> For the complete documentation index, see [llms.txt](https://laurence-wilse-samson.gitbook.io/textbooks/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://laurence-wilse-samson.gitbook.io/textbooks/financial-economics-claims-prices-holders/part-i-foundations/chapter_03_pricing_equation.md). # Chapter 3: Value, Arbitrage, and the Pricing Equation *Part I: Foundations — Financial Economics: Claims, Prices, and Holders* *** ## Opening Episode: Negative Twenty Billion Dollars, March 2000 On Thursday, 2 March 2000, 3Com Corporation sold shares in its subsidiary to the public. 3Com made networking equipment — modems, network cards, the unglamorous plumbing of corporate computing. Its subsidiary, Palm Inc., made the PalmPilot, the handheld organizer that was, briefly, the most desirable object in American business travel. 3Com sold about five percent of Palm in the offering, priced at $38 a share, and kept the rest. The retained stake was not going to stay retained. 3Com had announced that it intended to distribute its remaining Palm shares to 3Com's own shareholders later in the year, subject to a favorable tax ruling. Each 3Com share would receive roughly 1.5 Palm shares. Anyone buying one share of 3Com was therefore buying a claim to about 1.5 shares of Palm, plus everything else 3Com owned: the networking business, the factories, the patents, and well over a billion dollars of cash. Palm's first day of trading was a spectacle. The stock opened far above the $38 offering price, traded as high as roughly $165 intraday, and closed near $95. 3Com closed the same day near $82. Put those two numbers together. If a share of 3Com entitles its owner to about 1.5 shares of Palm, and Palm is worth $95, then the Palm stake inside a 3Com share is worth about $143. A 3Com share cost $82. The market was therefore assigning a value of about **negative $60 per share** to everything 3Com owned other than Palm — to the networking business, the patents, and the cash. Multiply by 3Com's roughly 350 million shares and the implied value of the non-Palm company was negative, on the order of twenty billion dollars. This is not a subtle mispricing. It is not a disagreement about growth rates or the equity premium. It is an arithmetic impossibility. A share of 3Com is a claim to a bundle; the bundle contains 1.5 Palm shares and a pile of cash; a claim to a pile of cash cannot be worth less than nothing, because its owner can always walk away. Two portfolios — "one 3Com share" and "1.5 Palm shares plus a lottery ticket on the networking business" — had payoffs that could be ordered, and their prices were ordered the wrong way. Someone should have bought 3Com, sold 1.5 Palm shares short, waited for the distribution, delivered the Palm shares they received, and kept the networking company and the cash for free. The trade was obvious. Practically every arbitrage desk on Wall Street could see it. It did not go away for months. The reason is the second half of this chapter's subject. Only five percent of Palm's shares had been sold to the public; the other ninety-five percent sat inside 3Com and could not be lent. The shares available to borrow were therefore few, and the cost of borrowing them was enormous — stock-loan fees on Palm ran to tens of percent per year, an extraordinary rate in a market where a normal borrow costs a fraction of one percent. Short interest ran up against the entire lendable float. Lenders could recall shares at will, which meant an arbitrageur could be forced to close the position at the worst moment. And the distribution itself was conditional: if the Internal Revenue Service refused the ruling, the trade's convergence date disappeared. Note what did *not* fail. Investors were not confused about the arithmetic; Lamont and Thaler (2003), who documented this episode and five others like it, found the mispricing hiding in plain sight in the financial press. Nor was the mispricing small enough to sit inside transaction costs — the numbers in Problem 4 show it was not. What failed was the machinery that is supposed to convert a known mispricing into a trade. The price of Palm was set by the investors who could buy it. The investors who thought it was too expensive mostly could not sell it, because you cannot sell what you cannot borrow. Two propositions come out of this episode, and the chapter is built on both. The first is that **arbitrage disciplines prices**: where a claim can be replicated by other claims, its price is pinned down by theirs, and the whole apparatus of modern valuation is built on that fact. The second is that **the discipline has limits, and the limits depend on who can trade**. The stub value went negative and stayed negative for a quarter of a year because of a fact about the balance sheets and stock-loan agreements of a few dozen hedge funds. Sections 3.2 through 3.6 develop the first proposition into the equation that organizes the rest of this book. Section 3.7 asks the question the second proposition raises, and the last four parts of the book answer it. *** ## 3.1 Present Value A financial claim is a promise of payments at future dates, generally uncertain. Everything else — the paperwork, the covenants, the ticker — is machinery for describing and enforcing that promise. To value a claim is to attach a number today to a stream of payments tomorrow. The master tool is present value. A claim promising a certain payment $$C\_t$$ at each date $$t$$ is worth $$ P = \sum\_{t=1}^{T} \frac{C\_t}{(1 + r\_t)^t} $$ where $$r\_t$$ is the discount rate appropriate to a payment arriving at date $$t$$. Two familiar special cases: a level perpetuity paying $$C$$ forever at a constant rate $$r$$ is worth $$C/r$$; one growing at rate $$g < r$$ is worth $$C/(r-g)$$. This chapter takes the mechanics of compounding as known and asks the question the mechanics do not answer. That question is: where does $$r$$ come from? Notice first that $$r$$ carries a subscript. Discount rates differ across dates, and they differ across claims arriving on the *same* date. Three reasons, and they are logically distinct: **Horizon.** Money in ten years is not money in one year, and the market quotes those two prices separately. The term structure of interest rates (Chapter 9) is the schedule of prices for riskless payments at different horizons, and it is rarely flat. The subscript on $$r\_t$$ is not decoration. **Risk.** A payment that arrives whether or not the economy is in recession is a different object from a payment of the same expected size that arrives only in good times. The second is worth less, and the difference is a risk premium. This is the reason discount rates differ that most of asset pricing is about, and §3.4 shows exactly where it comes from. **Everything institutional.** Two claims with the same horizon and the same risk can trade at different discount rates because one is tax-exempt and one is not (Chapter 10's municipal-bond puzzle), because one can be posted as collateral and one cannot (Chapter 9's convenience yield on Treasuries), or because one is held by insurers whose capital charges depend on its rating (Chapter 10's reaching-for-yield). These are the wedges this book takes seriously; incumbent texts tend to list them and move on. Now the warning that motivates the rest of the chapter. Present value looks like a machine for producing prices out of forecasts, and it is not. A discount rate *is* a price — the price of a dated, state-contingent dollar, quoted in a different unit. To assert that a claim's discount rate is 8 percent is to assert something about what dollars in those states at that date are trading for. Present value is therefore an accounting identity that reorganizes prices, not a theory that generates them. Used well, it is the most useful accounting identity in finance: it lets you take prices you can observe and infer the value of a claim you cannot observe. Used badly — a spreadsheet with a hurdle rate typed into cell B4 and applied to every project a firm might undertake — it launders an assumption into a valuation. Table 3.1 makes the arithmetic point that the chapter's data exercise turns into an empirical one. **Table 3.1: Present value of $100, by horizon and discount rate** | Payment arrives in | at 2% | at 5% | at 8% | | ------------------ | ------ | ------ | ------ | | 1 year | $98.04 | $95.24 | $92.59 | | 5 years | $90.57 | $78.35 | $68.06 | | 10 years | $82.03 | $61.39 | $46.32 | | 30 years | $55.21 | $23.14 | $9.94 | *Source: Author's calculation.* Read the table down a column and you see discounting. Read it across a row and you see something more consequential: moving the discount rate from 2 percent to 5 percent costs a one-year claim 3 percent of its value and a thirty-year claim 58 percent of its value. Long-duration claims are levered bets on discount rates. That single fact explains why growth stocks fell harder than value stocks in 2022, why a pension fund's liabilities are more rate-sensitive than its assets, and why the long end of the Treasury market is where duration risk is actually borne. Chapter 9 gives it a name — duration — and Chapters 16 and 19 show what happens when the holders of long-duration claims are levered. Figure 3.2 draws the table as a family of curves, on which reading across a row is the natural motion. ![Figure 3.2: What a discount rate does](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-4f1f64e0a98b9a30ec20180ef47f0038fa4a91b4%2Ffig_03_02_present_value.png?alt=media) **Figure 3.2: What a discount rate does.** Table 3.1 drawn as a family of curves: the present value of a $100 payment against the horizon at which it arrives, at discount rates of 2, 5 and 8 percent, with the table's four printed horizons marked on the axis. Read down a column of the table and you see discounting. Read across a row — which is the reading the curves make visible and the table does not — and you see that the same three-point rise in the rate costs a one-year claim 3 percent of its value and a thirty-year claim 58 percent of it. The two brackets are those two losses. That convexity is what makes long-duration claims levered bets on discount rates; it is the elasticity the chapter's data exercise measures numerically, and Chapter 9 gives it a name. *Source: Author's construction from the present-value arithmetic of Section 3.1, Table 3.1.* So the task is to price dated, state-contingent dollars. The rest of this chapter does that with almost no machinery, using one idea. *** ## 3.2 The Law of One Price, Arbitrage, and Dominance The idea is that identical things must cost the same. Stated with enough care, it does an enormous amount of work. Write $$x$$ for a payoff — a list of numbers, one for each state of the world at the next date — and $$p(x)$$ for its price today. Three conditions, from weakest to strongest. **The law of one price (LOOP).** If two portfolios deliver the same payoff in every state, they have the same price: $$ x = y \implies p(x) = p(y) $$ An immediate and powerful consequence: if LOOP holds and portfolios can be freely combined, then the pricing function is **linear**, $$ p(a x + b y) = a p(x) + b p(y) $$ for any weights $$a$$ and $$b$$. Linearity is the whole of the technical content. It says that a bundle costs what its parts cost, that there is no quantity discount on payoffs, and that the price of a complicated claim can be computed by decomposing it into simple ones. Every valuation formula in this book — bond pricing, Black-Scholes, the CAPM — is an application of linear pricing. **No arbitrage.** An arbitrage is a portfolio that costs nothing (or less) today and delivers a payoff that is never negative and is strictly positive in at least one state. Formally, a portfolio with payoff $$x$$ such that $$ p(x) \le 0, \quad x\_s \ge 0 \text{ for all } s, \quad x\_s > 0 \text{ for some } s $$ This is the free lunch: no money in, no possibility of loss, some possibility of gain. Textbooks sometimes separate the two flavors — a strictly negative price with a non-negative payoff, and a zero price with a strictly positive payoff — but the economics is the same. **No dominance.** Payoff $$x$$ dominates $$y$$ if $$x\_s \ge y\_s$$ in every state with strict inequality somewhere. No dominance requires that $$p(x) > p(y)$$ whenever $$x$$ dominates $$y$$: more is worth more, always. The relations among the three are worth getting right, because students routinely treat them as synonyms. No arbitrage implies LOOP. If two portfolios had the same payoff at different prices, buy the cheap one, sell the dear one, and collect the difference today against a payoff of exactly zero in every state — an arbitrage. So LOOP is the weaker requirement. LOOP does **not** imply no arbitrage. Suppose the only traded claim is a ticket paying $1 if the market rises and nothing otherwise, and the ticket trades at zero. No two portfolios have identical payoffs at different prices, so LOOP is not violated. But buying the ticket is a free lunch. LOOP disciplines *relative* prices among claims that replicate each other; it says nothing about whether the whole price system is sane. Once prices are linear, no dominance and no arbitrage coincide: a dominance violation is exactly the trade "buy $$x$$, sell $$y$$", which costs nothing and pays something. Without linearity — with bid-ask spreads, short-sale constraints, or position limits — the three come apart, and the dominance argument survives longest, because it needs only that someone, somewhere, prefers more to less. Two comments on how to use these conditions. First, no arbitrage is a very weak assumption about behavior and a very strong assumption about markets. It does not require that investors be rational, informed, or even sane; it requires only that *if* a free lunch exists, *someone* takes it, and that taking it moves prices. It says nothing about the level of the market, whether stocks are overvalued, or whether the equity premium is too high. It disciplines relative prices only. Second, and this is the chapter's second theme in its most compact form: **the assumption is about a trade, and trades are executed by holders with balance sheets.** The Palm arbitrage was an arbitrage in exactly the sense defined above. Its persistence was not a failure of the definition. It was a fact about the supply of borrowable shares. > **Box 3.1 — What a short sale actually is** > > Selling short is not the mirror image of buying. To sell a share you do not own, you must first **borrow** it: your broker locates a lender — typically a custodian acting for an index fund, pension, or insurer that holds the stock and does not intend to sell it — and delivers the borrowed share to your buyer. You post the sale proceeds with the lender as collateral, and the lender pays you interest on that cash at the **rebate rate**. > > When a stock is easy to borrow, the rebate rate sits just under the market interest rate, and shorting is nearly costless. When it is hard to borrow, the rebate rate falls, and for genuinely scarce names it goes **negative**: you pay the lender for the privilege. Rates of tens of percent a year are rare but real, and Palm was one of them. > > Two further asymmetries. The lender can **recall** the shares at any time, forcing you to buy back at whatever price prevails. And your position is marked to market daily against posted margin, so a mispricing that widens before it converges can end the trade before the convergence arrives. Every one of these frictions bites hardest exactly where mispricing is largest, because the same scarcity that makes a stock hard to borrow is what let its price run. > > Chapter 15 develops the general point (Shleifer-Vishny); Chapter 16 states it canonically for constrained capital; Chapter 11 supplies the market-structure detail. *** ## 3.3 Replication The law of one price is useful because of a single move, and the move is worth naming: **to price a claim you do not understand, build a portfolio of claims you do understand that delivers the same payoff in every state.** Then LOOP hands you the price. The portfolio is called a replicating portfolio, and constructing one is what "pricing by arbitrage" means in practice. Take the standard example, a forward contract. A forward on a stock is an agreement made today to buy one share at date 1 at a price $$F$$ fixed today, with no money changing hands until date 1. What is the fair $$F$$? Suppose the stock trades at 100 dollars a share, so $$S\_0 = 100$$; the one-year riskless rate is 5 percent, and the stock pays no dividend. Consider two ways to own a share of stock at date 1. *Route A.* Enter the forward. Pay nothing today; pay $$F$$ at date 1; receive a share. *Route B.* Borrow $100 at 5 percent and buy the share today. Pay nothing on net today; owe $105 at date 1; own the share. Both routes deliver exactly one share at date 1 and require no outlay today. Their date-1 obligations must therefore be equal: $$ F = S\_0 R\_f = 100 \times 1.05 = 105 $$ No probabilities appeared. No view about the stock appeared. The forward price is not a forecast of the stock price; it is the spot price carried forward at the interest rate, and it would be $105 whether every investor expects the stock to double or to halve. Now suppose the forward is quoted at $108. Table 3.2 shows what a trader does about it. **Table 3.2: Cash-and-carry arbitrage when the forward is rich** | | Cash flow today | Cash flow at date 1 | | ------------------------------------- | --------------- | ------------------- | | Sell one forward at 108 dollars | 0 | $$108 - S\_1$$ | | Buy one share | $$-100$$ | $$S\_1$$ | | Borrow 100 dollars for one year at 5% | $$+100$$ | $$-105$$ | | **Total** | **0** | **+3** | *Source: Author's calculation.* The strategy costs nothing today and pays 3 dollars at date 1 no matter what the stock does — the $$S\_1$$ terms cancel by construction. That is an arbitrage in the exact sense of §3.2, and its existence is what forces $$F$$ to $105. If the forward is instead quoted below $105, run the table backwards: buy the forward, short the stock, lend the proceeds. Three qualifications, each of which becomes a chapter. If the stock pays a dividend $$D$$ at date 1, Route B's owner receives it and the forward's does not, so $$F = S\_0 R\_f - D$$. If the underlying is a commodity, the carry includes storage costs and a convenience yield, and the sign of the adjustment determines whether the futures curve slopes up or down (Chapter 8). If the underlying cannot be shorted, the reverse arbitrage is unavailable and no-arbitrage delivers only an upper bound on $$F$$, not an equality. That last one deserves emphasis, because it is the general lesson and it reappears in every chapter of Parts III and IV. Read the cash-and-carry table again and notice what it assumes: that you can short the stock, borrow the stock, borrow cash at $$R\_f$$, and hold the position to maturity without being closed out. Those are assumptions about the arbitrageur's balance sheet, not about the securities. When they hold, replication pins prices exactly. When they fail — Palm in 2000, the Treasury basis in March 2020 (Chapter 19's opening episode, with the safe-asset side in Chapter 9), the ETF-NAV gap in stressed credit (Chapter 17) — replication delivers a band instead of a point, and something else has to determine where inside the band the price sits. That something else is holder demand, and it is Chapter 20's subject. *** ## 3.4 A Two-Date, Two-State World, Fully Worked Everything in this book's theoretical apparatus is visible in an economy with two dates and two states. This section builds that economy, prices an option in it three different ways, and extracts the risk premium. Nothing here is harder than arithmetic. Every general result in §3.5 is a restatement of something derived here. ### 3.4.1 The economy There are two dates: today (date 0) and next year (date 1). At date 1 one of two states occurs: the good state $$u$$ or the bad state $$d$$. The true probability of the good state is $$\pi\_u = 0.6$$, so $$\pi\_d = 0.4$$. Two assets trade. A riskless bond pays one dollar at date 1 in either state; the one-year riskless gross return is $$R\_f = 1.05$$, so the bond costs $$1/1.05 = 0.9524$$. A stock trades at 100 dollars today and will be worth 130 in state $$u$$ and 90 in state $$d$$. **Table 3.3: The two-date, two-state economy** | Asset | Price today ($$p$$) | Payoff in $$u$$ | Payoff in $$d$$ | | ------------------------ | ------------------- | ---------------- | ---------------- | | Bond (per $1 of face) | 0.9524 | 1 | 1 | | Stock | 100 | 130 | 90 | | Call option, strike $100 | ? | 30 | 0 | | — | | | | | Probability | | $$\pi\_u = 0.6$$ | $$\pi\_d = 0.4$$ | *Source: Author's construction.* The call option is the claim to be priced. It gives its owner the right, not the obligation, to buy the stock at date 1 for 100 dollars. In state $$u$$ the stock is worth 130, so the owner exercises and collects 30. In state $$d$$ the stock is worth 90, exercising would be a loss, so the option expires worthless and pays nothing. Its payoff is $$\max(S\_1 - K, 0)$$ with $$K = 100$$. ### 3.4.2 State prices Define $$q\_u$$ as the price today of a claim paying one dollar in state $$u$$ and nothing in state $$d$$, and $$q\_d$$ symmetrically. These are **state prices**, or Arrow-Debreu prices. They are the primitive objects of asset pricing: the prices of dated, state-contingent dollars that §3.1 said we needed. They are not quoted anywhere. They are inferred, from the prices of assets that are quoted. Every traded asset is a bundle of state-contingent dollars, so its price must be the sum of the state prices of the dollars in the bundle: $$ p = q\_u x\_u + q\_d x\_d $$ Apply this to both traded assets. The bond delivers $1 in each state: $$ 0.9524 = q\_u + q\_d $$ The stock delivers $130 and $90: $$ 100 = 130 q\_u + 90 q\_d $$ Two linear equations, two unknowns. Substitute $$q\_d = 0.9524 - q\_u$$ into the second: $$ 100 = 130q\_u + 90(0.9524 - q\_u) = 40q\_u + 85.71 $$ so $$q\_u = 14.29/40 = 0.3571$$ and $$q\_d = 0.9524 - 0.3571 = 0.5952$$. Stop and read those two numbers, because they are the chapter. A dollar delivered in the *bad* state costs $0.5952. A dollar delivered in the *good* state costs $0.3571 — barely more than half as much. Yet the good state is the *more likely* of the two, with probability 0.6 against 0.4. The market charges far more for a dollar in the unlikely bad state than for a dollar in the likely good one. That is not a puzzle; it is the entire economics of risk premia, and it is visible before any preferences, utility functions, or models have been introduced. Dollars are worth more where they are scarce. A claim that pays off in bad states is insurance, and insurance is expensive. A claim that pays off in good states is a bet on more of what you already have, and it is cheap. Everything Part II does is elaborate this observation. ### 3.4.3 Pricing the call, route one: replication Build a portfolio of stock and bonds with the same payoff as the call. Let $$\Delta$$ be the number of shares and $$B$$ the face value of bonds (negative $$B$$ means borrowing). The portfolio must match the call in both states: $$ 130\Delta + B = 30 \qquad (\text{state } u) $$ $$ 90\Delta + B = 0 \qquad (\text{state } d) $$ Subtract the second from the first: $$40\Delta = 30$$, so $$\Delta = 0.75$$. Then $$B = -90 \times 0.75 = -67.5$$. The replicating portfolio is: **buy 0.75 shares, borrow 67.50 dollars of face value.** Check it. In state $$u$$ the shares are worth $$0.75 \times 130 = 97.50$$ and the debt costs 67.50, leaving 30. In state $$d$$ the shares are worth $$0.75 \times 90 = 67.50$$ and the debt costs 67.50, leaving nothing. It matches. What does it cost today? The shares cost $$0.75 \times 100 = 75$$; borrowing 67.50 of face raises $$67.50/1.05 = 64.29$$. So $$ p\_{\text{call}} = 75 - 64.29 = 10.71 $$ By the law of one price, the call must trade at $10.71. If it traded at $12, sell the call and buy the replicating portfolio: $1.29 today, and the payoffs cancel exactly at date 1. If it traded at $9, do the reverse. Note the two facts this construction exposes. The hedge ratio $$\Delta = 0.75$$ is the ratio of the spread in the option's payoffs to the spread in the stock's, $$(30-0)/(130-90)$$ — in Chapter 8 it becomes the option's delta. And the replicating portfolio is *levered*: $75 of stock financed with $64.29 of borrowing against $10.71 of equity, a leverage ratio of exactly 7. Hold that number. ### 3.4.4 Pricing the call, route two: state prices Now use the state prices. The call is a bundle: 30 dollars delivered in state $$u$$, none in state $$d$$. Its price is therefore $$ p\_{\text{call}} = 30 q\_u + 0 \times q\_d = 30 \times 0.3571 = 10.71 $$ The same number, from three lines of arithmetic instead of a simultaneous equation. This is not a coincidence, and it is not a second theory. State prices *are* the replication argument, solved once and stored. Having done the work of extracting $$q\_u$$ and $$q\_d$$ from the bond and the stock, you can now price any claim in this economy — a put, a digital option, a convertible bond, a levered equity tranche — by one multiplication and one addition. That is the payoff to the state-price representation and the reason the rest of asset pricing is written in its language. Figure 3.3 draws both routes in the plane of date-1 payoffs: panel (a) builds the call by adding vectors, panel (b) reads its price off a line. ![Figure 3.3: The two-date, two-state economy](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-ecd1ee36664cb0f43ecee43c303a31fc4d80715c%2Ffig_03_03_two_state_economy.png?alt=media) **Figure 3.3: The two-date, two-state economy.** Section 3.4's worked economy drawn in the plane of date-1 payoffs, with the payoff in the bad state $$d$$ on the horizontal axis and the payoff in the good state $$u$$ on the vertical. Panel (a) is replication, Section 3.4.3: one share of stock is the vector (90, 130), 0.75 of a share is the open point on that ray, and borrowing 67.50 of bond face — a move of 67.50 back down the riskless direction (1, 1) — lands exactly on the call's payoff (0, 30). The recipe costs 75.00 less 64.29, which is 10.71. Panel (b) is Section 3.4.4: the same call priced by the state prices those two traded assets imply, $$q\_d = 0.5952$$ and $$q\_u = 0.3571$$. Every payoff on a given line costs the same, so the lines are the level sets of the pricing functional and $$q$$ is the vector normal to them; one share of stock and 105 of riskless face both lie on the line costing 100, and the call lies on the line costing 10.71. The strict positivity of $$q$$ is what no arbitrage buys: it is why the functional is increasing in both coordinates, and why every payoff in the positive quadrant has a positive price. Two notations, one argument. *Source: Author's construction from the worked economy of Section 3.4, Table 3.3.* ### 3.4.5 Risk-neutral probabilities State prices carry two pieces of information at once: the probability of a state and the value of a dollar in it. Separating them yields the most useful change of variables in finance. The state prices sum to $$q\_u + q\_d = 0.9524 = 1/R\_f$$, which is no accident — that sum is the price of a dollar delivered *for certain*, which is the bond. So dividing each state price by their sum produces two positive numbers that sum to one. They look like probabilities. Define $$ \pi^{\ast}\_s = \frac{q\_s}{\sum\_j q\_j} = q\_s R\_f $$ In our numbers: $$ \pi^{\ast}\_u = 0.3571 \times 1.05 = 0.375, \qquad \pi^{\ast}\_d = 0.5952 \times 1.05 = 0.625 $$ These are the **risk-neutral probabilities**: normalized state prices. They are not the probabilities of anything. Nobody believes the good state has probability 0.375; the true probability is 0.6. They are prices wearing the costume of probabilities, and the costume is worth putting on because it turns pricing into an expectation. Rewriting $$p = q\_u x\_u + q\_d x\_d$$ using $$q\_s = \pi^{\ast}\_s / R\_f$$: $$ p = \frac{1}{R\_f}\left(\pi^{\ast}\_u x\_u + \pi^{\ast}\_d x\_d\right) = \frac{E^{\ast}\[x]}{R\_f} $$ **Price equals the expected payoff, discounted at the riskless rate — provided the expectation is taken with the wrong probabilities.** Check the call: $$E^{\ast}\[x] = 0.375 \times 30 + 0.625 \times 0 = 11.25$$, and $$11.25/1.05 = 10.71$$. Check the stock: $$E^{\ast}\[x] = 0.375 \times 130 + 0.625 \times 90 = 105$$, and $$105/1.05 = 100$$. Both hold exactly, as they must. The risk adjustment has not disappeared. It has moved out of the discount rate and into the probabilities. Every claim in this economy is discounted at the same rate, 5 percent, and the compensation for risk is delivered by tilting the probabilities toward the bad state. > **Box 3.2 — Three routes to the same number** > > The call is worth $10.71, and there are three ways to see it: > > 1. **Replication.** Build the payoff from stock and bonds; the cost of the recipe is the price. Buy 0.75 shares, borrow $64.29. > 2. **State prices.** Decompose the payoff into state-contingent dollars and multiply by their prices: $$30 \times 0.3571$$. > 3. **Risk-neutral valuation.** Take the expected payoff under $$\pi^{\ast}$$ and discount at $$R\_f$$: $$11.25/1.05$$. > > They are the same argument in three notations, and each is the natural one somewhere. Replication is how a trading desk hedges. State prices are how you see the economics of risk. Risk-neutral valuation is how you compute, and it is the one that scales to many periods and continuous time — which is why Chapter 8 §8.3.2 develops it formally, once the binomial tree has made the many-period version necessary. ### 3.4.6 The wedge is the risk premium We now have two sets of probabilities over the same two states: the true ones, $$\pi = (0.6, 0.4)$$, and the risk-neutral ones, $$\pi^{\ast} = (0.375, 0.625)$$. Compute expected returns under each. **Table 3.4: Expected gross returns under the two measures** | Asset | Price | Payoff $$u$$ | Payoff $$d$$ | $$E\[R]$$ under $$\pi$$ | $$E^{\ast}\[R]$$ under $$\pi^{\ast}$$ | Risk premium | | ----- | ------ | ------------ | ------------ | ----------------------- | ------------------------------------- | ------------ | | Bond | 0.9524 | 1 | 1 | 1.05 | 1.05 | 0 | | Stock | 100 | 130 | 90 | 1.14 | 1.05 | 9.0% | | Call | 10.71 | 30 | 0 | 1.68 | 1.05 | 63.0% | *Source: Author's calculation from Table 3.3.* Work through the stock. Under the true probabilities, the expected payoff is $$0.6 \times 130 + 0.4 \times 90 = 114$$, so the expected gross return is $$114/100 = 1.14$$: 14 percent expected, against a riskless 5 percent. The stock carries a 9-percentage-point risk premium. Under the risk-neutral probabilities, the expected return is exactly 5 percent, as it is for every asset in the table. This is the definition of the risk-neutral measure made operational: $$\pi^{\ast}$$ **is the probability distribution under which every asset would have to earn the riskless rate.** The gap between what an asset actually earns in expectation (14 percent) and what it earns under $$\pi^{\ast}$$ (5 percent) is the risk premium, and the gap between $$\pi$$ and $$\pi^{\ast}$$ is where it lives. Nothing has been assumed about anyone's utility function. The 9-percentage-point premium was implicit in the observed prices of the bond and the stock from the moment they were quoted; §3.4.2's state prices merely extracted it. The call's row is the payoff to having done the option in full. Its expected return is 68 percent — an enormous number, and one no forecast produced. It follows mechanically from the replicating portfolio. The call *is* 0.75 shares financed with borrowing, at a leverage ratio of exactly 7, so it carries exactly 7 times the stock's risk premium: $$7 \times 9 = 63$$ percent. An option is a levered position in the underlying, and its expected return is the underlying's premium scaled by the leverage the replication requires. This is why "options have high expected returns" is not evidence that options are attractive, and why Chapter 8 measures option risk in units of delta rather than in units of expected return. Figure 3.4 sets the two measures and the two sets of expected returns they generate side by side. ![Figure 3.4: The wedge is the risk premium](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-ede856cdf66baa4f91751734e79534e39735340a%2Ffig_03_04_risk_premium_wedge.png?alt=media) **Figure 3.4: The wedge is the risk premium.** Panel (a): the two probability measures over Section 3.4's two states — the true one, $$\pi = (0.6, 0.4)$$, and the risk-neutral one, $$\pi^{\ast} = (0.375, 0.625)$$, which is the same distribution reweighted state by state by $$m\_sR\_f$$. The bad state is made to look more likely, in exact proportion to what a dollar is worth there. Panel (b): Table 3.4's expected returns under each measure, with the wedge between them marked. Under $$\pi^{\ast}$$ every asset earns the riskless 5 percent; under $$\pi$$ the bond earns 5, the stock 14 and the call 68. The gaps — nothing, 9 points and 63 points — are the risk premia, and the call's is exactly seven times the stock's because its replicating portfolio is levered seven to one. Nothing here required a utility function: the premium was implicit in the quoted prices of the bond and the stock from the moment they were quoted. *Source: Author's construction from the worked economy of Section 3.4, Tables 3.3 and 3.4.* ### 3.4.7 Why this worked, and when it stops working The two-state example was solvable because the number of independent traded assets — two, the bond and the stock — equalled the number of states. That made the linear system in §3.4.2 square and non-singular, so the state prices existed and were **unique**. A market with this property is **complete**: every conceivable payoff over the state space can be built out of traded assets, and every claim therefore has exactly one no-arbitrage price. Completeness is a strong condition and it is doing more work than it advertises. Suppose we add a third state — say the stock could also be worth $100 — while keeping only two traded assets. Then §3.4.2 becomes two equations in three unknowns. The state prices are no longer pinned down; a whole interval of them is consistent with the observed prices of the bond and the stock. The call's payoff can no longer be replicated, so no-arbitrage delivers a *range* of admissible prices rather than a number. Problem 6 works this case out and finds a call price anywhere between $5.00 and $14.40 — an interval too wide to be useful. The interval is not a defect of the theory. It is the theory telling the truth about what arbitrage can and cannot do. Where a claim can be replicated, its price is determined by other prices and the identity of the buyer is irrelevant. Where it cannot, arbitrage gives bounds, and something else — who wants the claim, how badly, and what constrains them — determines where in the bounds the price settles. Real markets are incomplete. That is why Part IV of this book exists. *** ## 3.5 ★ The Stochastic Discount Factor *This section is starred: it states in general notation what §3.4 established in numbers. A reader who skips it loses generality, not the argument.* ### From state prices to $$p = E\[mx]$$ Return to the complete two-state economy, and to the pricing relation $$ p = \sum\_s q\_s x\_s $$ This is correct but awkward, because state prices mix together two things that we usually want to keep apart: how likely a state is, and how much a dollar is worth there. Divide and multiply each term by the state's physical probability: $$ p = \sum\_s \pi\_s \left(\frac{q\_s}{\pi\_s}\right) x\_s $$ Define the object in parentheses as the **stochastic discount factor**: $$ m\_s \equiv \frac{q\_s}{\pi\_s} $$ and the sum becomes an expectation under the true probabilities. This is the **fundamental pricing equation**: $$ \boxed{p = E\[mx]} $$ Every asset-pricing model in this book is a specification of $$m$$. In the numbers of §3.4: $$m\_u = 0.3571/0.6 = 0.5952$$ and $$m\_d = 0.5952/0.4 = 1.4881$$. A dollar in the good state is discounted by a factor of 0.60; a dollar in the bad state by a factor of 1.49 — it is worth *more* than a certain dollar today. Verify the stock: $$E\[mx] = 0.6(0.5952)(130) + 0.4(1.4881)(90) = 46.43 + 53.57 = 100$$. Verify the bond: $$E\[m] = 0.6(0.5952) + 0.4(1.4881) = 0.9524 = 1/R\_f$$, so a riskless claim is discounted at the riskless rate, as required. Dividing through by price gives the equation's most-used form, $$ 1 = E\[m R] $$ which holds for every traded asset, riskless and risky alike. ### What $$m$$ is, economically The equation so far is a change of variables: $$m$$ was defined from prices, so it cannot by itself explain them. The economics enters when $$m$$ is tied to a decision. Consider a household choosing how much to consume today and how much to invest. It can buy a claim at price $$p$$ and receive payoff $$x$$ next period. At an optimum, the household is indifferent to buying a marginal unit: the utility given up today by spending $$p$$ must equal the expected utility gained tomorrow from receiving $$x$$. Writing $$u'(c)$$ for marginal utility and $$\delta$$ for the household's subjective discount factor, this first-order condition is $$pu'(c\_0) = E\[\delta u'(c\_1)x]$$, or $$ p = E\left\[\delta \frac{u'(c\_1)}{u'(c\_0)} x\right] \quad\Longrightarrow\quad m = \delta\frac{u'(c\_1)}{u'(c\_0)} $$ **The stochastic discount factor is marginal utility growth.** It is high in states where consumption is low, because an extra dollar matters more when you have less. That is why $$m\_d = 1.49 > m\_u = 0.60$$ in our example, and why the state price of a bad-state dollar exceeded that of a good-state dollar even though the bad state was less likely. A claim that pays when $$m$$ is high — when times are bad — is insurance, is expensive, and therefore earns a low expected return. A claim that pays when $$m$$ is low earns a high one. Chapter 5 takes this seriously as an empirical proposition, measures $$m$$ from aggregate consumption data, and finds that it fails by an order of magnitude; that failure is the equity premium puzzle. ### Complete and incomplete markets: when is $$m$$ unique? If markets are complete, the state prices are unique, so $$m$$ is unique. Every claim has one price and the model has no slack. If markets are incomplete — the realistic case — there are many stochastic discount factors consistent with observed prices. They agree on the value of every traded payoff and disagree about everything else. Formally, if $$m$$ prices all traded assets, so does $$m + \varepsilon$$ for any $$\varepsilon$$ orthogonal to the space of traded payoffs, since $$E\[\varepsilon x] = 0$$ by construction for every traded $$x$$. Among this family there is exactly one member that lies *in* the payoff space, the projection $$m^{\ast} = \mathrm{proj}(m \mid X)$$, and it is the one empirical work uses: it is the unique discount factor that can be written as a portfolio of traded assets. Appendix D gives the existence and uniqueness results (the fundamental theorem of asset pricing: no arbitrage holds if and only if there exists a strictly positive $$m$$; markets are complete if and only if it is unique). The practical content is the one §3.4.7 already delivered in arithmetic. Incompleteness means arbitrage does not finish the job, and the residual is settled by whoever holds the claim. ### The beta representation The pricing equation contains the risk-return tradeoff, and one line of algebra extracts it. Expand the covariance in $$1 = E\[mR]$$: $$ 1 = E\[m]E\[R] + \mathrm{Cov}(m, R) \Longrightarrow E\[R] = \frac{1 - \mathrm{Cov}(m,R)}{E\[m]} = R\_f\big(1 - \mathrm{Cov}(m,R)\big) $$ using $$E\[m] = 1/R\_f$$. Therefore $$ E\[R] - R\_f = -R\_f\mathrm{Cov}(m, R) $$ **An asset's risk premium is proportional to the negative of the covariance between its return and the discount factor.** Not its variance. Not its total risk. Only the part of its return that moves with marginal utility is priced; idiosyncratic risk earns nothing, because it can be diversified away and therefore does not affect anyone's consumption at the margin. Check it against §3.4. The stock's return is 1.30 in state $$u$$ and 0.90 in state $$d$$, with $$E\[R] = 1.14$$. Then $$E\[mR] = 1$$ by construction and $$E\[m]E\[R] = 0.9524 \times 1.14 = 1.0857$$, so $$\mathrm{Cov}(m, R) = 1 - 1.0857 = -0.0857$$, and $$-R\_f \mathrm{Cov}(m,R) = -1.05 \times (-0.0857) = 0.09$$. Nine percent, exactly the risk premium in Table 3.4. Rearranged into the form Chapter 4 uses, $$ E\[R] - R\_f = \underbrace{\frac{\mathrm{Cov}(m,R)}{\mathrm{Var}(m)}}\_{\beta \_{R,m}} \times \underbrace{\left(-\frac{\mathrm{Var}(m)}{E\[m]}\right)} \_{\lambda\_m} $$ a quantity of risk ($$\beta$$) times a price of risk ($$\lambda$$), common to all assets. Every factor model in Part II is this expression with a particular guess about what $$m$$ depends on. One further consequence, since it costs a line: because a correlation cannot exceed one in absolute value, the equation implies $$|E\[R] - R\_f| / \sigma(R) \le \sigma(m)/E\[m]$$ — the maximum attainable Sharpe ratio is bounded by the volatility of the discount factor. In §3.4's economy, $$\sigma(m)/E\[m] = 0.459$$ and the stock's Sharpe ratio is $$0.09/0.196 = 0.459$$: the bound binds exactly, because the market is complete. Chapter 5 turns this inequality on real data and gets the Hansen-Jagannathan bound. ### Change of measure: a preview The risk-neutral probabilities of §3.4.5 have a compact expression in this notation. Since $$\pi^{\ast}\_s = q\_s R\_f$$ and $$q\_s = m\_s \pi\_s$$, $$ \pi^{\ast}\_s = \pi\_s m\_s R\_f $$ The risk-neutral measure is the physical measure reweighted by marginal utility: states are made to look more likely in proportion to how much a dollar is worth in them. The reweighting factor $$m R\_f$$ has expectation one, so $$\pi^{\ast}$$ is a legitimate probability distribution, and pricing becomes $$p = E^{\ast}\[x]/R\_f$$. This is as far as this chapter takes it. The formal apparatus — equivalent martingale measures, the Radon-Nikodym derivative, and the fundamental theorem in its measure-theoretic form — is developed in **Chapter 8 §8.3.2**, where the multi-period binomial model creates a genuine need for it. Introducing it here, four chapters before the payoff diagrams that motivate it, would be machinery in search of a problem. *** ## 3.6 Three Specializations, Previewed The pricing equation $$p = E\[mx]$$ is empty until $$m$$ is specified, and it is compatible with almost any set of prices, which is both its strength and the standard complaint against it. What makes it a research program is that the field's three canonical models are all restrictions on $$m$$, and the rest of this book's theory chapters are the restrictions worked out. **The CAPM (Chapter 4) restricts** $$m$$ **to be linear in the market return.** Write $$m = a - b R\_M$$ for constants $$a, b > 0$$. Substituting into the beta representation and grinding through the algebra yields $$E\[R\_i] - R\_f = \beta\_i (E\[R\_M] - R\_f)$$, where $$\beta\_i$$ is the regression coefficient of asset $$i$$'s return on the market's. The economics of the restriction is that the only thing making times bad is the market portfolio falling; anything uncorrelated with the market carries no premium. Chapter 4 derives the CAPM the classical way, from mean-variance portfolio choice, and then shows in a starred section that the two derivations are the same result. **The term structure (Chapter 9) applies** $$m$$ **across horizons rather than across states.** A zero-coupon bond maturing at date $$t$$ is a claim to one dollar, so its price is $$P\_t = E\[m\_{0 \to t}]$$, the expected value of the discount factor compounded to that horizon. The yield curve is the term structure of $$E\[m]$$; the expectations hypothesis is the claim that $$m$$ and future short rates are uncorrelated; term premia are exactly the failure of that claim, and their decomposition is the empirical core of the chapter. **Black-Scholes (Chapter 8) is §3.4's replication argument taken to the limit.** Let the period shrink and the number of them grow. Each two-state step is priced exactly as in §3.4.3, by a hedge ratio $$\Delta$$ and a borrowing position, and the hedge is rebalanced at each node. In the continuous-time limit the tree becomes geometric Brownian motion, the discretely rebalanced hedge becomes continuous, and the price converges to a closed-form expression. The formula's content is not the normal distribution; it is that continuous rebalancing makes the replicating portfolio exact, so an option is redundant and its price is the cost of manufacturing it. Two more specializations belong to this list even though they are not usually taught as such: consumption-based pricing (Chapter 5), which takes $$m = \delta u'(c\_1)/u'(c\_0)$$ literally and measures it, and intermediary asset pricing (Chapter 19), which replaces the household's marginal utility with a leveraged intermediary's. That last one is where §3.7 is heading. *** ## 3.7 Whose $$m$$? The derivation in §3.5 slid past a question. The first-order condition that gave $$m = \delta u'(c\_1)/u'(c\_0)$$ belonged to *a household* — one household, with one consumption stream and one utility function. The economy has hundreds of millions of households, plus pension funds, insurers, dealers, sovereign wealth funds, and index providers. Whose marginal utility is in the equation? The standard answer is the **representative agent**. Under conditions that are precise and demanding — complete markets, and preferences that aggregate — the entire economy prices claims as though a single agent consuming aggregate consumption were doing so. Individual heterogeneity washes out because complete markets let households insure away their idiosyncratic risk, leaving only aggregate risk to be priced, and everyone's marginal utility moves together with aggregate consumption. Grant those conditions and $$m$$ is a function of aggregate consumption growth, measurable from national accounts data. This is the assumption underlying most of the asset-pricing canon, and it is why Cochrane's exposition can treat $$m$$ as a single well-defined object throughout. It is also the assumption this book declines to make. Three reasons, each of which becomes a section later in the book. **Participation and heterogeneity.** The representative-agent aggregation requires that everyone hold the risky assets at the margin. Most households do not. A large share of American households own no equities at all outside a retirement account, and equity ownership is concentrated in the top decile of the wealth distribution to a degree that makes "aggregate consumption" a poor proxy for the consumption of the people actually bearing equity risk. If a small, wealthy, high-participation group holds the risk, the relevant $$m$$ is *their* marginal utility growth, which is far more volatile than aggregate consumption growth. Chapter 14 documents who holds what; Chapter 5 shows that limited participation is one of the more successful resolutions of the equity premium puzzle precisely because it changes whose $$m$$ is in the equation. **Constrained arbitrageurs.** The Euler equation $$p = E\[mx]$$ holds for an investor who is *unconstrained at the margin* — free to buy a little more or a little less. An investor at a corner does not satisfy it. A hedge fund that wants to short Palm and cannot borrow the shares has a first-order condition with a Lagrange multiplier in it, and the multiplier — the shadow price of the constraint — enters the pricing relation alongside marginal utility. When the constraint binds, the price reflects the constraint, not the preference. Palm's price in March 2000 was set by the marginal *buyer*, because the marginal seller was locked out of the market. This is not an exotic case. It is the mechanism behind the Treasury basis in March 2020, the CDS-bond basis in 2008, and covered-interest-parity deviations in every year since the crisis. Chapter 16 states the general case canonically; Chapter 15 supplies the noise-trader complement. **Segmentation and mandate.** Even unconstrained institutions optimize against objectives that are not consumption. A life insurer's asset demand is driven by the duration and convexity of its liabilities and by risk-based capital charges that vary with a bond's rating (Chapter 10). A pension fund running a liability-driven mandate buys thirty-year gilts because its liabilities are thirty years long, not because it forecasts thirty-year yields (Chapter 9's preferred-habitat section, and the 2022 UK episode in Chapter 16). An index fund buys Tesla because a committee added it (Chapter 17). Each of these holders has a demand curve, and none of them is a household's Euler equation. If those holders are the marginal ones in a market, they are the ones whose shadow prices appear in $$m$$. So the question "whose $$m$$?" has an answer, and the answer is empirical: $$m$$ **belongs to the marginal investor in that claim, and the identity of the marginal investor is a fact about market segments, balance sheets, and mandates — not a modeling convenience.** In a frictionless complete market this question is uninteresting, because everyone is marginal in everything and all the $$m$$'s coincide. In actual markets, the marginal holder of agency MBS is not the marginal holder of municipal bonds, and neither is a representative household. Three chapters make the answer operational. **Chapter 16** establishes the canonical statement that constrained capital moves prices, with fire sales, margin spirals, and the leverage cycle. **Chapter 19 §19.5** replaces the household with the intermediary: He-Krishnamurthy and Adrian-Etula-Muir specify $$m$$ as a function of the net worth or leverage of the levered financial sector, and test it, and it works better across asset classes than consumption does. **Chapter 20** inverts the problem entirely: demand-system asset pricing (Koijen-Yogo) estimates the demand curves of every institutional holder from their reported holdings and derives prices from the market-clearing condition, which is the most literal possible statement of this book's title. The equation survives all of this. $$p = E\[mx]$$ is an implication of no arbitrage, and it holds in every economy in this book, including the ones with constrained investors, segmented markets, and inelastic demand. What changes is what $$m$$ is a function of, and therefore what one has to know about the world to say anything about prices. The canonical answer is "aggregate consumption". This book's answer is "the marginal investor, and you had better find out who that is." Parts III and IV are the search. *** ## Elsewhere in the Series * **What a promise to pay actually is, and how promises settle** — *International Finance*, Chapter 2. The money view treats a claim as a promise embedded in a hierarchy of settlement obligations, which is a complementary lens on the object this chapter prices as a payoff vector. The two treatments do not overlap: this chapter asks what a claim is worth; that one asks how it is discharged. * Everything else in this chapter is this book's own. The pricing equation, state prices, and the "whose $$m$$?" question are stated here and used from Chapter 4 to Chapter 27. *** ## Summary 1. **A discount rate is a price, not a preference.** Present value reorganizes observed prices into a valuation; it does not manufacture value from forecasts. Discount rates differ across claims for three distinct reasons — horizon, risk, and institutional wedges such as tax, collateral eligibility, and capital charges. 2. **Discount-rate changes move long-duration claims far more than short ones.** A move from 2 to 5 percent costs a one-year claim 3 percent of its value and a thirty-year claim 58 percent (Table 3.1). This single fact organizes Chapter 9 and much of what Chapters 16 and 19 say about levered holders. 3. **Three no-arbitrage conditions, properly ordered.** No dominance is strongest, no arbitrage next, the law of one price weakest. No arbitrage implies LOOP but not conversely: LOOP disciplines relative prices among replicating portfolios and says nothing about the level of the price system. Under linear pricing, no arbitrage and no dominance coincide. 4. **Replication is the core move.** Build a portfolio of understood claims with the same payoff as the claim you want to price, and LOOP hands you the price. The forward price $$F = S\_0 R\_f$$ is not a forecast; it is a carry cost, enforced by the cash-and-carry trade in Table 3.2. 5. **State prices are the primitive.** In a two-date, two-state economy, the prices of a bond and a stock imply unique prices for state-contingent dollars: $$q\_u = 0.3571$$ and $$q\_d = 0.5952$$. A dollar in the unlikely bad state costs nearly twice a dollar in the likely good state. That inequality *is* the risk premium, visible before any preferences are specified. 6. **Three routes, one number.** The call is worth 10.71 dollars by replication (0.75 shares less 64.29 of borrowing), by state prices ($$30 \times 0.3571$$), and by risk-neutral valuation ($$11.25/1.05$$). They are one argument in three notations. 7. **Risk-neutral probabilities are normalized state prices**, $$\pi^{\ast}\_s = q\_s R\_f$$, not beliefs. Under $$\pi^{\ast}$$ every asset earns the riskless rate; the wedge between $$\pi$$ and $$\pi^{\ast}$$ is where the risk premium lives. In §3.4's economy the stock earns 14 percent under $$\pi$$ and 5 percent under $$\pi^{\ast}$$, a 9-point premium; the call earns 68 percent, which is exactly 7 times the stock's premium because its replicating portfolio is levered 7 to 1. 8. **The fundamental pricing equation is** $$p = E\[mx]$$, where $$m\_s = q\_s/\pi\_s$$ is the stochastic discount factor and, at a household's optimum, equals marginal utility growth $$\delta u'(c\_1)/u'(c\_0)$$. Equivalently $$1 = E\[mR]$$, and $$E\[R] - R\_f = -R\_f\mathrm{Cov}(m,R)$$: only covariance with $$m$$ is priced. 9. **Completeness decides whether arbitrage finishes the job.** With as many independent assets as states, $$m$$ is unique and every claim has one price. With fewer, no-arbitrage delivers an interval — $5.00 to $14.40 for the call in Problem 6 — and holder demand determines where inside it the price sits. 10. **The open question is whose** $$m$$**.** Textbook pricing assumes a representative agent whose marginal utility tracks aggregate consumption. Limited participation, constrained arbitrageurs, and mandate-driven institutions each break that assumption. Palm's price was set by the investors who could buy it because the investors who wanted to sell could not borrow the shares. Chapters 16, 19, and 20 make the identity of the marginal investor an object of measurement. *** ## Key Terms * **Present value**: The price today of a stream of future payments, computed by discounting each at the rate appropriate to its date and risk * **Law of one price (LOOP)**: The condition that portfolios with identical payoffs in every state have identical prices; equivalently, that the pricing function is linear * **Arbitrage**: A portfolio with non-positive cost today and a payoff that is non-negative in every state and strictly positive in some * **Dominance**: The relation between payoffs when one is at least as large in every state and larger in some; ruling it out requires the dominating payoff to cost more * **Replicating portfolio**: A combination of traded assets that reproduces a target claim's payoff state by state; its cost is the claim's no-arbitrage price * **Cash-and-carry**: The arbitrage that enforces the forward price — borrow, buy the underlying, carry it to delivery * **State price (Arrow-Debreu price)** $$q\_s$$: The price today of one dollar delivered in state $$s$$ and nothing otherwise; the primitive object from which all other prices are built * **Complete market**: A market in which every payoff over the state space can be replicated by traded assets; equivalently, one in which state prices are unique * **Risk-neutral probability** $$\pi^{\ast}\_s$$: A state price normalized by the price of a riskless dollar, $$\pi^{\ast}\_s = q\_s R\_f$$; the distribution under which every asset earns the riskless rate * **Stochastic discount factor (SDF)** $$m$$: The ratio of a state price to its physical probability, $$m\_s = q\_s/\pi\_s$$; equal at a household's optimum to marginal utility growth * **Fundamental pricing equation**: $$p = E\[mx]$$, equivalently $$1 = E\[mR]$$ — the statement that every asset-pricing model is a specification of $$m$$ * **Hedge ratio (delta)**: The number of shares in the replicating portfolio, equal to the ratio of the spread in the derivative's payoffs to the spread in the underlying's * **Marginal investor**: The holder who is unconstrained at the margin in a given claim, and whose $$m$$ therefore appears in its price; an empirical object, not a modeling convenience * **Stub value**: The implied value of a parent company net of its holdings in a listed subsidiary; negative stubs are the sharpest observable violations of the law of one price *** ## Readings ### Required * Cochrane, J. (2005). *Asset Pricing*, revised edition. Princeton University Press. Chapters 1-3. *The canonical modern statement of the pricing equation and the source of this chapter's organizing claim that every asset-pricing model is a specification of the discount factor; read Chapter 1 before Chapter 4 of this book.* * Ross, S. (1978). "A Simple Approach to the Valuation of Risky Streams." *Journal of Business* 51(3): 453-475. *Establishes that absence of arbitrage is equivalent to the existence of a positive linear pricing rule — the formal content of §3.2 and §3.4.2, written before the machinery became standard.* * Dybvig, P. and S. Ross (1987). "Arbitrage." In *The New Palgrave: A Dictionary of Economics*, ed. Eatwell, Milgate and Newman. Macmillan. *Five pages that get the definitions right, including the distinctions among arbitrage, dominance, and the law of one price that §3.2 follows.* ### Recommended * Lamont, O. and R. Thaler (2003). "Can the Market Add and Subtract? Mispricing in Tech Stock Carve-outs." *Journal of Political Economy* 111(2): 227-268. *The Palm paper. Documents six negative-stub episodes and shows the mispricing survived both publicity and the presence of sophisticated arbitrageurs; the short-sale cost data are the empirical heart of this chapter's opening episode.* * Mitchell, M., T. Pulvino and E. Stafford (2002). "Limited Arbitrage in Equity Markets." *Journal of Finance* 57(2): 551-584. *The complementary study of negative stubs, with an accounting of why the trades were not taken; read it as the transition from §3.2's definitions to §3.7's question.* * Shleifer, A. and R. Vishny (1997). "The Limits of Arbitrage." *Journal of Finance* 52(1): 35-55. *Names the mechanism §3.7 gestures at: arbitrage is performed by specialists using other people's money, and the agency relation is what holds a published mispricing open. Chapter 15 §15.5 develops it.* * De Long, J. B., A. Shleifer, L. Summers and R. Waldmann (1990). "Noise Trader Risk in Financial Markets." *Journal of Political Economy* 98(4): 703-738. *The companion to §3.2's separation of arbitrage from dominance: what stops the trade is horizon risk rather than missing information, and the mispricing survives in equilibrium.* * Lucas, R. (1978). "Asset Prices in an Exchange Economy." *Econometrica* 46(6): 1429-1445. *The general-equilibrium step §3.5 skips: the discount factor is not posited but produced by a household consuming an endowment, which is where §3.6's consumption specialization comes from.* * Hansen, L. P. and S. Richard (1987). "The Role of Conditioning Information in Deducing Testable Restrictions Implied by Dynamic Asset Pricing Models." *Econometrica* 55(3): 587-613. *The formal home of §3.5's beta representation, and the source of the caveat this chapter states informally — a model that holds conditionally need not hold unconditionally.* * Martin, I. (2012). "On the Valuation of Long-Dated Assets." *Journal of Political Economy* 120(2): 346-358. *Carries §3.1's present-value arithmetic out to horizons at which the choice of discount rate dominates every other input; read it with Discussion Question 6.* * Dimson, E. and M. Mussavian (1999). "Three Centuries of Asset Pricing." *Journal of Banking and Finance* 23(12): 1745-1769. *A compact intellectual history of where the pricing equation came from, useful placed against §3.6, since each of the three specializations arrived as a separate research program.* * Duffie, D. (2003). "Intertemporal Asset Pricing Theory." In *Handbook of the Economics of Finance*, Volume 1B, ed. G. Constantinides, M. Harris and R. Stulz. Elsevier. *The measure-theoretic version of §3.5, including the change of measure previewed there and the existence results Appendix D defers to.* *** ## Discussion Questions 1. **Whose** $$m$$**?** Pick one claim: a thirty-year municipal general-obligation bond, an agency mortgage-backed security, or a single-name credit default swap. Who is plausibly the marginal investor in it? What would you need to observe to find out, and what would change about the claim's price if that investor's balance sheet came under stress? State which chapter of this book you would go to for the evidence. 2. **Why Palm persisted.** The negative stub was public, published, and enormous. Decompose the failure into its parts: the cost of borrowing shares, the recall risk, the margin requirement, and the conditionality of the distribution. Which of these is a *transaction cost* (a wedge that a large enough mispricing overcomes) and which is a *risk* (something that can make the trade lose money)? Why does the distinction matter for whether we should call the episode a market failure? 3. **Is a LOOP violation evidence of irrationality?** Suppose you observe two claims with identical payoffs trading at different prices. List three explanations that do not require any investor to be irrational. What additional observation would let you distinguish them? 4. **One hurdle rate.** A CFO applies a single 10 percent discount rate to every project the firm considers. Which of §3.1's three reasons for differing discount rates is she ignoring, and in which direction does the error bias the firm's investment — toward long-lived projects or short-lived ones, toward risky or safe ones? (Chapter 22 gives the corporate-finance treatment; answer it here from Table 3.1.) 5. **What does "the price" mean when markets are incomplete?** In Problem 6 the no-arbitrage price of a call is any number between $5.00 and $14.40. It nevertheless trades at one price. What determines which? Sketch the kind of evidence that would let you answer that empirically, and say why the answer is not available from the no-arbitrage conditions of §3.2. 6. **Fifty years out.** Section 3.1 presents present value as settled arithmetic, and at short horizons it is. Value a claim to one dollar delivered in fifty years at 3, 5, and 7 percent, and report the ratio of the largest of the three answers to the smallest. Then say what a water authority financing a reservoir, a life insurer reserving against annuities, and a climate cost-benefit study would each have to agree on before they could agree on that claim's value. Do §3.1's three reasons for differing discount rates settle the disagreement, or only relabel it? *** ## Problems **Problem 1 — State prices from observed prices.** A two-date, two-state economy. A riskless bond paying one dollar at date 1 trades at 0.9524. Asset A trades at 50 dollars and pays 80 in state $$u$$, 40 in state $$d$$. The true probability of state $$u$$ is 0.5. (a) Find the riskless gross return $$R\_f$$. (b) Find the state prices $$q\_u$$ and $$q\_d$$. (c) Find the risk-neutral probabilities $$\pi^{\ast}\_u$$ and $$\pi^{\ast}\_d$$, and verify that asset A earns the riskless rate under them. (d) Find asset A's expected gross return under the true probabilities and its risk premium. (e) Price a digital claim paying 100 dollars in state $$u$$ and nothing in state $$d$$. **Problem 2 — Pricing by replication and by state prices.** Continue with the economy of Problem 1. (a) Price a call on asset A with strike $50 by constructing its replicating portfolio. Report the hedge ratio and the amount borrowed. (b) Price the same call using the state prices from Problem 1, and confirm the two answers agree. (c) Price a put on asset A with strike 50 dollars, and verify put-call parity, $$C - P = p\_A - K/R\_f$$. (d) What is the call's expected gross return under the true probabilities? Express its risk premium as a multiple of asset A's, and explain the multiple. **Problem 3 — Find the arbitrage.** In a two-date, two-state economy the following three assets trade: a bond at $0.95 paying $1 in both states; a stock at $100 paying $130 and $90; and asset C at $40 paying $60 and $30. (a) Using the bond and the stock, find the implied state prices. (b) What should asset C cost? (c) Construct an explicit arbitrage: state the position in each of the three assets, the cash flow today, and the cash flow in each state at date 1. Report the profit per unit. (d) Suppose asset C cannot be sold short. Does an arbitrage still exist? Does your answer change if instead the *stock* cannot be sold short? **Problem 4 — The negative stub.** Parent company P has 350 million shares outstanding and owns 95 percent of listed subsidiary S. P has announced that in nine months each P share will receive 1.5 S shares, subject to a tax ruling. S trades at $95; P trades at $82. P also holds cash and a profitable operating business. (a) Compute the implied stub value per P share and in aggregate. What does the sign tell you? (b) Construct the trade that captures the mispricing. State every leg, the horizon, and what you own at the end. (c) Borrowing S costs 35 percent per annum, charged on the value of the borrowed shares. Ignoring all other frictions, does the trade still make money over nine months? At what annualized borrow cost does it break even? (d) List the risks that remain even for an arbitrageur who *can* borrow the shares at 35 percent. Which of them would show up as a loss, and which merely as an unpleasant path? **Problem 5 — Cash-and-carry with a dividend.** A stock trades at $60. The one-year riskless rate is 4 percent. The stock will pay a $2 dividend just before the one-year date. A one-year forward on the stock is quoted at $62.50. (a) What is the no-arbitrage forward price? (b) Construct the arbitrage, with a cash-flow table in the format of Table 3.2. Report the profit. (c) The stock is hard to borrow, at an annualized fee of 6 percent. Does that change your answer? Would it change your answer if the forward were quoted at $57.00 instead? **Problem 6 ★ — Incomplete markets and pricing bounds.** Now suppose there are three states. A bond at 0.95 pays one dollar in all three. A stock at 100 pays 130, 100, and 80 in states $$u$$, $$m$$, $$d$$. Only these two assets trade. (a) Write the two pricing equations in the three unknowns $$q\_u, q\_m, q\_d$$ and show that the system does not determine them. (b) Impose $$q\_s \ge 0$$ for all $$s$$ and find the admissible range of $$q\_u$$. (c) A call with strike $100 pays $30, $0, $0. Find the range of prices consistent with no arbitrage. (d) Explain in one paragraph what would have to be added to the model to price the call exactly, and what it would mean economically to add it. *** ## Selected Solutions *Solutions to Problems 1 and 2 follow. Solutions to the remainder are in the instructor materials.* **Problem 1.** (a) $$R\_f = 1/0.9524 = 1.05$$, so the riskless rate is 5 percent. (b) The two pricing equations are $$q\_u + q\_d = 0.9524$$ (the bond) and $$80q\_u + 40q\_d = 50$$ (asset A). Substituting the first into the second: $$80q\_u + 40(0.9524 - q\_u) = 40q\_u + 38.10 = 50$$, so $$q\_u = 11.90/40 = 0.2976$$ and $$q\_d = 0.9524 - 0.2976 = 0.6548$$. Note again that the bad-state dollar is more than twice as expensive as the good-state dollar although the two states are equally likely. (c) $$\pi^{\ast}\_u = q\_u R\_f = 0.2976 \times 1.05 = 0.3125$$ and $$\pi^{\ast}\_d = 0.6875$$; they sum to one. Under $$\pi^{\ast}$$, asset A's expected payoff is $$0.3125 \times 80 + 0.6875 \times 40 = 25 + 27.5 = 52.50$$, and $$52.50/50 = 1.05 = R\_f$$, as required. (d) Under the true probabilities the expected payoff is $$0.5 \times 80 + 0.5 \times 40 = 60$$, so $$E\[R\_A] = 60/50 = 1.20$$. The risk premium is $$20 - 5 = 15$$ percentage points. (e) The digital pays 100 in state $$u$$ only, so $$p = 100 \times 0.2976 = 29.76$$. **Problem 2.** (a) The call pays $$\max(x - 50, 0)$$, which is 30 in state $$u$$ and nothing in state $$d$$. The replicating portfolio $$(\Delta, B)$$ solves $$80\Delta + B = 30$$ and $$40\Delta + B = 0$$. Subtracting, $$40\Delta = 30$$, so $$\Delta = 0.75$$ and $$B = -40 \times 0.75 = -30$$. The recipe is: **buy 0.75 units of A, borrow 30 dollars of face value**, which raises $$30 \times 0.9524 = 28.57$$ today. The cost is $$0.75 \times 50 - 28.57 = 37.50 - 28.57 = 8.93$$. (b) By state prices, $$p = 30 \times q\_u = 30 \times 0.2976 = 8.93$$. The two agree, as §3.4.4 requires — the state prices were extracted from precisely the replication argument used in (a). (c) The put pays $$\max(50 - x, 0)$$: nothing in state $$u$$ and 10 in state $$d$$. So $$P = 10 \times 0.6548 = 6.55$$. Parity: $$C - P = 8.93 - 6.55 = 2.38$$, and $$p\_A - K/R\_f = 50 - 50 \times 0.9524 = 50 - 47.62 = 2.38$$. They agree. Parity is itself a law-of-one-price result and does not depend on the two-state structure. (d) Under the true probabilities the call's expected payoff is $$0.5 \times 30 = 15$$, so $$E\[R\_{\text{call}}] = 15/8.93 = 1.68$$: 68 percent expected, a risk premium of 63 percentage points against asset A's 15. The multiple is $$63/15 = 4.2$$, which is exactly the leverage of the replicating portfolio: 37.50 dollars of asset A supported by 8.93 dollars of equity, or $$37.50/8.93 = 4.2$$. An option's expected return is the underlying's premium scaled by the leverage that replication requires; it is not information about whether the option is cheap. *** ## Data Exercise: What a Discount Rate Does **Part A — Discount rates and horizon (free data: FRED).** Download the daily constant-maturity Treasury yield series `DGS1` (one-year) and `DGS10` (ten-year) from the Federal Reserve Bank of St. Louis's FRED database, over the longest available sample. 1. Pick four dates spanning very different rate environments — for example a business day in each of January 2000, January 2010, August 2020, and October 2023 — and record the `DGS1` and `DGS10` yields on each. 2. Value a fixed stream: $100 a year for thirty years, plus a $1,000 balloon at year 30. Compute its present value on each date twice: once discounting every cash flow at that date's `DGS10` yield (a flat-curve approximation), and once using a crude two-point curve that applies `DGS1` to cash flows in years 1-3 and `DGS10` to the rest. 3. Report the four valuations as a table and compute the percentage change between the highest- and lowest-rate dates. Then decompose it: how much of the change comes from the coupon stream and how much from the balloon? Verify Table 3.1's claim by computing the same percentage change for a five-year version of the same instrument. 4. Compute the elasticity of value with respect to the discount rate, $$-(dP/P)/(dr)$$, numerically at each date by repricing at $$r \pm 0.25$$ percentage points. This number is (modified) duration. Note that it *changes with the level of rates* and explain why from the arithmetic of Table 3.1. Chapter 9 gives the closed form. **Part B — What was actually paid, and what was actually received (free data: Shiller).** Download Robert Shiller's long-run US series (monthly S\&P Composite price, dividends, earnings, the consumer price index, and the long-term interest rate, available from his Yale website), which begins in 1871. 1. Choose a starting year at least fifty years back. Compute the *realized* present value of the real dividend stream actually paid on the index from that year forward, discounting at (i) the long-term interest rate prevailing in the starting year and (ii) that rate plus a 5-percentage-point risk premium. Truncate at the end of the sample and state how you handle the terminal value. 2. Compare both figures to the real price of the index in the starting year. Which discount rate would have made the observed price "correct" ex post? Repeat for four or five different starting years and report the implied discount rates as a series. 3. Discuss what the variation in that implied series does and does not establish. Shiller's argument is that prices move too much to be explained by news about subsequent dividends; the discount-rate response is that the variation belongs in the denominator. Chapter 7 treats the disagreement properly. State here which object in §3.5's notation each side is claiming moves. **Part C ★ (if you have WRDS).** Using CRSP's Treasury files, construct an actual zero-coupon curve on each of your four dates rather than the flat approximation of Part A, and reprice the same instrument. Report how much of the valuation difference in Part A was an artifact of the flat-curve assumption. Then, using OptionMetrics, extract the risk-neutral density for the S\&P 500 at a one-year horizon on one of those dates and compare it with the empirical distribution of one-year returns over the preceding twenty years. The gap between the two is $$\pi^{\ast}$$ against $$\pi$$ — §3.4.6's wedge, measured.