Options: European, American, Bermudan and exotics; binomial, Black–Scholes, the Greeks

Level 1 Plain words

For every reader: the idea in plain words, no formulas.

A right without an obligation

A call option gives its holder the right to buy something at a set price, the strike price, on or before a set date. A put option gives the right to sell at the strike. The buyer pays the seller a premium up front. If the right is worth using at the end, the buyer uses it; if not, the buyer lets it lapse and loses only the premium.

Think of a call on a share at a strike of $100. Should the share end at $130, the holder buys at $100 and could sell at $130, a gain of $30 less the premium paid. With the share at $80 at the end, nobody wants to pay $100 for it, so the call expires worthless. The seller kept the premium and owes nothing.

Puts work the other way. With a strike of $100, a put lets its holder sell at $100. Should the share fall to $70, the holder gains $30 less the premium. Should it rise, the put expires unused.

Why people buy and sell options

1. Insurance. An investor who owns shares and buys puts has set a floor under the loss, the way a homeowner insures a house. The premium is the insurance cost.
2. Income. An investor who owns shares and sells calls collects premiums in exchange for giving up gains above the strike.
3. Leverage. A call costs a fraction of the share price but gains dollar for dollar above the strike, so a small outlay can make or lose a large percentage.
4. A view on volatility. Options gain value when the market expects bigger moves. Traders buy and sell options to bet on how much prices will move, whichever way.

Intrinsic value and time value

Before expiry, an option's premium has two parts. Intrinsic value is what the option would pay if exercised right now: for a call, the share price minus the strike when that is positive, otherwise zero. Time value is the rest, the value of the chance that things get better before expiry. A call struck at $100 on a share at $110 has $10 of intrinsic value; if the call trades at $17, the other $7 is time value. Time value shrinks as expiry approaches and is zero on the last day.

Level 2 How it works

For readers who want the mechanism, with the diagrams.

Beginner path stops here

Payoffs at expiry

With strike $K$ and a final share price $S_T$, the payoffs at expiry are:

A seller of calls that does not own the share, a naked call writer, faces losses with no limit. That is why brokers ask for more margin from option sellers than from buyers.

Moneyness

An option whose exercise would pay something today is in the money. A call is in the money when the share is above the strike; a put when it is below. An option at the strike is at the money, and one that would pay nothing is out of the money. Out-of-the-money options are all time value.

For a one-year call struck at 100, with a 5 % interest rate and 20 % volatility, the Black–Scholes values of Level 3 give these splits:

European, American, Bermudan

An option's style says when the holder may exercise it.

1. European options allow exercise only on the expiry date. Most index options, such as those on the S&P 500 listed by Cboe, are European.
2. American options allow exercise on any day up to expiry. Options on individual US stocks are American.
3. Bermudan options allow exercise on a set list of dates, such as once a quarter. They are common in interest rate markets, for example a right to cancel a swap on each coupon date, and the callable feature of many bonds works the same way.

An American option is worth at least as much as a European one with the same terms, because it carries every right the European one has and more. Level 3 shows that the extra right is sometimes worth nothing (calls on stocks with no dividend) and sometimes worth a lot (deep in-the-money puts).

Exotics, in a paragraph each

Asian options pay on the average price over a period instead of the final price. Averaging lowers the volatility of the payoff, so they cost less, and they suit a buyer who purchases a commodity steadily over a year.

Barrier options switch on or off when the price touches a level. A down-and-out call dies if the share falls below the barrier; a down-and-in call comes alive only if it does. They cost less than plain options because they pay in fewer states of the world.

Digital (binary) options pay a fixed amount if the price ends above or below the strike, and nothing otherwise. Their value jumps at the strike, which makes them hard for dealers to hedge near expiry.

Lookback options pay on the best price over the life, the highest for a call or the lowest for a put. They remove the regret of bad timing and cost a lot for it.

Compound and chooser options are options on options. A chooser lets the holder decide later whether the option is a call or a put, a bet on a large move in either direction.

Options on futures, rates and currencies

Options exist on almost anything with a price. Options on futures (on oil, grains, Treasury bonds, stock index futures) deliver a futures position when exercised; Fischer Black adapted the pricing formula for them in 1976. In interest rate markets, a cap pays when a floating rate rises above a strike in any period, a floor when it falls below, and a swaption gives the right to enter a swap at a set fixed rate. Many swaptions are Bermudan, exercisable on each coupon date, and a callable corporate bond is a plain bond with a Bermudan call sold by the investor to the issuer. Currency options let an importer cap the cost of foreign money while keeping the gain if the currency weakens. The pricing ideas below carry over to all of them; what changes is the underlying and how it is assumed to move.

What moves an option's value

Volatility is the input nobody can observe. The other five are on the screen.

Implied volatility, the smile and the VIX

Given an option's market price, traders solve the pricing formula backwards for the volatility that produces it. That number is the implied volatility. It is the market's price of uncertainty, expressed as a yearly standard deviation.

If the Black–Scholes assumptions held, every option on the same share and expiry would show the same implied volatility. Since the 1987 crash, equity index options have not behaved that way. Out-of-the-money puts carry higher implied volatility than at-the-money options, and out-of-the-money calls lower, a downward slope traders call the skew or smile. Mark Rubinstein documented the change in "Implied Binomial Trees" (Journal of Finance, 1994). Investors pay up for crash insurance because they know crashes happen more often than a normal curve allows, as Chapter 3 explains.

Cboe's VIX index turns the prices of a strip of S&P 500 options into a single number, the market's implied volatility for the next 30 days. FRED publishes its daily close as VIXCLS. A VIX of 20 means the options are priced as if the index will move with a yearly standard deviation of about 20 %.

Three strategies investors use

All three start with a share bought at 100. A covered call adds a sold one-year call with a strike of 110, for 6.04. For the protective put we add a bought one-year put with a strike of 95, for 3.71. A collar does both, so the call premium pays for the put with 2.33 to spare. Premiums are Black–Scholes values at 20 % volatility and a 5 % rate; we ignore interest on the premiums. Profit per share at expiry:

Read across the rows. When the share goes nowhere, the covered call earns more than the stock, and it gives up everything above 110. Insurance through the put costs a little in every good outcome and caps the loss below 95. Doing both narrows the range of outcomes at both ends. None of them is a free improvement; each trades some outcomes for others.

The risk drivers of an options position

Level 3 The math

For Series 65 and CFA readers: formulas and worked examples.

Series 65 path stops here

Put-call parity

Take a European call and a European put on the same stock, same strike $K$, same expiry $T$, with no dividends. Buying the call and selling the put gives a payoff of $S_T - K$ in every state: if the share ends above $K$ the call pays and the put expires; if below, the put holder exercises against you and you buy at $K$. That is the payoff of a forward. So the pair must cost what the stock costs, less the present value of the strike:

In words, a call minus a put with the same strike and expiry is worth the share price minus the strike discounted at the risk-free rate.

Early exercise

For an American call on a share with no dividend, parity gives a lower bound on the call's value. Since the put is worth at least zero:

In words, a call is worth at least the share price minus the discounted strike.

When $r$ is positive, $K e^{-rT}$ is less than $K$, so the right-hand side is larger than $S_0 - K$, the amount early exercise would pay. Selling the call always beats exercising it.

Puts behave differently. A put's best case is the strike itself, reached if the share goes to zero, and money received earlier earns interest.

Dividends change the call result. When a share goes ex-dividend its price drops by about the dividend, and a call holder does not receive the dividend. Exercising just before the ex-date captures the dividend but pays the strike early. That trade pays only if the dividend is larger than the interest saved by paying the strike later, $K(1 - e^{-r \tau})$, where $\tau$ is the time left to expiry after the ex-date. With a strike of 100, rates at 5 % and three months left, the threshold is 1.24. A dividend of 3 clears it; a dividend of 1 does not. So early exercise matters for calls on shares paying large dividends near expiry, and for puts in general.

One-step binomial

Assume the share, now at 100, will in one year be either 120 (up by $u = 1.2$) or 80 (down by $d = 0.8$). The rate is 5 % with continuous compounding. A call struck at 100 pays 20 in the up state and 0 in the down state.

We can copy that payoff with shares and borrowing. Holding $\Delta$ shares and borrowing $B$:

In words, the number of shares in the copy equals the spread of the option's payoffs divided by the spread of the share's prices.

Here $\Delta$ = 0.50. Holding half a share is worth 60 in the up state and 40 in the down state; borrowing the present value of 40, which is 38.05, makes the down state pay zero and the up state pay 20. The copy costs 50 minus that loan, 11.95, so the call must cost the same or traders would buy one and sell the other.

The same answer comes out of a neat shortcut. Define the risk-neutral probability:

In words, $q$ is the probability that makes the share's expected growth equal the risk-free rate.

With these numbers $q$ = 0.6282, and the call is the discounted expected payoff under $q$:

In words, the option's value is its average payoff using the risk-neutral probabilities, discounted at the risk-free rate.

Two-step binomial, European and American

Add a second year with the same up and down factors. The share can end at 144, 96 or 64.

John Cox, Stephen Ross and Mark Rubinstein set out this method in "Option Pricing: A Simplified Approach" (Journal of Financial Economics, 1979). With many short steps and $u$ and $d$ set from volatility, the tree converges to Black–Scholes for European options, and it handles American and Bermudan exercise by checking at each node, which the closed-form formula cannot.

Black–Scholes

For a European call on a share with no dividend, with volatility $\sigma$:

In words, the call is worth the share price times one probability-like factor, minus the discounted strike times another.

In words, $d_1$ measures how far the share is above the strike, in units of volatility over the option's life, adjusted for interest and volatility, and $d_2$ is $d_1$ less one volatility unit.

$N(x)$ is the chance that a standard normal variable is below $x$. $N(d_2)$ is the risk-neutral probability that the call ends in the money. $N(d_1)$ is the call's delta, the number of shares in the copy. The put follows from parity:

In words, the put is worth the discounted strike times the chance of ending in the money, minus the share price times the put's share-equivalent.

The assumptions are strong: the share follows a lognormal random walk with constant volatility, trading never stops, there are no transaction costs, and the rate is constant. Real markets break each of these, the smile is the visible result, and traders use the formula as a way to quote prices in units of implied volatility more than as a forecast.

The Greeks

Each Greek measures how the option's value changes with one input, holding the others fixed.

In words, delta is the slope of the option's value against the share price, and gamma is how fast that slope changes.

The Greeks are tied together by the Black–Scholes equation. For any option on a share with no dividend:

In words, the time decay plus the gain from curvature plus the interest on the hedge equals the risk-free return on the option's value.

Implied volatility from a price

Black–Scholes runs one way: inputs in, price out. Traders mostly run it the other way. Given a market price, they search for the volatility that reproduces it.

Reading the VIX as a daily move

A yearly volatility converts to a daily one by dividing by the square root of the number of trading days, about 252:

In words, the daily standard deviation the options imply is the VIX divided by the square root of 252.

A VIX of 20 implies a typical daily move of about 1.26 %. A VIX of 82.69 implies about 5.21 % a day.

Level 4 History, analysed

For readers who want past episodes run through the Level 3 math.

1973: a market and a formula

Within a few years traders on the exchange floor carried printed sheets of Black–Scholes values. Scholes and Merton received the Nobel Memorial Prize in Economic Sciences in 1997; Black had died in 1995. The formula's key idea, that an option can be copied by trading the share and a bond, also gave traders a recipe for creating options that did not exist, which led to portfolio insurance.

19 October 1987 and portfolio insurance

Portfolio insurance promised a pension fund a floor without buying puts. A manager would copy a put by selling index futures, using the put's delta from Black–Scholes, and selling more as the market fell. Take a fund that wants a one-year floor at 95 on an index at 100, at 20 % volatility. At 100 the copy holds short futures equal to 27 % of the stock position. If the index falls to 80, the put's delta says the short must grow to 69 %.

Now add up many funds following the same rule. A fall in the market forces them all to sell, which pushes prices lower, which forces more selling. Their model assumed that the insurer could sell smoothly at each price on the way down, with nobody's selling affecting the price. On 19 October futures traded at large discounts to the cash index, the cash market could not absorb the orders, and the gaps in trading meant the copies failed to track the puts they were meant to imitate. The insurers' own delta rule made the fall worse.

Option traders learned. A skew appeared after 1987 and has stayed: a lasting premium on crash protection, by which the market admits that a normal curve understates the left tail. A one-day fall of 20.5 % against a typical daily standard deviation of about 1 % is a move of about 20 standard deviations, which a normal curve treats as all but impossible (Chapter 3).

1997 and 1998: the prize and the fund

Scholes and Merton were partners in Long-Term Capital Management, the hedge fund that needed a $3.6 billion recapitalisation in September 1998, less than a year after their prize (Chapter 15). The fund's trouble did not come from the option formula. LTCM sold large amounts of long-dated equity index volatility in 1998, betting that implied volatility, then high, would fall back. When markets panicked after Russia's default, implied volatility rose further and stayed there, and the fund's short vega positions lost money at the same time as its bond trades. Being right about the long-run level of volatility does not help a seller who runs out of capital before the market comes back.

5 February 2018: short volatility

In one day the VIX jumped +115.6 %, the largest one-day rise in its history to that point. A product that pays minus the daily return of VIX futures loses one dollar for each dollar the futures gain; a rise of about 96 % in the futures index on one day leaves almost nothing. For years before, the VIX futures curve had been in steep contango, and selling volatility had earned money nearly every month through the same roll arithmetic that hurts commodity funds (Chapter 13). Investors read that steady record as low risk. A seller of vega earns small premiums often and pays a large amount rarely, the same profile as the carry trade of Chapter 14. Sellers' own need to buy back VIX futures as prices rose fed the jump.

16 March 2020: a VIX of 82.69

The Level 3 conversion turns 82.69 into an implied daily standard deviation of 5.21 %, against about 1.26 % for a VIX of 20. Options were priced for daily moves four times normal, and the index delivered: the S&P 500 fell about 12 % that day. Dealers who had sold puts were short gamma; as the market fell, their hedges required selling more futures, the 1987 loop in a smaller form. Buyers of puts in February 2020 earned many times their premium. Sellers who had collected premiums for years gave back much of what they had earned in three weeks.