 in the downstate. Setting the portfolio value equal to the call value in each state gives two equations in two unknowns.

Subtracting the first line from the second isolates F¹: 0.004902 times F¹ equals 2.555831, so F¹ = 521.4375, and the first line then gives F^0.5 = -515.0000. The replicating portfolio is long 521.4375 face of one-year zeros and short 515.0000 face of six-month zeros. Its cost today, using the date-0 bond prices, is the option price.

Why arbitrage pins the price

The law of one price says two claims with identical payoffs must cost the same, and arbitrage enforces it. If the call traded below 0.504, a trader could buy the call, short the replicating portfolio, pocket the difference and owe nothing on date 1, since the two positions cancel in both states. If it traded above 0.504, the trader would do the reverse. So, the only price that rules out riskless profit is 0.504.

Averaging would mislead badly here. Discounting the average payoff, 0.5 × 0 + 0.5 × 2.555831, at 2% gives 1.2653, more than double the arbitrage price. The call is priced lower because its holder bears rate risk, and that risk discount is inherited from the one-year zero: the replicating portfolio is long that bond, and the bond already trades below its expected discounted value. The framework is general. A five-year bond also evolves with the six-month rate, so an option on it could be priced the same way.

Look back at the replication equations and the cost calculation. Nowhere did the probability of the upstate appear. The portfolio was chosen so that it equals the option in both states, and that requirement says nothing about how likely each state is. Whether the chance of a rise is 20%, 50% or 80%, the same face amounts of 521.4375 and -515.0000 do the job. With the composition fixed and the bond prices given, the cost of the portfolio, and therefore the option price, cannot move with the probabilities.

The indirect channel

That can feel wrong: if a rise to 2.50% became ever more likely, a right to buy the bond at 990 ought to be worth less. The resolution is that probabilities do affect the option, but only through the bond prices taken as inputs. Suppose the market suddenly assessed a higher chance of the upstate. The one-year zero, which is worth less in that state, would fall in price today. The call’s replicating portfolio is long the one-year zero, so its cost would drop and the call price would drop with it.

A derivative feels the probabilities only by way of today’s bond prices. Once those bond prices are fixed, no probability is needed to find the arbitrage price. So two investors with opposite views on rates still agree on a bond option’s value, provided they share the tree of possible rates and the bond prices.

The probabilities of one half describe what the modeller believes about the real world, so they are called true, or real-world, probabilities. Weighted by them, the one-year zero discounts to 980.302, above its 978.842 price. Risk-neutral pricing replaces them with a different pair of probabilities that make expected discounted value hit the market price exactly. Once the tree is adjusted, any number of claims can be valued by straight discounting, with no replicating portfolio needed.

Call the risk-neutral probability of the upstate p and that of the downstate 1 – p, and require the one-year zero to be fairly priced by discounted expectation.

Solving the linear equation gives p = 0.8009 and 1 – p = 0.1991. Under these weights the one-year zero is worth exactly 978.842, and the call from the previous section comes out at its arbitrage price.

Reading the difference as drift

Under the true probabilities the six-month rate is as likely to rise by 50 basis points as to fall by 50, so its expected change, or drift, is zero. Switch to the risk-neutral weights, 80.09% on the rise and 19.91% on the fall, and the expected change becomes 0.8009 × 50 less 0.1991 × 50 = 30.09 basis points. Risk-neutral pricing therefore behaves as if rates drift upward. Higher expected rates mean lower expected bond values, and that is exactly how the risk penalty in the one-year zero’s price is absorbed into the probabilities rather than into the discount rate.

The general recipe has two steps. Calibrate first: choose probabilities under which every underlying security is worth its discounted average. Then value any claim by discounting its average payoff with those calibrated weights. In the one-step setting just one underlying bond is risky, the one-year zero, so a single equation fixes p.

Matching 0.504 by two routes is not luck, and a four-step argument explains why.

Step one: arbitrage prices ignore risk preferences. A security is priced by arbitrage when some portfolio of underlying securities reproduces its cash flows. If everybody agrees on how the underlying prices can evolve, everybody agrees on that portfolio, and so on its cost. A cautious retiree and a professional gambler would both conclude that the call is reproduced by holding 521.4375 face in one-year zeros while shorting 515.0000 face in six-month zeros. Their attitudes to risk never enter. The bond option qualifies under the assumed rate process; an individual common stock usually does not, since no bond portfolio can mimic the firm-specific swings in its value.

Step two: imagine a risk-neutral economy. Picture a second economy with the same current bond prices and the same possible future six-month rates as the real one, but populated by investors who are indifferent to risk. Such investors price every security by expected discounted value. For the one-year zero to be priced at 978.842 in that economy, the upstate probability there must be the p that solves the calibration condition, namely 0.8009. So the risk-neutral probabilities are simply the probabilities that hold in the imaginary economy.

Step three: price the option in the imaginary economy. In that economy the call, like everything else, is worth its expected discounted value using 0.8009 and 0.1991, which is 0.504.

Step four: carry the price back. By step one, given bond prices and possible rates, the option price does not depend on attitudes to risk. The real and the imaginary economies share bond prices and possible rates and differ only in attitudes to risk. So the option must have the same price in both, and the real-economy price is also 0.504. More broadly, any derivative that can be replicated may be valued in the real economy by expected discounted value under risk-neutral probabilities.

Adding a second six-month step raises a design choice. In a recombining tree, a rise followed by a fall lands on the same node as a fall followed by a rise, so the up-down and down-up paths share a state. In a nonrecombining tree those paths end at different nodes. Nonrecombining dynamics can be economically sensible, for example when volatility itself depends on the path taken, but they are usually avoided because they are hard, and sometimes impossible, to implement.

Why growth rules out nonrecombining trees

Without recombination the number of states doubles every step: two after six months, four after one year, and 2^N after N semiannual steps. Pricing a 10-year security needs 20 steps, so the final column alone holds 2²⁰ = 1,048,576 nodes, well over 500,000. A 20-year security needs 40 steps and more than 500 billion final nodes. Since practical trees also use much shorter steps, such growth defeats even modern computers. A recombining tree, by contrast, has only N + 1 nodes in its final column. Effects such as state-dependent volatility can still be modelled, but they must be built in more efficient ways than an exploding tree.

Naming the nodes

A simple labelling convention helps as trees grow. Dates are the columns, numbered from 0 on the left. States are the rows on each date, numbered from 0 at the bottom. In the two-step rate tree used below, the node at date 2, state 0 carries a rate of 1.00%, and the node at date 1, state 1 carries 2.50%.

Extend the rate tree one more step. From 2.50% the six-month rate can go to 3.00% or 2.00%, and from 1.50% it can go to 2.00% or 1.00%. A third underlying security now enters: a 1.5-year zero priced off a 1.5-year spot rate of 2.25%. The task is to find the risk-neutral probability q of an up-move on date 1, under the simplifying assumption that q is the same from both date-1 states.

Nodes that can be written down at once

On date 3 the bond matures at 1,000 in every state. On date 2 it is a six-month zero, so in states 2, 1 and 0 it is worth 1,000 discounted for a half-year at 3%, 2% and 1%. On date 0 the given spot rate fixes its market price.

Solving for q

The two date-1 prices, P_1,1 and P_1,0, are unknown, but each must equal the discounted risk-neutral expectation of its date-2 values. They in turn must reproduce the date-0 price when weighted by 0.8009 and 0.1991, the first-step probabilities already found.

Solving gives q = 0.6520, so 1 – q = 0.3480. Putting q back into the first two lines completes the bond’s price tree, shown in Table 2.

From date 1 to date 2 the true drift is again zero, while the risk-neutral drift is 0.6520 × 50 less 0.3480 × 50 = 15.20 basis points.

What the completed tree reveals

On date 1 the 1.5-year zero has exactly one year left, so its two possible values imply the one-year spot rate six months from now. Solving 974.735 = 1,000 / (1 + r/2)² gives r = 2.5754%, and solving 984.430 = 1,000 / (1 + r/2)² gives r = 1.5754%. That may be surprising, since nothing was assumed about the one-year rate. It follows because this is a one-factor model: once the six-month rate process and today’s spot rates are fixed, arbitrage determines every bond price on every node, and with them every longer rate. A modeller who has written down the short-rate tree is not free to add separate assumptions about the one-year rate.

Two further points hold for any number of steps. Each new date needs assumed values for the short rate and new risk-neutral probabilities chosen to recover a given set of bond prices. And the replicating portfolio for a derivative changes with date and state: the mix that works on date 0 generally differs from the one needed on date 1 in state 0, and again from the one in state 1. Rebalancing as time passes and rates move is called dynamic replication. It contrasts with static replication, such as reproducing a coupon bond with an unchanging pair of same-maturity coupon bonds.

A constant-maturity Treasury (CMT) swap exchanges, at each payment date, the yield on a Treasury of a fixed maturity for a fixed rate. Consider a stylised one-year version on 1,000,000 notional, struck at 2%, with payments every six months and the floating leg set to the six-month yield. In practice, CMT swaps are usually written on the yields of liquid two-year, five-year or 10-year Treasuries, but the six-month version keeps the arithmetic on our tree.

A semiannually compounded six-month yield is just the six-month spot rate, so the payoffs come straight off the rate tree in Figure 3. On date 1 they are 1,000,000 × (2.50% less 2%) / 2 = 2,500 in state 1 and 1,000,000 × (1.50% less 2%) / 2 = -2,500 in state 0. On date 2, in states 2, 1 and 0, they are 5,000, 0 and -5,000.

Rolling back through the tree

Valuation runs from the final date backwards. The value on a date-1 node is the risk-neutral expected value of the date-2 payoffs, discounted at that node’s six-month rate, plus the payment made on date 1 itself. The date-0 value then discounts the date-1 values at 2% using 0.8009 and 0.1991.

Why the swap is worth more than zero

A value of 3,702.11 can look odd. The payments are zero when the rate is 2%, and under the true probabilities the expected rate on every date is exactly 2%. The point is that the arbitrage price is set by the risk-neutral probabilities, not the true ones, and under the risk-neutral process rates drift upward, which favours the side receiving the CMT yield. If the same rollback is done with 0.5 on every branch, the result is -6.07, close to zero, as intuition based on true probabilities would expect.

Option-adjusted spread is the single number that, when added to every discount rate in the model, makes the model value equal to the observed market price. It is a widely used gauge of relative value, because it expresses how far a security’s market price sits from its model price as a yield spread.

Suppose the CMT swap trades at 3,699.18, which is 2.92 below the model price of 3,702.11. Its OAS is 10 basis points, which can be confirmed by discounting at the tree rates plus 10 basis points: the date-1 rates of 2.5% and 1.5% become 2.6% and 1.6%, and the date-0 rate of 2% becomes 2.10%.

One rule must be kept straight. The spread shifts the rates used for discounting, and only those. Cash flows are still computed from the unshifted tree, so the swap payments remain 2,500, -2,500, 5,000, 0 and -5,000.

Cheap, rich and the return an OAS implies

A positive OAS means the market price is below the model price, so the security trades cheap. A negative OAS means it trades rich. There is a second way to see the same number. Discounting expected values at a given rate per period is the same as saying the security earns that rate per period. So, under the risk-neutral process, a security with an OAS is expected to earn the short-term rate plus the OAS.

The fairly priced swap earns 1.00% over six months, half of the 2% short rate. The cheap swap earns 1.05%, half of 2.10%, which is the short rate plus the 10 basis point OAS.

In a one-factor model the price of a security at time t depends on the factor r, often a rate, and on its OAS, so it can be written P(t, r, OAS). A first-order Taylor approximation splits a small price change into the part due to the factor, the part due to time, and the part due to the spread. Each partial derivative holds the other two inputs fixed.

Divide by P to work in returns and take expectations. Because an OAS valuation treats the spread as constant over the life of the security, the expected change in OAS is set to zero.

Under the risk-neutral process a security priced exactly by the model is expected to earn the short rate, r₀. A security with a nonzero OAS is discounted at the short rate plus the OAS, so its expected return rises by the same amount.

The return decomposition

Use the expected-return equation to replace the time derivative term, substitute the general expected return, and rearrange. Multiplying through by P then gives the dollar version.

Read left to right, the three pieces are carry-roll-down, the gain or loss from rates moving differently from their expected path, and the gain or loss from a change in spread. For simplicity, coupons and other direct cash flows are left out of this decomposition. When a model has genuine predictive power, a security’s OAS tends to converge to zero over time, meaning its price drifts toward model value.

Using the decomposition for relative value

If the model is right, a long position in a cheap security wins twice: it collects the OAS during periods when the price does not converge, and it gains its OAS sensitivity times any convergence toward zero. A relative value trader who spots a cheap or rich security takes the position and hedges away all factor risk, so that the partial derivative of P with respect to r is zero. What remains is the short rate, the OAS and any convergence. If the trade is financed by borrowing P at r₀, the expected return is just the OAS plus the convergence gain. Cash used or generated by the hedge earns the short rate, and any OAS on the hedge also flows into P&L. A risk premium for bearing factor risk could add a further term to these equations, but with factor risk hedged out such terms cancel.

Nothing in the method requires six-month steps. With a step of Δt years, the tree carries rates of term Δt and every node discounts over that interval: a rate r for a period of Δt means dividing by 1 + rΔt. With monthly steps Δt = 1/12, or .0833, and a one-month rate of 2% discounts by dividing by 1 + 0.02/12.

Why go smaller

Two practical reasons push the step below six months. First, real portfolios seldom pay on an exact six-month grid from the valuation date, and a step of a month, a week or a day keeps every cash flow close to some date on the tree. Second, a tree in which the short rate takes two values after six months and three after a year is far too coarse to describe the distribution of rates; finer steps fill the tree with enough possible rates to price claims accurately.

Why not always go smallest

Finer steps cost something: numerical issues need closer attention and the calculation may become too slow. The right step depends on the job. For a 30-year callable bond, weekly or even monthly steps may already give a realistic enough distribution of rates to produce reliable prices. A one-month bond option, on the other hand, needs a far smaller step to be priced with any precision. Nor must the step be constant: well-built implementations vary it across dates to balance realism against computing cost.

The Black-Scholes-Merton (BSM) analysis of stock options rests on two assumptions: the stock price follows a specified random process, and the short-term interest rate is constant. Under those assumptions a mix of the stock and short-term bonds replicates the option, so arbitrage sets the option price equal to the cost of that mix.

Three problems with modelling the bond price directly

Now carry that logic over to an option on a five-year bond, beginning with an assumed process for the bond price. Three difficulties appear. The bond price must converge to face value at maturity, while a stock price faces no such pull. Because of that pull, the volatility of the bond price has to shrink as maturity approaches, so a constant volatility, often acceptable for a stock, fits a bond poorly. And stock volatility is large compared with short-rate volatility, which is why holding the short rate fixed is tolerable for stock options; for a bond it is internally inconsistent to let the bond price move randomly with rates while holding the short-term rate constant.

These objections are why researchers model the random evolution of interest rates rather than of bond prices. Bond prices then pull to par on their own, price volatility falls to zero at maturity, and no rate is assumed to be fixed.

Which rate to model

Choosing the rate is the next question. Modelling the five-year spot rate alone is not enough, for two reasons. A five-year coupon bond also depends on all the shorter spot rates that discount its coupons. And an option on that bond soon depends on a bond with less than five years left, so its relevant rate keeps changing. Pricing bond derivatives therefore requires assumptions about the whole term structure. In a one-factor model, as this lesson has shown, a process for the short-term rate is enough to generate the entire curve.

So there are good reasons to go beyond BSM for fixed income. Even so, simplified BSM variants remain in everyday use for pricing and hedging several kinds of fixed income derivative, because they are simple.