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# Chapter 8: Derivatives and Option Pricing

*Part II: Asset Pricing — Financial Economics: Claims, Prices, and Holders*

***

## Opening Episode: A Smoking Lounge in Chicago, April 26, 1973

The Chicago Board of Trade had a smoking lounge on the fourth floor, off the grain-futures floor, and by the spring of 1973 it had been cleared of its furniture and fitted with a trading ring. On Thursday, April 26, the Chicago Board Options Exchange opened for business there. Sixteen stocks were listed. Only calls were traded; puts would not be listed until 1977. About nine hundred contracts changed hands on the first day, in a market that had until then been conducted by telephone among a few dozen put-and-call brokers in New York, who matched buyers and sellers one contract at a time and quoted whatever the traffic would bear.

There was no accepted way to say what a call was worth. Practitioners used rules of thumb and a feel for the underlying. The academic literature had a formula from Bachelier in 1900 and several from the 1960s, all of which required knowing the stock's expected return and the investor's risk aversion — the two things nobody knew.

Weeks later, in the May-June issue of the *Journal of Political Economy*, Fischer Black and Myron Scholes published "The Pricing of Options and Corporate Liabilities." The paper had been rejected by that journal and by the *Review of Economics and Statistics* before Merton Miller and Eugene Fama pressed the editors to reconsider it. In the same season Robert Merton published a companion in the *Bell Journal* that generalized the argument and supplied the continuous-time machinery. The formula they arrived at contains five inputs: the stock price, the strike, the time to expiration, the interest rate, and the volatility of the stock. It does not contain the stock's expected return. It does not contain anyone's risk aversion. Two investors who disagree completely about whether the stock is going up must nonetheless agree on the price of the option, or one of them can be arbitraged.

The reason is the argument of Chapter 3 §3.3, applied harder. An option can be manufactured out of the underlying stock and borrowing, if you are willing to adjust the recipe continuously as the stock moves. If it can be manufactured, its price is the cost of manufacturing it, and a cost of manufacture involves no forecast. Chapter 3 priced a call in two states with a hedge ratio of 0.75 and a loan of $64.29. This chapter takes that argument to its limit.

The second thing that happened is less often taught and matters as much. Within a few years the formula was not merely describing the market but running in it. Black sold sheets of model prices by subscription; Texas Instruments sold a handheld calculator with the equation built in; floor traders learned to quote not in dollars but in implied volatility, which is a quote in the model's own units. By the early 1980s a dealer who quoted a price far from the formula's was assumed to have made an error rather than to hold a view. Donald MacKenzie's evidence, which Chapter 7 §7.6 introduced, is that the fit between the model and the market improved because the model was adopted, not only because it was right — and that the fit broke on a single day in October 1987 and has never returned. Section 8.5 is about the shape that break left in the data.

Both themes run through the chapter. Derivatives are priced by replication, which is why their pricing theory is the cleanest in finance. And the model that prices them became infrastructure, which is why its failures are not academic.

***

## 8.1 Payoffs, Bounds, and Parity

A derivative is a claim whose payoff is a function of the price of something else. The something else — the **underlying** — is written $$S\_t$$, with $$S\_0$$ its price today and $$S\_T$$ its price at the derivative's expiration date $$T$$. Everything in this chapter follows from writing that function down.

**Forwards and futures.** A forward contract obliges its holder to buy one unit of the underlying at date $$T$$ for a price $$F$$ agreed today; nothing changes hands at date 0. The payoff at $$T$$ is $$S\_T - F$$: linear, unbounded in both directions, symmetric. A futures contract has the same economics with different plumbing (§8.2).

**Calls and puts.** A **call** gives its holder the right, not the obligation, to buy one unit at the **strike** $$K$$ at date $$T$$; a **put** gives the right to sell. Their payoffs are

$$
C\_T = \max(S\_T - K, 0), \qquad P\_T = \max(K - S\_T, 0)
$$

The kink at $$K$$ is the whole of the difference between an option and a forward, and it is the reason options cost money up front while forwards do not. The right to walk away is worth something, so it is sold, not given.

Table 8.1 evaluates the four basic positions at three terminal prices, with $$K = 100$$.

**Table 8.1: Payoffs at expiration, strike $100**

| Position                    | $$S\_T = 80$$ | $$S\_T = 100$$ | $$S\_T = 130$$ | Shape                                                        |
| --------------------------- | ------------- | -------------- | -------------- | ------------------------------------------------------------ |
| Long forward at $$F = 100$$ | $$-20$$       | 0              | $$+30$$        | Straight line through $$(100, 0)$$, slope $$+1$$             |
| Long call                   | 0             | 0              | $$+30$$        | Flat at zero to the left of 100, slope $$+1$$ to the right   |
| Long put                    | $$+20$$       | 0              | 0              | Slope $$-1$$ to the left of 100, flat at zero to the right   |
| Short call                  | 0             | 0              | $$-30$$        | Flat at zero, then slope $$-1$$: the mirror of the long call |

*Source: Author's calculation.*

Read the first three rows together. Long a call and short a put at the same strike, and you hold a straight line of slope one — a forward: to the right of the strike the call supplies the upside and the put is dead, to the left the put supplies the downside and the call is dead. That observation, priced, is put-call parity.

Figure 8.1 draws all four, and adds the one thing a payoff table cannot carry: where each position breaks even once the premium is paid for.

![Figure 8.1: Payoff diagrams](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-b160b227313a80eb0dde071a3d62a0ff6387ee25%2Ffig_08_01_payoff_diagrams.png?alt=media)

**Figure 8.1: Payoff diagrams.** The four basic positions at expiry, struck at 100, with the premiums taken from the benchmark option of §8.4 — a call at 10.45 dollars and, by parity, a put at 5.57. The solid line in each panel is the payoff of Table 8.1: flat then sloped for the call, sloped then flat for the put, and the short positions their reflections in the horizontal axis. Nothing in the solid lines depends on what the option cost, which is why the table can be written without a price. The dashed line is the same payoff net of the premium, carried forward to expiry at the riskless rate, and the black marker is where it crosses zero — 110.99 for the call and 94.14 for the put. Note that the breakeven is not the strike plus the premium: the premium is paid a year earlier than the payoff arrives, and at 5 percent that costs another 54 cents. The long positions have unbounded gains and a loss capped at the premium; the short positions have the mirror, which is a capped gain against a loss that is unbounded for the call and bounded only by zero for the put. Read panels (a) and (b) together and the sense in which an option is a transfer rather than a creation of risk is visible: the two curves sum to zero at every terminal price.

**No-arbitrage bounds.** Before any model, several facts follow from dominance (Chapter 3 §3.2). A call's payoff is never negative, so $$C \ge 0$$; it never exceeds $$S\_T$$, so $$C \le S\_0$$. More usefully, a call is worth at least the forward's value when that is positive:

$$
C \ge \max\left(S\_0 - \frac{K}{R\_f}, 0\right)
$$

The argument is a portfolio comparison. Buy the call and lend $$K/R\_f$$; at $$T$$ you hold $$\max(S\_T - K, 0) + K = \max(S\_T, K)$$, which is at least $$S\_T$$ in every state and more in some. The package dominates the stock and must cost more: $$C + K/R\_f > S\_0$$. With $$S\_0 = K = 100$$ and $$R\_f = 1.05$$, the bound is $$100 - 95.24 = 4.76$$; a one-year at-the-money call on a non-dividend-paying stock cannot trade below 4.76 dollars however low volatility is. A parallel argument gives $$P \ge \max(K/R\_f - S\_0, 0)$$.

One corollary is a result, not the rule students usually meet: an American call on a non-dividend-paying stock is never exercised early, because exercising converts an asset worth at least $$S\_0 - K/R\_f$$ into $$S\_0 - K$$. Puts are different, and §8.3.5 explains why.

**Put-call parity.** Compare two portfolios. Portfolio A: one call plus a riskless bond paying $$K$$ at date $$T$$. Portfolio B: one put plus one share. At expiration A is worth $$\max(S\_T - K, 0) + K = \max(S\_T, K)$$, and B is worth $$\max(K - S\_T, 0) + S\_T = \max(S\_T, K)$$. Identical in every state. By the law of one price (Chapter 3 §3.2) they cost the same today:

$$
C + \frac{K}{R\_f} = P + S\_0 \qquad\Longleftrightarrow\qquad C - P = S\_0 - \frac{K}{R\_f}
$$

Notice what is absent. No distribution for $$S\_T$$, no volatility, no preferences, no model of any kind — the argument needs only that the two payoff vectors coincide. Parity is the purest law-of-one-price result in finance, and it holds in any economy in which the four instruments trade and can be held to expiry. It is also the standard first test of an options dataset: quotes that violate it are almost always stale, mismatched in expiration, or American.

**A parity violation, worked.** A stock trades at $$S\_0 = 100$$ dollars and pays no dividend. One-year calls and puts struck at $$K = 100$$ trade at $$C = 8.00$$ and $$P = 2.50$$. The one-year riskless gross return is $$R\_f = 1.05$$. Parity requires $$C - P = 100 - 100/1.05 = 4.762$$. The observed difference is $5.50. The call is rich relative to the put by $0.738, and Table 8.2 collects it.

**Table 8.2: A conversion arbitrage on an $0.738 parity violation**

| Leg                          | Cash flow today | Cash flow at $$T$$       |
| ---------------------------- | --------------- | ------------------------ |
| Sell one call                | $$+8.00$$       | $$-\max(S\_T-100,0)$$    |
| Buy one put                  | $$-2.50$$       | $$+\max(100-S\_T,0)$$    |
| Buy one share                | $$-100.00$$     | $$+S\_T$$                |
| Borrow $$100/1.05 = 95.238$$ | $$+95.238$$     | $$-100.00$$              |
| **Total**                    | $$+0.738$$      | $$0$$ **in every state** |

*Source: Author's calculation.*

Check the last column at three prices. At $$S\_T = 60$$: the call expires worthless, the put pays $$40$$, the share is worth $$60$$, and the loan takes $$100$$ — net zero. At $$S\_T = 140$$: the short call costs $$40$$, the put is worthless, the share is worth $$140$$, the loan takes $$100$$ — net zero. At $$S\_T = 100$$ everything is $$100$$ and the loan takes it. The position is money today against nothing ever, which is an arbitrage in the sense of Chapter 3 §3.2. Desks call the package a **conversion**; run in reverse, with the call bought and the put sold, it is a **reversal**. The trade is close to riskless in practice, which is why observed parity deviations in liquid listed markets are small, short-lived, and roughly the size of the bid-ask spread and the cost of the stock borrow — and why they widen exactly when that borrow becomes expensive, as Chapter 3's Palm episode showed.

Figure 8.2 is the argument in two pictures, with the arithmetic of the violation written beside them.

![Figure 8.2: Put-call parity as a payoff identity](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-0d8cab5a2a477a7782504526bc143d373837c2f2%2Ffig_08_02_put_call_parity.png?alt=media)

**Figure 8.2: Put-call parity as a payoff identity.** Panel (a) is a call plus a bond paying the strike; panel (b) is a put plus a share. In each the two components are dashed and the package is solid, and the two packages are the same function of the terminal price — the upper envelope of the strike and the share, kinked at 100. The three markers pick out arbitrary terminal prices at which the two packages agree; they agree at every other one as well, which is the whole of the argument. No distribution for the terminal price has been assumed, no volatility, no preferences, and no model: two payoff vectors coincide, so two prices must. The block on panel (b) prices §8.1's violation. Quoted at 8.00 and 2.50, the call-less-put difference is 5.50 where parity requires 4.762, so the call is rich by 0.738 and the conversion of Table 8.2 collects exactly that, today, against a terminal cash flow of zero in every state. Parity is the purest law-of-one-price result in finance and the first test any options dataset should be made to pass.

***

## 8.2 Forwards, Futures, and the Cost of Carry

Chapter 3 §3.3 priced a one-year forward on a non-dividend-paying stock at $$F = S\_0 R\_f = 105$$ dollars and made the point that no probability entered. This section extends the same cash-and-carry argument to underlyings that pay something, cost something, or are useful to own.

Let $$y$$ be the **net convenience yield**: the flow of benefits accruing over the life of the contract to whoever holds the physical asset rather than the contract, expressed as a rate on the spot price and measured net of the cost of holding it. (The same letter is a bond's yield to maturity in Chapter 9 and the learnable part of a payoff in Chapter 7 §7.2; the three senses never co-occur.) For a dividend-paying stock, $$y$$ is the dividend yield. For a coupon bond it is the coupon yield; for a foreign currency it is the foreign interest rate; for a commodity it is the value of having the barrel or the bushel on hand, less storage, insurance and spoilage. One symbol covers all four cases because the arbitrage is identical in all four: the holder of the physical asset receives $$y$$, the holder of the forward does not, and the forward price must adjust by exactly that much or the cash-and-carry trade of Chapter 3's Table 3.2 is available. With $$k$$ written separately for the proportional storage cost, so that $$y = y^{\text{gross}} - k$$, the one-period relation is

$$
F\_{0,T} = S\_0\left(1 + r\_f - y\right) = S\_0\left(1 + r\_f + k - y^{\text{gross}}\right)
$$

and over a horizon $$T$$ in continuous compounding, $$F\_{0,T} = S\_0 e^{(r\_f - y)T}$$. This is the **cost-of-carry** relation. The forward price is the spot price plus the cost of carrying the asset to delivery, where the cost of carrying is interest plus storage minus whatever the asset throws off along the way.

The sign of $$r\_f - y$$ determines the shape of the futures curve, and the two shapes have names.

**Table 8.3: Carry and the shape of the futures curve**

| Condition    | Curve                                     | Name              | Typical setting                                       |
| ------------ | ----------------------------------------- | ----------------- | ----------------------------------------------------- |
| $$y < r\_f$$ | Futures above spot, rising with maturity  | **Contango**      | Ample inventory; financing and storage dominate       |
| $$y > r\_f$$ | Futures below spot, falling with maturity | **Backwardation** | Scarce inventory; the physical asset is worth holding |

*Source: Author's construction.*

**The theory of storage** supplies the economics of $$y$$ for commodities, and it is the reason commodity futures are not simply a stock with a funding cost. Kaldor and Working's argument, developed empirically by Brennan and Telser and formalized since, is that inventory is held for a reason: a refiner with crude in the tank can meet an unexpected order, keep the plant running through a delivery disruption, and avoid the cost of shutting down and restarting. That option has value, and the value is high exactly when inventories are low, because that is when running out is likely. So $$y$$ is a decreasing function of the stock on hand. When inventories are ample, the marginal barrel supplies no such convenience, $$y$$ collapses toward zero, and the curve reverts to full carry — the contango case, in which the futures price is bounded above by $$S\_0(1 + r\_f + k)$$ because anyone can buy the spot, store it, and sell the future. When inventories are scarce, $$y$$ exceeds $$r\_f$$ and the curve inverts. There is no corresponding bound below: you cannot store a negative barrel, so the reverse cash-and-carry requires borrowing the physical commodity, which for most commodities cannot be done at any price. **Backwardation is therefore not an arbitrage violation. It is what an unenforceable inequality looks like.**

![Figure 8.7: Contango and backwardation](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-4e8d916954377e7b1f04c3f015ae549237a1809e%2Ffig_08_07_contango_and_backwardation.png?alt=media)

**Figure 8.7: Contango and backwardation.** WTI spot and the first four NYMEX contracts on three dates. On 12 February 2009 the curve slopes up from 34 dollars at the spot to 48 four months out: inventories were ample, the marginal barrel in the tank supplied no convenience, and the futures price is the spot plus the cost of carrying it. On 2 March 2022 it slopes down, from 111 to 98: inventories were scarce, the convenience yield exceeded the interest rate, and no arbitrage stops it, because the trade that would — sell the physical, buy the future — needs a barrel you do not have and generally cannot borrow. The third date is the one that shows why the two bounds are not the same kind of object. On 20 April 2020 the tanks at Cushing were full. The cost of carry stopped being a number, so the upper bound the cash-and-carry argument enforces stopped existing, and the front contract settled at −37.63 dollars: holders paid others to take delivery. The arbitrage argument was not violated. Its premise was. *Source: US Energy Information Administration, petroleum navigator daily series (RWTC, RCLC1-RCLC4). The EIA stopped publishing the NYMEX contracts in April 2024, so all three dates are historical.*

A worked case. Crude trades at $$S\_0 = 80$$ dollars, the one-year riskless rate is 5 percent, storage costs 3 percent of spot per year, and the one-year future trades at 78. Then $$y = 1 + r\_f - F/S\_0 = 1.05 - 0.975 = 0.075$$, a net convenience yield of 7.5 percent, or 10.5 percent gross of storage. The market is paying an implicit 10.5 percent a year for the privilege of holding the barrel rather than a claim to it — a statement about inventory, not about forecasts.

This is the point at which the empirical material of Chapter 6 §6.6 acquires its mechanism. That section sorted commodities by the slope of the futures curve and found that backwardated commodities outperform contangoed ones, and left the reason here. The reason is the theory of storage: the basis is a noisy but genuine read on inventory, low inventory produces backwardation, and a buyer of a backwardated future is being paid to supply storage capacity the market does not have. The premium is compensation for bearing the risk that the shortage resolves.

**Futures versus forwards.** A forward settles once, at delivery. A futures contract is marked to market daily: the exchange debits and credits each account with the day's price change, so that a long futures position is economically a sequence of one-day forwards, each initiated at the previous close. The consequence is that a futures holder receives cash gains early and pays losses early, and those cash flows are reinvested or financed at rates that may themselves be random. When interest rates are uncorrelated with the futures price the two prices coincide; when the correlation is positive — gains arrive when reinvestment rates are high — the futures price exceeds the forward price, a result due to Cox, Ingersoll and Ross (1981). For most contracts the wedge is negligible, which is why the two words are used interchangeably in pricing and never in risk management: marking to market converts a paper loss into a margin call, and the machinery around that — clearing houses, initial and variation margin, and what happens when the margin cannot be met — is developed in *International Finance* Chapter 13.

***

## 8.3 The Binomial Model

This is the chapter's engine. Everything before it is bounds; everything after it is this argument taken to a limit.

### 8.3.1 One period, again

Return to Chapter 3 §3.4 and its numbers, because the bridge should be exact. A stock trades at $$S\_0 = 100$$ dollars and will be worth 130 or 90 at date 1. The riskless gross return is $$R\_f = 1.05$$. The true probability of the up state is 0.6. A call struck at 100 pays 30 or nothing.

Write $$u$$ and $$d$$ for the gross up and down multipliers on the stock, so here $$u = 1.30$$ and $$d = 0.90$$. Let $$\Delta$$ be the number of shares in the replicating portfolio and $$B$$ the face value of riskless borrowing. Matching the option in both states,

$$
\Delta S\_0 u + B = C\_u, \qquad \Delta S\_0 d + B = C\_d
$$

Subtracting gives the general hedge ratio,

$$
\Delta = \frac{C\_u - C\_d}{S\_0(u - d)} = \frac{30 - 0}{130 - 90} = 0.75
$$

and back-substitution gives $$B = C\_d - \Delta S\_0 d = -67.50$$. The recipe costs $$\Delta S\_0 + B/R\_f = 75 - 64.29 = 10.71$$, and by the law of one price that is the option's price. Chapter 3 established this. What follows is new.

### 8.3.2 Risk-neutral valuation, in general

Chapter 3 §3.4.5 defined the risk-neutral probabilities as normalized state prices, $$\pi^{\ast}\_s = q\_s R\_f$$, and computed $$\pi^{\ast}\_u = 0.375$$ in this economy. The binomial model turns that definition into a recipe that needs no state prices at all.

Substitute the solutions for $$\Delta$$ and $$B$$ back into the cost of the recipe and collect terms. The algebra is two lines and the result is

$$
C\_0 = \frac{1}{R\_f}\Big\[\pi^{\ast}C\_u + (1-\pi^{\ast})C\_d\Big], \qquad \pi^{\ast} \equiv \frac{R\_f - d}{u - d}
$$

Check it: $$\pi^{\ast} = (1.05 - 0.90)/(1.30 - 0.90) = 0.375$$, and $$(0.375 \times 30 + 0.625 \times 0)/1.05 = 10.71$$. The same number, from a formula with no simultaneous equations in it.

Three properties of $$\pi^{\ast}$$ deserve to be stated as such.

First, $$\pi^{\ast}$$ lies strictly between 0 and 1 **if and only if** $$d < R\_f < u$$. If the riskless return were above $$u$$, the bond would dominate the stock and the no-arbitrage condition would fail; if it were below $$d$$, the stock would dominate the bond. So the condition that makes $$\pi^{\ast}$$ look like a probability is precisely the condition of no arbitrage. This is the fundamental theorem of asset pricing in its smallest instance: prices admit no arbitrage if and only if a risk-neutral measure exists.

Second, under $$\pi^{\ast}$$ the stock itself earns the riskless rate. Its expected gross return is $$\[\pi^{\ast} u + (1-\pi^{\ast})d]$$, and substituting $$\pi^{\ast}$$ gives exactly $$R\_f$$. Every asset in the economy does, which is what Chapter 3 §3.4.6 showed in its table and is the reason for the name.

Third, $$\pi^{\ast}$$ is not a belief. It is $$(R\_f - d)/(u-d)$$ — a ratio of quoted prices. Nobody in the economy assigns probability 0.375 to the up state; the true probability is 0.6. The measure is an accounting device that stores the market's risk adjustment in a place where it is convenient to compute with.

**The general recipe**, which is what this chapter contributes to Chapter 3's apparatus, is now stated in three steps and applies to any claim on any underlying in any number of periods:

1. Replace the true probabilities of the underlying's movements with risk-neutral ones, chosen so that the underlying itself earns the riskless rate.
2. Take the expected payoff of the claim under those probabilities.
3. Discount at the riskless rate.

In symbols, $$p\_0 = E^{\ast}\[x\_T]/R\_f^{T}$$. Formally, the deflated price process $$p\_t/R\_f^{t}$$ is a martingale under $$\pi^{\ast}$$ — its expected future value equals its current value — which is why the measure is also called an equivalent martingale measure. "Equivalent" means it agrees with the true measure about which states are possible, which is exactly what no arbitrage requires and all it requires.

### 8.3.3 Why the true drift drops out

The most important fact in this chapter is that $$\pi = 0.6$$ appears nowhere in the calculation. Two investors, one convinced the stock will rise and one convinced it will fall, must agree that the call is worth $10.71. They may act on their disagreement by trading the stock. They cannot act on it by trading the option against the stock, because the option is a package of stock and borrowing and its price is the cost of the package.

Why should the expected return of the underlying be irrelevant to the value of a claim whose payoff is a function of that return? The answer is that it is not irrelevant — it has already been counted. The expected return of the stock is embedded in $$S\_0$$. A more optimistic view about the stock's prospects is a view that $$S\_0$$ is too low, and it is expressed by buying stock. The option is being priced *relative to the observed* $$S\_0$$, and relative pricing takes the underlying's price, and therefore everything that determines it, as data. Ask what the option is worth given that the stock costs $100 and can go to $130 or $90, and the answer is a statement about manufacturing cost, not about the future. The drift would matter if we were pricing the option and the stock jointly from primitives — from preferences, endowments and technology. We are not. We are pricing one against the other.

The same argument disposes of risk aversion, and this is where the phrase "risk-neutral valuation" earns and forfeits its name. It does not assume anyone is risk-neutral. It observes that the risk premium the market actually charges is already visible in $$S\_0$$ relative to $$R\_f$$, so that discounting under $$\pi^{\ast}$$ double-counts nothing. In Chapter 3's numbers the stock earns a 9-percentage-point premium under the true probabilities and the call earns 63 points, seven times as much because the replicating portfolio is levered seven to one — and both numbers are consequences of the pricing, not inputs to it. This is also the sharpest available answer to a common student objection, that the model must be wrong because it ignores whether the stock is a good investment. It ignores it in the same way that the price of a sandwich ignores whether you are hungry: the ingredients have prices, and the sandwich costs what the ingredients cost.

The limitation is the mirror image of the strength. Because the argument is relative, it is silent on everything absolute. It cannot tell you whether the stock is fairly priced, whether the equity premium is too high, or whether volatility is what the market says it is. It converts one price into another, exactly, and it converts nothing into anything.

### 8.3.4 Two periods, worked

Extend the tree by letting the same multipliers apply again. A stock at $$S\_0 = 100$$ dollars moves by $$u = 1.2$$ or $$d = 0.8$$ in each of two periods; the riskless gross return is $$R\_f = 1.05$$ per period; a European call is struck at $$K = 100$$ and expires at the end of period two. Then

$$
\pi^{\ast} = \frac{1.05 - 0.8}{1.2 - 0.8} = 0.625
$$

The terminal nodes are $$S\_{uu} = 144$$, $$S\_{ud} = 96$$ and $$S\_{dd} = 64$$, with call payoffs $$44$$, $$0$$ and $$0$$.

Work backwards. At the upper node after one period the stock is at $$120$$, and the one-period formula gives

$$
C\_u = \frac{0.625 \times 44 + 0.375 \times 0}{1.05} = \frac{27.50}{1.05} = 26.19
$$

At the lower node the stock is at $$80$$ and both continuations are worthless, so $$C\_d = 0$$. Apply the formula once more at the root:

$$
C\_0 = \frac{0.625 \times 26.19 + 0.375 \times 0}{1.05} = 15.59
$$

Equivalently, in one step, weight each terminal payoff by its risk-neutral binomial probability and discount twice: $$\left(0.625^2 \times 44\right)/1.05^2 = 17.1875/1.1025 = 15.59$$.

**Table 8.4: The two-period tree,** $$S\_0 = 100$$**,** $$u = 1.2$$**,** $$d = 0.8$$**,** $$R\_f = 1.05$$**,** $$K = 100$$

| Node             | Stock  | Call value     | $$\Delta$$ | Borrowing (face) |
| ---------------- | ------ | -------------- | ---------- | ---------------- |
| Root             | 100.00 | 15.59          | 0.6548     | $$-52.38$$       |
| After one up     | 120.00 | 26.19          | 0.9167     | $$-88.00$$       |
| After one down   | 80.00  | 0.00           | 0          | 0                |
| Two ups          | 144.00 | 44.00 (payoff) | —          | —                |
| One up, one down | 96.00  | 0.00 (payoff)  | —          | —                |
| Two downs        | 64.00  | 0.00 (payoff)  | —          | —                |

*Source: Author's calculation.*

The $$\Delta$$ column is the point of the exercise. The replicating portfolio is not bought once and held; it is **rebalanced** at every node. At the root the recipe is 0.6548 shares against 52.38 dollars of face borrowing, costing $$65.48 - 49.89 = 15.59$$. If the stock rises, the hedge must be increased to 0.9167 shares, financed by borrowing more; if it falls, the position is liquidated entirely. And each rebalancing is **self-financing**: at the up node the old portfolio is worth $$0.6548 \times 120 - 52.38 = 26.19$$, exactly what the new one costs. No money is added and none is taken out. The strategy is a machine that turns $15.59 today into the option's payoff at expiry, and it works state by state, not on average. Figure 8.3 draws the tree, with that recipe written under every node at which it still has to be held.

![Figure 8.3: The binomial tree](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-945126012d74d03f69ee7bce7e8c691d82b30fcb%2Ffig_08_03_binomial_tree.png?alt=media)

**Figure 8.3: The binomial tree.** Section 8.3.4's two-period tree, at that section's parameters — $$S\_0 = 100$$, $$u = 1.2$$, $$d = 0.8$$, $$R\_f = 1.05$$, and a call struck at 100 — with Table 8.4's replicating recipe written under every interior node. The risk-neutral probability is derived at the side out of quoted prices alone, $$\pi^{\ast} = (R\_f - d)/(u - d) = 0.625$$, and the option values are filled in from the right: the terminal payoffs are 44, nothing and nothing, so $$C\_u = 26.19$$, $$C\_d = 0$$ and $$C\_0 = 15.59$$, which the single-pass binomial formula reproduces in one line. The true probability of an up move plays no part in any of it. The recipe is not bought once and held: it changes at every node, and it is self-financing between them, since at the up node the portfolio bought at the root is worth 26.19, exactly what the new one costs. *Source: Author's construction from the two-period tree of Section 8.3.4, Table 8.4.*

That is the whole of the multi-period extension. Any number of periods is the same operation repeated: risk-neutral expectation of the next node's value, discounted one period, rolled back to the root. A tree with $$n$$ periods has $$n+1$$ terminal nodes and prices any European claim in a single pass, and the European price has the closed form

$$
C\_0 = \frac{1}{R\_f^{n}}\sum\_{j=0}^{n}\binom{n}{j}(\pi^{\ast})^j(1-\pi^{\ast})^{n-j}\max\left(S\_0 u^j d^{n-j} - K, 0\right)
$$

which is Cox, Ross and Rubinstein's (1979) formula and the reason their paper is on the required list.

### 8.3.5 American options and early exercise

An American option may be exercised at any node, not only at expiry. The tree handles this with one extra line in the backward recursion: at every node, the value is the larger of the continuation value and the immediate exercise value.

$$
V\_t = \max\Big(\text{exercise value},\ \tfrac{1}{R\_f}\big\[\pi^{\ast} V\_{t+1}^u + (1-\pi^{\ast})V\_{t+1}^d\big]\Big)
$$

Take a put on the same tree, struck at 100. Terminal payoffs are $$0$$, $$4$$ and $$36$$. The European put is worth $$\left(2 \times 0.625 \times 0.375 \times 4 + 0.375^2 \times 36\right)/1.1025 = 6.9375/1.1025 = 6.29$$, which put-call parity confirms: $$15.59 - 6.29 = 9.30 = 100 - 100/1.05^2$$.

Now the American version. At the down node the stock is at $$80$$. The continuation value is $$(0.625 \times 4 + 0.375 \times 36)/1.05 = 16/1.05 = 15.24$$. Immediate exercise is worth $$100 - 80 = 20$$. **Exercise.** At the up node, continuation is worth $$(0.375 \times 4)/1.05 = 1.43$$ against an exercise value of zero, so the option is held. Rolling back to the root with the corrected down value,

$$
P^{\text{Am}}\_0 = \frac{0.625 \times 1.43 + 0.375 \times 20}{1.05} = \frac{8.393}{1.05} = 7.99
$$

against $6.29 for the European. The early-exercise premium is $1.70, and it is not small: 27 percent of the European value.

Why puts and not calls? Exercising a put converts the option into cash — the strike — which then earns interest; exercising a call pays cash out early. Deep in the money, a put's remaining optionality is nearly worthless while the interest on the strike is real, so at some point the interest wins. Volatility opposes it, by making the surrendered optionality valuable; dividends push the other way and are what make early exercise of *calls* rational just before an ex-dividend date. The general lesson is that an American option has no closed-form price, because the exercise boundary is part of the solution. The tree finds it numerically, which is a large part of why the binomial model survives in practice long after continuous-time methods became standard.

***

## 8.4 Black-Scholes, Stated and Interpreted

Let the period in the tree shrink and the number of periods grow, holding the horizon fixed. Set $$u = e^{\sigma\sqrt{\Delta t}}$$ and $$d = 1/u$$ so that the tree's per-period variance matches an annual volatility $$\sigma$$. As $$\Delta t \to 0$$ the multiplicative random walk converges to geometric Brownian motion, the discretely rebalanced hedge becomes continuous, and the binomial price converges to a closed form. For a European call on a non-dividend-paying stock,

$$
C\_0 = S\_0N(d\_1) - K e^{-rT} N(d\_2)
$$

$$
d\_1 = \frac{\ln(S\_0/K) + \left(r + \tfrac{1}{2}\sigma^2\right)T}{\sigma\sqrt{T}}, \qquad d\_2 = d\_1 - \sigma\sqrt{T}
$$

where $$N(\cdot)$$ is the standard normal cumulative distribution function and $$r$$ is the continuously compounded riskless rate. The put follows from parity: $$P\_0 = C\_0 - S\_0 + Ke^{-rT}$$.

★ *The derivation — Itô's lemma, the hedged-portfolio argument, the Black-Scholes partial differential equation and its solution — is in Appendix D.* The logic, in words, is the logic of §8.3 with the periods run together. Hold $$\Delta$$ shares against a short option; over an instant, the portfolio's value does not depend on which way the stock moves, because the option's exposure is locally linear and $$\Delta$$ cancels it. A portfolio with no exposure must earn the riskless rate, or there is an arbitrage. That requirement, imposed at every instant and every stock price, is a partial differential equation, and Black-Scholes is its solution subject to the boundary condition $$C\_T = \max(S\_T - K, 0)$$. Nothing in the argument is statistical. The normal distribution enters only because it is what a limit of many small independent multiplicative shocks looks like.

Table 8.5 shows the convergence, for a benchmark option used through the rest of the chapter: $$S\_0 = K = 100$$ dollars, $$r = 5$$ percent, $$\sigma = 20$$ percent, $$T = 1$$ year.

**Table 8.5: The binomial price converges to Black-Scholes**

| Steps $$n$$   | Binomial call price |
| ------------- | ------------------- |
| 1             | 12.16               |
| 2             | 9.54                |
| 10            | 10.25               |
| 50            | 10.41               |
| 100           | 10.43               |
| 500           | 10.45               |
| 2,000         | 10.45               |
| Black-Scholes | **10.4506**         |

*Source: Author's calculation, Cox-Ross-Rubinstein parametrization* $$u = e^{\sigma\sqrt{\Delta t}}$$, $$d = 1/u$$.

The oscillation at small $$n$$ is characteristic: terminal nodes straddle the strike differently as $$n$$ changes parity, and convergence is $$O(1/n)$$ rather than smooth.

**What each input does.** Five inputs, of which four are observable and one is not.

* $$S\_0$$, the spot. The call is increasing in it; the derivative is $$N(d\_1) = 0.637$$ for the benchmark option, which is the delta.
* $$K$$, the strike. Decreasing, obviously.
* $$T$$, the time to expiry. Increasing for a call, because more time is more opportunity and the downside is already truncated.
* $$r$$, the riskless rate. Increasing, because the call defers payment of the strike, and $$Ke^{-rT}$$ is what a call holder saves.
* $$\sigma$$, the volatility. Increasing, and this is the input that matters. The benchmark call is worth $10.45 at 20 percent volatility and $14.23 at 30. Volatility is the only input that is not quoted somewhere, which is why the market's real content is a view about it.

Figure 8.4 shows what those five inputs produce, and isolates the part of the value that is not intrinsic.

![Figure 8.4: Black-Scholes value and intrinsic value](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-62d89a02942bb33c2aa22ab0d513a255cd92457b%2Ffig_08_04_black_scholes_value.png?alt=media)

**Figure 8.4: Black-Scholes value and intrinsic value.** Panel (a) is the benchmark call's value against the spot price at three maturities, drawn against the intrinsic value it would be worth if it expired now. The curve lies strictly above the kinked black line everywhere, and the gap is the time value: at the strike, where intrinsic value is zero, the whole of the 10.45 is time value. Panel (b) plots that gap alone, and makes three things visible that panel (a) hides. Time value is largest at the strike and falls away in both directions, because the option's payoff is most uncertain exactly where a small move changes whether it pays at all. It does not go to zero deep in the money — a one-year call struck at 100 with the stock at 145 still carries 5.03 dollars of it, which is the interest saved by deferring payment of the strike. And it collapses as the clock runs down: the same hump falls from 10.45 at a year to 4.62 at three months to 1.91 at eighteen days, converging onto the kink, which is the payoff. That collapse is the whole content of the statement that an option is a wasting asset, and it is why the time decay of a hedged book is a position and not an accounting entry.

$$N(d\_2)$$ **and** $$N(d\_1)$$**.** The two terms are not arbitrary. $$N(d\_2)$$ is the **risk-neutral probability that the option finishes in the money** — the probability, under $$\pi^{\ast}$$ rather than $$\pi$$, that $$S\_T > K$$. For the benchmark option it is 0.560. So $$Ke^{-rT}N(d\_2)$$ is the present value of paying the strike, weighted by the chance you will have to. The first term, $$S\_0 N(d\_1)$$, is the present value of receiving the stock in exactly those states, and $$N(d\_1)$$ is the corresponding probability under a measure in which the stock rather than the money-market account is the unit of account. Read that way the formula says something simple: **the call is worth what you get, times the chance you get it, minus what you pay, times the chance you pay it.**

**Assumption by assumption.** Each assumption buys something, and each fails in a known way.

*Constant, known volatility.* Buys the closed form. Fails in every direction: volatility is stochastic, clusters, and rises when prices fall. This is the assumption whose failure §8.5 is about.

*Lognormal returns with continuous paths.* Buys the exact hedge. Fails because prices jump; on 19 October 1987 the S\&P fell about 20 percent in a day, an event the lognormal model with 1987's estimated volatility assigns a probability so small as to be meaningless. When the stock can jump, the local hedge does not protect the hedger, the option is not replicable, and the model's central claim — redundancy — is false.

*Continuous, costless trading.* Buys self-financing replication. Fails with transaction costs, which make the hedge approximate and give the "price" a band whose width grows with the rebalancing frequency.

*Frictionless borrowing and shorting at a single rate* $$r$$. Buys the discounting. Fails for hard-to-borrow names, where the borrow cost enters as a negative dividend.

*No dividends.* Removable, not an assumption. Replace $$S\_0$$ by $$S\_0 e^{-yT}$$ with $$y$$ the payout yield of §8.2, and the formula prices options on indices, currencies (with $$y$$ the foreign rate) and futures.

*Price-taking hedger.* Buys the assumption that hedging does not move the underlying. This is the assumption that failed catastrophically in 1987 and that §8.6 shows still fails in miniature every month.

**Implied volatility.** Invert the formula. Take the observed market price of an option and solve numerically for the $$\sigma$$ that reproduces it; the answer is the **implied volatility**, $$\sigma\_{\text{imp}}$$. Because the call price is strictly increasing in $$\sigma$$, the inversion is unique. If the benchmark call trades at $12.00 rather than $10.45, its implied volatility is 24.1 percent; at $8.60 it is 15.0 percent.

Implied volatility is best understood as a change of units, not as a forecast. It is the price of the option restated in the model's language, in the same way that a bond's yield is its price restated in the language of a discount rate. Quoting in implied volatility is convenient because it strips out the mechanical dependence on the spot, the strike and the clock, leaving the one number traders disagree about. It also has the property that makes the next section possible: if the model were true, every option on the same underlying with the same expiry would have the same implied volatility, because there is only one $$\sigma$$ in the world. They do not.

***

## 8.5 The Smile

Take all the options on a single underlying with a single expiration date, invert each one for its implied volatility, and plot the results against the strike. Under Black-Scholes the plot is flat. It is not flat, and the shape it takes is the sharpest empirical result in derivatives.

**Table 8.6: The shape of the index-option implied volatility curve, one-year expiry (stylized)**

| Strike, as percent of spot     | Before October 1987 | After |
| ------------------------------ | ------------------- | ----- |
| 80 (deep out-of-the-money put) | ≈ 18                | ≈ 26  |
| 90                             | ≈ 18                | ≈ 22  |
| 100 (at the money)             | ≈ 18                | ≈ 19  |
| 110                            | ≈ 18                | ≈ 17  |
| 120 (out-of-the-money call)    | ≈ 18                | ≈ 16  |

*Source: Author's construction. Levels are illustrative and the shape is the object; the empirical pattern is documented for S\&P 500 index options by Rubinstein (1994) and the subsequent literature, which dates the regime change to the October 1987 crash.*

For equity indices the post-1987 curve is monotonically downward-sloping in the strike — a **skew** or **smirk** rather than a symmetric smile. In currency options the curve is closer to a genuine smile, symmetric and U-shaped, and in single-name equity options it is intermediate. The equity-index skew has been continuously present since 1987, across market regimes, in every liquid index in the world.

![Figure 8.5: The volatility smile](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-d801a39ecd80d186c1ad426cd8da31a758196da9%2Ffig_08_05_the_volatility_smile.png?alt=media)

**Figure 8.5: The volatility smile.** A live S\&P 500 chain, out-of-the-money options only — puts below the index level and calls above it, which is the liquid side of every strike — for a single December expiry. The curve is the skew this section is about, and its magnitude is easier to feel in numbers than in adjectives: a put twenty percent out of the money implies 28 percent volatility, the same index at the money implies 14, and a call ten percent out of the money implies 12. One asset, one horizon, three volatilities. Under the model that generated those quotes that is impossible; the parameter σ is a property of the asset and does not know what strike it is being asked about. The dashed line is what Black-Scholes requires — a flat profile at the at-the-money level — and, on Rubinstein's evidence, roughly what the market quoted before October 1987. Box 8.1 tells that story. Two cautions on reading the chart. Deep out-of-the-money quotes are wide, so the extreme left of the curve is a mid-point between prices at which little trades. And this is one chain on one date: the *level* of the whole curve moves with the market every day, while the *shape* — down and to the right, steeper on the put wing — has been there every day for nearly forty years. *Source: Cboe delayed option quotes for the S\&P 500 index, snapshotted on the date the figure states. Only contracts with a live bid are drawn.*

**What the skew says.** Because the Black-Scholes price is a monotone map from volatility to price, an implied volatility above the at-the-money level is simply a price above the lognormal benchmark. With a 4 percent riskless rate, an 80-strike one-year put on a 100 index is worth $0.50 under a flat 18 percent volatility and $1.85 under 26 percent — the market charges nearly four times the lognormal price for that insurance. Read as a distribution rather than as a set of prices — Breeden and Litzenberger showed that the second derivative of the option price with respect to the strike *is* the risk-neutral density, so a full set of quotes is a full distribution — the skew says that the risk-neutral distribution of index returns has a fat left tail and negative skewness. Under a flat 18 percent lognormal, the risk-neutral probability of the index falling more than 20 percent over a year is about 8.5 percent; at the 26 percent volatility the market charges for that strike, it is about 19 percent.

Three readings compete, and they are not mutually exclusive.

*Crash risk and fat tails.* The physical distribution really does have a fatter left tail than the lognormal, because prices jump and volatility rises as prices fall. The model is wrong and the market has corrected it. This is certainly part of the answer, but it cannot be all of it: the risk-neutral probabilities the skew implies are substantially larger than the historical frequencies of the corresponding declines.

*A crash risk premium.* The gap between the risk-neutral and physical tail probabilities is a price of risk, not a probability. A claim that pays in a crash is insurance, is expensive for exactly the reason Chapter 3 §3.4.2 gave — a dollar in a bad state costs more — and the skew is where that state price is quoted. The measured wedge is large: index options as a class have delivered substantially negative average returns to their buyers over long samples, which is what a large insurance premium looks like from the seller's side. Chapter 5 §5.5 makes the complementary point from the other direction: the skew is large, but not large enough to support the disaster calibration that would resolve the equity premium puzzle on its own.

*Demand.* The people who buy out-of-the-money index puts are not indifferent agents equating marginal utilities; they are institutions with mandates, and the dealers on the other side cannot hedge perfectly. That reading is §8.6's.

The deepest point is the second-order one. The smile is the market pricing the model's own inadequacy *within the model's own language*. Traders did not abandon Black-Scholes when they stopped believing it. They kept the formula as a quoting convention and moved the disagreement into the input, so that a single number per strike now carries everything the model omits — jumps, stochastic volatility, illiquidity, and risk premia. The formula survives as a coordinate system after it has failed as a theory. That is an unusual fate for a scientific model and it is not an accident. What the coordinate system cannot carry becomes visible as soon as a payoff depends on more than the terminal price of a single asset: a barrier option, an average-rate option, or a contract on the better of two assets is sensitive to the path, to the shape of the whole distribution, and to a correlation the surface never quotes — which is why desks price such contracts from models calibrated to the smile rather than from the smile itself.

> **Box 8.1 — Black-Scholes and the pre-1987 smile**
>
> The flatness of the pre-1987 column in Table 8.6 is the single most cited piece of evidence in the sociology of finance, and it is worth being precise about what it shows.
>
> When the CBOE opened in 1973, traded prices departed substantially from Black-Scholes values. Within roughly a decade the departures had largely closed. MacKenzie and Millo's account, drawn from interviews with floor traders and from the exchange's own records, traces the mechanism rather than asserting it: Black sold sheets of theoretical values; Texas Instruments sold a calculator with the formula in it; the CBOE's own materials taught it; and quoting shifted from dollars to implied volatility, which cannot be done at all without a model. By the early 1980s the model was the medium in which the market's participants communicated. MacKenzie calls this **Barnesian performativity** — use of a theory making the theory more nearly true — and Chapter 7 §7.6 places it in the general argument.
>
> Two guards against over-reading. First, this is not a claim that the model was empty. The replication argument is real, and it is why traders found the formula worth adopting. Second, the adoption was never complete: the fit was good, not perfect, and there is evidence of a mild skew in some pre-1987 samples.
>
> The decisive observation is what happened next. On 19 October 1987 the S\&P 500 fell about 20 percent in a day. Portfolio insurance — a synthetic put manufactured by exactly the dynamic hedging argument that underlies Black-Scholes, covering something on the order of sixty to ninety billion dollars of US equity on the Brady Commission's estimate — required its users to sell into the decline, all at once, and the Commission identified that selling as a principal amplifier. The model's own success had destroyed its premise that hedgers are small. MacKenzie calls this **counterperformativity**.
>
> The skew appeared immediately afterward and has never left. Read together, the two halves of the episode make a claim that neither half makes alone: the market conformed to the model until an event demonstrated that the model's central assumption was false, and then it stopped, permanently, and encoded the refutation as a permanent deformation of the model's own output. Whether the skew's persistence is memory, or a correct assessment of a risk that was always there, is genuinely open — and it is the question Discussion Question 2 asks.

> **Box 8.2 — The Greeks**
>
> The partial derivatives of the option price have names, and a desk's entire risk report is a list of them. Definitions and uses only; the practice of running a hedged book is Chapter 26 §26.2's, which rebalances this table's benchmark call week by week.
>
> **Table 8.7: The Greeks for the benchmark call** ($$S\_0 = K = 100$$, $$r = 5$$ percent, $$\sigma = 20$$ percent, $$T = 1$$)
>
> *Source: Author's calculation from the Black-Scholes formula.*
>
> Delta is §8.3's hedge ratio and is what a dealer trades to become locally indifferent to the underlying. Gamma is why local indifference does not last: a delta-hedged short option loses when the underlying moves either way, by roughly half of gamma times the squared move. Theta is the offset — the short option earns time decay — so a hedged book is a bet that realized movement will be smaller than the implied volatility it sold. Vega is the exposure to the quote itself, and for a book of many strikes and maturities it dominates. Gamma is also the channel through which options trading feeds back into the underlying, which is §8.6's subject.

| Greek                | Definition                          | Value                      | Reads as                                                           |
| -------------------- | ----------------------------------- | -------------------------- | ------------------------------------------------------------------ |
| Delta $$\Delta$$     | $$\partial C/\partial S$$           | 0.637                      | Shares of stock in the replicating portfolio; $$\Delta = N(d\_1)$$ |
| Gamma $$\Gamma$$     | $$\partial^2 C/\partial S^2$$       | 0.0188                     | How fast delta changes; the curvature the hedge cannot capture     |
| Vega $$\mathcal{V}$$ | $$\partial C/\partial\sigma$$       | 0.375 per volatility point | Sensitivity to the one unobservable input                          |
| Theta $$\Theta$$     | $$-\partial C/\partial T$$, per day | $$-0.018$$                 | The cost of waiting; what a long option pays for its convexity     |

***

## 8.6 Who Holds the Option, and What Their Constraints Do to Its Price

Every argument so far has been a replication argument, and every replication argument makes the holder irrelevant. If the option is manufacturable, who wants it cannot matter: the price is a cost of production, and a change in demand changes the quantity manufactured, not the price. That is the theory. The evidence says the holder matters, and the reason is that the manufacturing process does not work as advertised.

**The mechanism.** Dealers are the counterparty to almost all end-user option demand, and their business model is to hedge the exposure and earn the spread. But the hedge fails in exactly the ways §8.4 catalogued. The underlying jumps, so a delta hedge does not eliminate the risk of a large move. Volatility is stochastic, so a book that is delta-neutral is still exposed to a change in the quote. Positions cannot be unwound instantly. What remains after hedging is inventory risk that the dealer must be paid to bear, and the payment comes out of the option's price. **Demand-based option pricing** — the framework of Gårleanu, Pedersen and Poteshman (2009) — writes this down: the deviation of an option's price from its frictionless value is proportional to the net demand of end users for that option, scaled by the unhedgeable risk of the dealer's resulting inventory. Where end users are net buyers, options are expensive; where they are net sellers, cheap; and the size of the effect depends on how badly the dealer is stuck.

**Who is on each side, and it is not the same people.** The empirical content comes from decomposing exchange data into end-user and market-maker positions.

*Institutions buying index puts.* Pension funds, insurers, endowments and long-only managers with drawdown limits are structurally short crash insurance through their portfolios and buy index puts to cap it. The demand is one-directional — mandates create buyers of protection, not sellers — and it is concentrated in the out-of-the-money strikes where the skew is steepest. Gårleanu, Pedersen and Poteshman document that end users are net long index options, and that the expensiveness of index options relative to their frictionless benchmark tracks that net demand over time.

*Retail buying single-stock calls.* The single-name market has the opposite sign. End users are net *short* single-stock options in aggregate, principally through covered call writing, while the retail flow that does buy is concentrated in short-dated out-of-the-money calls on volatile names. That is lottery demand, and Chapter 15 §15.1 supplies the preference behind it: probability weighting overvalues small chances of large payoffs, and a short-dated out-of-the-money call is the cleanest lottery ticket a brokerage account can buy. The prediction is that such options should be expensive relative to the model and should deliver poor average returns to their buyers, and that is what the cross-section of option returns shows.

The two facts together explain something a single-agent model cannot: index options and single-stock options have different smiles, and the difference lines up with who is buying which.

**Dealer gamma and hedging feedback.** The feedback runs the other way as well. A dealer who is short options is short gamma, and delta-hedging a short-gamma book requires selling the underlying as it falls and buying it as it rises — the mechanical amplification of portfolio insurance in miniature, executed every day. When the dealer community is net long gamma, the same hedging runs the other way and damps moves. The aggregate sign therefore matters for realized volatility in the underlying, and dealer positioning estimates have become a standard input for exactly that reason. The extreme case is the February 2018 unwind of the short-volatility complex, in which hedging flows from products whose exposure grew as volatility rose fed back into the volatility they were referencing; *International Finance* Chapter 13 tells that story, and the point retained here is only the mechanism, which is Box 8.2's gamma with a large sign attached.

**Reading the smile again.** Section 8.5 offered three readings of the skew: fat tails, a crash risk premium, and demand. The third is not an alternative to the second so much as its microfoundation. In a frictionless complete market the price of crash insurance is set by the marginal investor's marginal utility in the crash state, and the identity of that investor is irrelevant because everyone is marginal in everything. In the actual market the price of crash insurance is set by how much protection mandate-bound institutions want, against how much unhedgeable inventory dealers with finite capital will hold. The skew is therefore a joint measurement of a risk and a market structure, and no amount of care about the first will identify it without the second. That is this book's thesis applied to the one market where the replication argument was supposed to make it unnecessary — and it is why Chapter 20's demand-system machinery has an option-market counterpart at all.

The honest summary is a partition. Replication is exact for what can be replicated, and where it is exact the holder is irrelevant. Replication is approximate for everything else, and the gap between the approximation and the price is where the holders live. Options are unusual only in that the gap can be measured precisely, because the frictionless benchmark is known to four decimal places.

In Chapter 1 §1.2's terms, a move in the skew is rarely news about the distribution: it is a flow — mandate-driven demand for protection — arriving at a dealer whose constraint is binding, and quoted throughout in a model that survived as the market's convention after failing as its theory.

***

## Elsewhere in the Series

* **FX derivatives market structure** — *International Finance*, **Chapter 11**: the forwards, swaps and options market in currencies, its participants, its size, and the settlement machinery. Section 8.2's cost-of-carry relation with $$y$$ the foreign interest rate *is* covered interest parity; the phenomena, the parity conditions, and the carry-trade evidence are developed there and in §11.5, and are not re-derived here.
* **Non-FX derivatives markets** — *International Finance*, **Chapter 13**: rates, credit, equity, commodity and structured-credit derivatives as markets — instrument sizes, who uses them and why, central clearing, initial and variation margin, and the February 2018 short-volatility unwind. That chapter carries zero pricing theory and defers technical treatment to Hull; this chapter carries all of it. The division is clean: what a contract is worth is here, what the market for it looks like is there.
* **Commodity trade finance and physical trading** — *International Finance*, **Chapter 22**: trading houses, warehousing, and the financing of physical commodity flows. Section 8.2 prices the futures curve and explains convenience yield as a shadow price of inventory; the institutions that actually hold the inventory are there.
* Within this book: **Chapter 3 §§3.2-3.4** owns the law of one price, replication, state prices and the two-state economy this chapter extends; **Chapter 6 §6.6** owns commodity and currency factor portfolios, whose basis signal §8.2 supplies the mechanism for; **Chapter 7 §7.6** owns performativity, of which Box 8.1 is the canonical exhibit; **Chapter 15 §15.1** owns probability weighting and lottery demand; **Chapter 20** owns demand-system asset pricing; **Chapter 26** owns corporate risk management (§26.1), hedging practice and the Greeks in use (§26.2), and value at risk (§26.3); **Appendix D** carries the Black-Scholes derivation.

***

## Summary

1. **A derivative's payoff is a function, and the function is the whole specification.** A forward is linear and symmetric; a call and a put are the two halves of a forward split at the strike. The kink is what the buyer pays for.
2. **Bounds come before models.** Dominance alone gives $$C \ge \max(S\_0 - K/R\_f, 0)$$ and its put analogue, and implies that an American call on a non-dividend-paying stock is never exercised early.
3. **Put-call parity is pure law of one price.** $$C + K/R\_f = P + S\_0$$, because the two portfolios have identical payoffs in every state. No distribution, no volatility, no preferences. A $0.738 violation on a $100 stock is collected by a conversion (Table 8.2) that costs nothing and pays nothing thereafter.
4. **The cost-of-carry relation prices forwards on anything.** $$F\_{0,T} = S\_0(1 + r\_f - y)$$, with $$y$$ the net convenience yield: a dividend yield for a stock, a foreign interest rate for a currency, and for a commodity the shadow value of holding inventory less the cost of storing it. Contango and backwardation are the two signs of $$r\_f - y$$.
5. **Backwardation is not an arbitrage.** Full carry bounds the futures price above, because anyone can buy and store; nothing bounds it below, because a negative inventory cannot be held. The theory of storage makes $$y$$ decreasing in inventory, which supplies the mechanism behind Chapter 6 §6.6's commodity basis factor.
6. **The binomial model prices by replication, node by node.** $$\Delta = (C\_u - C\_d)/\[S\_0(u-d)]$$, and the recipe is rebalanced at every node and is self-financing between them. Two periods with $$u = 1.2$$, $$d = 0.8$$, $$R\_f = 1.05$$ and $$K = 100$$ give a call worth $15.59 and a European put worth $6.29, which satisfy parity exactly.
7. **Risk-neutral valuation is a general recipe, and this is where it is stated.** Replace the true probabilities with $$\pi^{\ast} = (R\_f - d)/(u - d)$$, take the expectation, discount at the riskless rate. $$\pi^{\ast}$$ lies in $$(0,1)$$ if and only if no arbitrage holds, which is the fundamental theorem in miniature; under $$\pi^{\ast}$$ every asset earns $$R\_f$$; and $$\pi^{\ast}$$ is a ratio of prices, not a belief.
8. **The true drift drops out because it is already in** $$S\_0$$**.** The option is priced relative to the observed spot, which impounds every view about the underlying and every attitude to risk. Investors who disagree about the stock must agree about the option. The same fact is the theory's limitation: it converts one price into another and says nothing absolute.
9. **Black-Scholes is the binomial limit**, $$C\_0 = S\_0 N(d\_1) - Ke^{-rT}N(d\_2)$$, with $$N(d\_2)$$ the risk-neutral probability of finishing in the money. Four of its five inputs are observable; the fifth, $$\sigma$$, is what the market trades. Inverting the formula gives implied volatility, which is the price restated in the model's units — a change of coordinates, not a forecast.
10. **The smile is the model pricing its own inadequacy.** Implied volatilities were roughly flat across strikes before October 1987 and have sloped downward in every liquid equity index since. The skew implies a fat left tail in the risk-neutral distribution — about a 19 percent one-year probability of a 20 percent decline, against 8.5 percent under a flat lognormal — and the wedge between that and the physical frequency is a crash risk premium.
11. **Who holds the option moves its price, because the manufacture is imperfect.** Dealers cannot hedge jumps or stochastic volatility, so end-user demand is priced: institutions buying index puts make index options expensive and steepen the skew, while retail lottery demand makes short-dated out-of-the-money single-stock calls expensive and their returns poor. Dealer gamma feeds back into the underlying's realized volatility.

***

## Key Terms

* **Underlying**: The asset whose price determines a derivative's payoff, written $$S\_t$$
* **Forward contract**: An obligation to trade at a price $$F$$ fixed today, settled once at maturity; payoff $$S\_T - F$$
* **Futures contract**: A forward marked to market daily through a clearing house, so gains and losses are settled in cash as they accrue
* **Call / put**: The right, not the obligation, to buy (respectively sell) the underlying at the strike $$K$$; payoffs $$\max(S\_T-K,0)$$ and $$\max(K-S\_T,0)$$
* **Strike price** $$K$$: The fixed exchange price written into an option contract
* **Put-call parity**: $$C + K/R\_f = P + S\_0$$; the law of one price applied to two portfolios with identical payoffs
* **Conversion / reversal**: The arbitrage packages that enforce parity — long stock, long put, short call against borrowing, and its mirror
* **Cost of carry**: Interest plus storage less the yield thrown off by holding the physical asset; the wedge between spot and forward
* **Net convenience yield** $$y$$: The flow of benefit from holding the physical asset rather than a claim to it, net of storage; a dividend yield, a foreign interest rate, or the shadow value of inventory
* **Contango / backwardation**: Futures above spot ($$y < r\_f$$) and futures below spot ($$y > r\_f$$)
* **Theory of storage**: The account of convenience yield as an option on unexpected demand, decreasing in the level of inventory
* **Replicating portfolio** $$(\Delta, B)$$: The position in the underlying and riskless borrowing that reproduces a derivative's payoff at every node
* **Self-financing**: The property that rebalancing the replicating portfolio requires no injection or withdrawal of cash
* **Risk-neutral probability** $$\pi^{\ast}$$: In the binomial model, $$(R\_f - d)/(u - d)$$; the measure under which every asset earns the riskless rate. Not a belief
* **Equivalent martingale measure**: The measure under which discounted prices are martingales; it exists if and only if there is no arbitrage, and is unique if and only if markets are complete
* **Early-exercise premium**: The excess of an American option's value over the otherwise identical European one, positive for puts and for calls on dividend-paying stocks
* **Black-Scholes formula**: $$C\_0 = S\_0N(d\_1) - Ke^{-rT}N(d\_2)$$, the continuous-time limit of the binomial model
* **Implied volatility** $$\sigma\_{\text{imp}}$$: The volatility input that makes the Black-Scholes price equal the observed price; the option's price restated in the model's units
* **Volatility smile / skew**: The dependence of implied volatility on the strike; flat under Black-Scholes, downward-sloping in equity indices since October 1987
* **The Greeks**: Delta, gamma, vega and theta — the sensitivities of the option price to the spot, to delta itself, to volatility, and to time
* **Demand-based option pricing**: The framework in which an option's deviation from its frictionless price is proportional to end-user net demand times the unhedgeable risk of the dealer's inventory

***

## Readings

### Required

* Black, F. and M. Scholes (1973). "The Pricing of Options and Corporate Liabilities." *Journal of Political Economy* 81(3): 637-654. *The paper. Read it for the hedged-portfolio argument in §II and for the closing sections on corporate liabilities, which show the authors understood from the start that equity in a levered firm is a call option — the idea Chapter 10 builds credit risk on.*
* Cox, J., S. Ross and M. Rubinstein (1979). "Option Pricing: A Simplified Approach." *Journal of Financial Economics* 7(3): 229-263. *The binomial model, and the source of §8.3. Written explicitly to make the argument accessible without stochastic calculus; it remains the clearest statement of why the drift drops out.*

### Recommended

* Merton, R. (1973). "Theory of Rational Option Pricing." *Bell Journal of Economics and Management Science* 4(1): 141-183. *The companion paper. Establishes the no-arbitrage bounds of §8.1 in generality, proves that an American call on a non-dividend-paying stock is never exercised early, and extends the formula to stochastic interest rates.*
* Gårleanu, N., L. H. Pedersen and A. Poteshman (2009). "Demand-Based Option Pricing." *Review of Financial Studies* 22(10): 4259-4299. *The holder section's source. Read §I for the model and the empirical sections for the decomposition of end-user demand into index and single-name flows, which is where §8.6's asymmetry comes from.*
* Duffie, D. and H. Zhu (2011). "Does a Central Clearing Counterparty Reduce Counterparty Risk?" *Review of Asset Pricing Studies* 1(1): 74-95. *The netting arithmetic behind an answer §8.6 does not give: whether interposing a clearinghouse reduces a dealer's exposure depends on how many contract classes it clears, and adding one clearinghouse per asset class can raise total exposure rather than lower it.*
* Acharya, V. (2014). "A Transparency Standard for Derivatives." In M. K. Brunnermeier and A. Krishnamurthy (eds.), *Risk Topography: Systemic Risk and Macro Modeling*. University of Chicago Press, 83-95; circulated as NBER Working Paper 17558. *A short policy argument about what observers must be able to see in a derivatives market, usable as a prompt against §8.5: the smile is only informative to someone who can observe the quotes across strikes in the first place.*
* MacKenzie, D. (2006). *An Engine, Not a Camera: How Financial Models Shape Markets.* MIT Press. *Chapters 5-6 on the options market and Chapter 7 on 1987. The evidence behind Box 8.1; read alongside Chapter 7 §7.6 of this book.*
* MacKenzie, D. (2010). "Models as Coordination Devices." In M. Akrich, Y. Barthe, F. Muniesa and P. Mustar (eds.), *Débordements: Mélanges offerts à Michel Callon*. Paris: Presses des Mines, 299-302. *The general form of Box 8.1's argument, and the harder case: a model can be retained as a shared quoting language after everyone has stopped believing it describes anything, which is exactly what §8.5 says implied volatility became.*
* Derman, E. Conference papers and retrospective essays on the use of models on a derivatives desk, including his 2009 appreciation of Fischer Black. *The practitioner's own epistemics of models, and the second witness Box 8.1 otherwise lacks: the same phenomenon reported from inside the trading operation rather than by an observer of it.*
* Hull, J. *Options, Futures, and Other Derivatives.* Pearson, current edition. *The technical reference for everything this chapter compresses: numerical methods, exotic payoffs, interest-rate derivatives, and the mechanics of margining. Both this chapter and International Finance Chapter 13 defer to it, from opposite directions.*
* Kohn, R. V. *Derivative Securities.* Course notes, syllabus, section handouts and problem sets, Courant Institute, New York University. *A complete graduate course running arbitrage to binomial to Black-Scholes to stochastic differential equations to rates and credit — that is, §§8.1-8.5 in order, continuing into the continuous-time route this chapter defers to Appendix D. Freely available and the obvious source of additional problems.*

***

## Discussion Questions

1. **Why the drift drops out.** A student objects: "The option pays off only if the stock rises, so a stock with a higher expected return must have a more valuable call. The model must be wrong." Answer the objection in your own words without using the phrase "risk-neutral." Then state precisely what would have to be true of the market for the objection to become *correct*, and identify which of §8.4's assumptions your answer relies on.
2. **The smile: probabilities or demand?** The equity-index skew implies a risk-neutral probability of a large decline that exceeds the historical frequency of such declines by a wide margin. Two explanations: the market is pricing a genuine risk premium for crash exposure, and the market is pricing the imbalance between institutions that must buy protection and dealers with finite capacity to sell it. Propose an observation that would distinguish them. Would a *fall* in the skew following an increase in dealers' capital be decisive?
3. **Performativity's limits.** Box 8.1 argues that the market conformed to Black-Scholes partly because the model was adopted. Name two other financial models that plausibly became infrastructure in the same sense, and one that plainly did not despite being widely taught. What distinguishes them? (Chapter 7 §7.6 and Chapter 26 supply two candidates.)
4. **Backwardation and the missing arbitrage.** A commodity trades in steep backwardation: the one-year future is 15 percent below spot while the riskless rate is 4 percent. A colleague proposes to buy the future and short the physical. Explain what is required to execute the second leg, why it is generally impossible, and what that asymmetry implies about which of the two bounds on the futures price is enforceable. Then say what the backwardation is telling you about inventory.
5. **What the option market knows.** Options are in zero net supply: every long is matched by a short. In what sense, then, can the option market "expect" anything? Distinguish between the risk-neutral distribution the quotes imply and the physical distribution, and explain what would have to be assumed to read the first as a forecast of the second.
6. **An option inside a liability.** A convertible bond is a straight bond plus the holder's right to exchange it for a fixed number of shares. Decompose the instrument: name the underlying, the strike, and the reason the effective terms move over the bond's life in a way an exchange-traded call's do not. Then name two features of the issuer that make §8.4's framework the wrong model for the embedded option — one about the underlying's volatility and one about who is on the other side of the default — and say which chapter of this book supplies each. Finally, ask why a firm would sell the package rather than issuing debt and equity separately; Chapter 23 has the corporate answer, and §8.1's bounds are enough to say what the buyer is paying for.

***

## Problems

**Problem 1 — Put-call parity and the conversion.** A non-dividend-paying stock trades at 50 dollars. One-year European options struck at 50 trade at $$C = 6.50$$ and $$P = 4.00$$. The one-year riskless gross return is $$R\_f = 1.04$$.

(a) State the parity relation and compute the value each side takes. (b) Which instrument is rich? By how much? (c) Construct the arbitrage in a cash-flow table in the format of Table 8.2. Report the profit today and verify the date-1 cash flow is zero at $$S\_T = 30$$, $$S\_T = 50$$ and $$S\_T = 70$$. (d) The stock is hard to borrow, at a fee of 4 percent per year on the value of the borrowed shares. Does that change your answer? Would it change your answer if the mispricing had the opposite sign?

**Problem 2 — One period: replication and risk-neutral pricing agree.** A stock trades at 40 dollars and will be worth 52 or 32 in one period. The riskless gross return is $$R\_f = 1.05$$.

(a) Price a call struck at 40 by constructing the replicating portfolio. Report $$\Delta$$, $$B$$, and the cost. (b) Compute $$\pi^{\ast}$$ and price the same call by risk-neutral valuation. Confirm the two agree. (c) Verify that the stock's expected gross return under $$\pi^{\ast}$$ is exactly $$R\_f$$. (d) Suppose the true probability of the up state is 0.7. Compute the call's expected gross return under the true probabilities, and express its risk premium as a multiple of the stock's. Explain the multiple in terms of the replicating portfolio's leverage. (e) The true probability is now revealed to be 0.3 instead. Which of your answers to (a)-(d) change?

**Problem 3 — Two periods, European and American.** A stock trades at 50 dollars and moves by $$u = 1.25$$ or $$d = 0.8$$ in each of two periods. The riskless gross return is $$R\_f = 1.04$$ per period. Options are struck at $$K = 50$$ and expire after two periods.

(a) Compute $$\pi^{\ast}$$ and draw the tree of stock prices. (b) Price the European call and the European put. Verify put-call parity. (c) Price the American put by backward induction, stating at each node whether exercise is optimal. (d) Report the early-exercise premium in dollars and as a percentage of the European put's value. Explain economically what the holder gains by exercising early at the node where you found it optimal. (e) Would an American *call* on this stock ever be exercised early? What single change to the problem would make it so?

**Problem 4 — Cost of carry with a convenience yield.** A commodity trades at a spot price of $75. The one-year riskless rate is 4 percent and storage costs 2 percent of spot per year, paid at year end.

(a) What is the full-carry one-year futures price — the price that would prevail if the convenience yield were zero? (b) The one-year future actually trades at $73.50. Compute the implied net convenience yield and the implied gross convenience yield. (c) Is the market in contango or backwardation? What does your answer imply about inventories? (d) A trader observes that the future is 6 dollars below full carry and proposes to buy the future and short the physical commodity. What exactly must she do to short the physical, and why is the trade generally unavailable? Which of the two no-arbitrage bounds on $$F$$ survives? (e) Inventories build over the following quarter. State the direction of the change in $$y$$, in the shape of the curve, and — using Chapter 6 §6.6 — in this commodity's expected excess return.

**Problem 5 — Reading a smile.** One-year options on an index trading at 100 show implied volatilities of 26 percent at a strike of 80, 19 percent at 100, and 16 percent at 120. The riskless rate is 4 percent.

(a) Using Black-Scholes, price the 80-strike put twice: once at 26 percent and once at the at-the-money 19 percent. Report the ratio. (b) The Black-Scholes risk-neutral probability of finishing below the strike is $$N(-d\_2)$$. Compute it for the 80 strike at each of the two volatilities. Interpret the difference in words. (c) A colleague concludes that the market assigns a 16 percent chance to a 20 percent decline. State two distinct reasons this reading is wrong, one about measures and one about the identity of the buyers. (d) The same exercise on single-name options on a mid-cap stock produces a curve that is high at *both* wings. Offer an explanation consistent with §8.6.

**Problem 6 ★ — Incompleteness and the limits of replication.** Take the one-period economy of Problem 2, but now suppose the stock can take three values at date 1: $52, $44 or $32. Only the stock and the riskless bond trade.

(a) Show that the call struck at $40 cannot be replicated, and that the pricing equations do not determine the state prices. (b) Impose $$q\_s \ge 0$$ and find the range of no-arbitrage call prices. (Compare Chapter 3, Problem 6.) (c) A dealer sells the call at the upper end of your range and delta-hedges using the two-state hedge ratio from Problem 2. Describe the profit and loss in each of the three states. Which state hurts, and what is the analogue in continuous time? (d) Explain how §8.6's demand-based framework selects a price within the interval, and what would have to be measured to implement it.

***

## Selected Solutions

*Solutions to Problems 1 and 3 follow. Solutions to the remainder are in the instructor materials.*

**Problem 1.**

(a) Parity is $$C - P = S\_0 - K/R\_f$$. The right side is $$50 - 50/1.04 = 50 - 48.077 = 1.923$$. The left side is $$6.50 - 4.00 = 2.50$$.

(b) The call is rich relative to the put by $$2.50 - 1.923 = 0.577$$. (Strictly, parity is a joint statement about four instruments; "the call is rich" is shorthand for the fact that the conversion package has a positive price.)

(c) Sell the call, buy the put, buy the stock, and borrow $$K/R\_f$$.

**Table 8.8: The conversion in Problem 1**

| Leg               | Today       | Date 1                  |
| ----------------- | ----------- | ----------------------- |
| Sell call         | $$+6.50$$   | $$-\max(S\_T - 50, 0)$$ |
| Buy put           | $$-4.00$$   | $$+\max(50 - S\_T, 0)$$ |
| Buy stock         | $$-50.00$$  | $$+S\_T$$               |
| Borrow $$48.077$$ | $$+48.077$$ | $$-50.00$$              |
| **Total**         | $$+0.577$$  | $$0$$                   |

*Source: Author's calculation.*

At $$S\_T = 30$$: $$0 + 20 + 30 - 50 = 0$$. At $$S\_T = 50$$: $$0 + 0 + 50 - 50 = 0$$. At $$S\_T = 70$$: $$-20 + 0 + 70 - 50 = 0$$.

(d) The borrow fee is irrelevant here, because this trade is *long* the stock — the arbitrageur is a potential lender of shares, not a borrower, and may in fact earn the fee. It matters entirely for the opposite sign. Had the call been cheap relative to the put, the reversal would require shorting the stock for a year at a 4 percent fee, costing $$0.04 \times 50 = 2.00$$ against a mispricing that would have to exceed that to be worth taking. This asymmetry is why observed parity violations are systematically one-sided in hard-to-borrow names, and it is Chapter 3's Palm episode in miniature.

**Problem 3.**

(a) $$\pi^{\ast} = (1.04 - 0.80)/(1.25 - 0.80) = 0.24/0.45 = 0.5333$$. The tree: $$S\_0 = 50$$; after one period $$62.50$$ or $$40$$; after two, $$78.125$$, $$50$$ and $$32$$.

(b) Call payoffs are $$28.125$$, $$0$$ and $$0$$. Weighting by the risk-neutral binomial probabilities $$((\pi^{\ast})^{2}, 2\pi^{\ast}(1-\pi^{\ast}), (1-\pi^{\ast})^{2}) = (0.2844, 0.4978, 0.2178)$$ and discounting twice:

$$
C\_0 = \frac{0.2844 \times 28.125}{1.04^2} = \frac{8.000}{1.0816} = 7.396
$$

Put payoffs are $$0$$, $$0$$ and $$18$$, so

$$
P\_0 = \frac{0.2178 \times 18}{1.0816} = \frac{3.920}{1.0816} = 3.624
$$

Parity: $$C\_0 - P\_0 = 7.396 - 3.624 = 3.772$$, and $$S\_0 - K/R\_f^2 = 50 - 50/1.0816 = 50 - 46.228 = 3.772$$. They agree.

(c) Backward induction on the put. At the up node ($$S = 62.50$$): continuation $$= (0.5333 \times 0 + 0.4667 \times 0)/1.04 = 0$$; exercise value $$= \max(50 - 62.50, 0) = 0$$. Hold (indifferently). At the down node ($$S = 40$$): continuation $$= (0.5333 \times 0 + 0.4667 \times 18)/1.04 = 8.400/1.04 = 8.077$$; exercise value $$= 50 - 40 = 10$$. **Exercise.** At the root: continuation $$= (0.5333 \times 0 + 0.4667 \times 10)/1.04 = 4.667/1.04 = 4.487$$; exercise value $$= 0$$. Hold. So $$P^{\text{Am}}\_0 = 4.487$$.

(d) The premium is $$4.487 - 3.624 = 0.863$$, or 23.8 percent of the European put's value. At the down node the holder gives up an option that is worth little — the stock would have to fall a great deal more for the remaining optionality to pay, and it can also rise back to 50, where the put dies — in exchange for $10 in cash a period early, which earns 4 percent. The interest on the strike exceeds the surrendered time value.

(e) No. There are no dividends, so exercising a call means paying the strike earlier than necessary while destroying the insurance embedded in the option; the bound $$C \ge S\_0 - K/R\_f^{n}$$ of §8.1 makes this precise. Introduce a dividend large enough that the stock's ex-dividend drop exceeds the interest saved by deferring payment of the strike, and early exercise immediately before the ex-dividend date can become optimal.

***

## Data Exercise: Implied Volatility, Realized Volatility, and the Variance Premium

**Part A — The variance premium (free data: FRED and Ken French's data library).** Download the daily VIX series `VIXCLS` from FRED, which begins in 1990. From Kenneth French's data library, download the daily **Fama/French 3 Factors**, and construct the daily market return as `Mkt-RF` + `RF`. (FRED's `SP500` series is an acceptable substitute but carries only about ten years of history.)

1. For each month, compute **realized volatility** as the standard deviation of daily market returns within the month, annualized by multiplying by $$\sqrt{252}$$. Plot it against the monthly average of `VIXCLS` on the same axes, 1990 to the present.
2. Compute the **variance premium**: the average of $$\mathrm{VIX}\_t$$ minus the realized volatility over the *following* month, in volatility points. Report the full-sample mean, the standard deviation, and the fraction of months in which it is positive. State the sign and explain what a persistently positive value means about the average return to selling index options.
3. The premium is a risk premium only if it is compensation for something. Compute the realized volatility of the market in the ten worst months of your sample, and compare the loss a seller of one-month at-the-money straddles would have taken in those months with the accumulated premium from the other months. Present the strategy's cumulative return path and its skewness. Write one paragraph on why an unconditionally profitable strategy with this shape is not evidence of mispricing.
4. Regress next month's realized volatility on this month's VIX and on this month's realized volatility jointly. Which forecasts better? What does the intercept tell you? Note that the coefficient on VIX confounds a forecast with a premium, and say what additional data would separate them.

**Part B — Put-call parity in public quotes.** Most exchanges and brokerage sites publish delayed option chains at no charge: for a single large index-tracking exchange-traded fund, record the mid-quote of the call and the put at three strikes bracketing the spot, for a single expiration roughly three months out, together with the spot price, the fund's expected dividend over the period, and a matched-maturity Treasury bill yield.

5. For each strike, compute $$C - P$$ and $$S\_0 - K/R\_f - \mathrm{PV}(\text{dividends})$$, and report the difference in dollars and as a percentage of the spot.
6. Report the bid-ask spread on each option. What fraction of your measured deviations lies inside the sum of the two spreads? Draw the conclusion §8.1 predicts, and state the three data problems — staleness, non-synchronous quotes, and American exercise — that make the exercise harder than it looks, along with what you did about each.

**Part C ★ (if you have WRDS).** Using OptionMetrics' `IvyDB` standardized volatility surface, extract the one-year implied volatilities at deltas of 0.10, 0.25, 0.50, 0.75 and 0.90 for the S\&P 500 index and for a matched set of large-capitalization single names, monthly since 1996.

7. Construct a skew measure — the 0.10-delta put volatility minus the at-the-money volatility — for the index and for the average single name, and plot both. Confirm §8.6's asymmetry: the index skew is steeper.
8. Merge with a measure of dealer inventory or intermediary capital (the CBOE's open-interest breakdowns, or the intermediary capital ratio of Chapter 19 §19.5) and regress the index skew on it, controlling for realized volatility. Gårleanu, Pedersen and Poteshman predict that the skew steepens when the constraint on the supply side tightens. Report whether your sample agrees, and say honestly what the endogeneity problem is.