An intuition for option pricing

OpeningAll at the money, a sevenfold price gap

Three stocks all closed at $100 today. On each one's option chain you find a call with a 100 strike. The quotes are $1.88, $7.01 and $13.75.

A and B have exactly the same stock price, strike and expiration, yet B costs 3.7 times as much. The difference is entirely in volatility: the market expects B to move four times as much as A over the next month. C has the highest dollar price, but it is buying a whole year of uncertainty; $13.75 on its own tells you nothing about whether it is expensive or cheap.

So an option's dollar price, by itself, doesn't tell you whether it is cheap or expensive. To judge that, you first need to know where the price comes from. This chapter answers three questions: which inputs determine the premium; how those inputs are combined into a price; and how to estimate a roughly reasonable price without opening a calculator.

3.1The six inputs and which way they push

Of the six inputs, the underlying price, the strike, the time to expiration, the interest rate and dividends can all be looked up directly; only volatility has to be backed out of market prices. Start with which way calls and puts move when each input increases.

The volatility row deserves a second thought. Once the stock falls below the strike, the most a call can lose is its premium, however far the stock falls; above the strike, it gains as much as the stock rises. So when volatility increases, the call holder just gets more chance of a “big rally,” while the cost of a “big drop” is already capped. The same goes for puts. That is why option prices always rise with volatility.

Checking with numbers

Below, the base case is “stock 100, strike 100, 30 days, IV 30%, rate 4%, no dividends,” and we change one input at a time.

Three observations:

• The stock price and the strike matter most. The change in the option's price for each $1 move in the stock is Delta (Chapter 4).
• IV raised the call and the put by exactly the same 0.570. This is no coincidence. Put–call parity (Section 3.5) shows that the call minus the put depends only on the stock price, the strike, the interest rate and time, not on volatility. So when volatility changes, the two have to rise and fall together.
• Interest rates and dividends have little effect on a 30-day option, just a few cents. For long-dated options of a year or two (LEAPS) you can't ignore them; Chapter 5 works through the numbers.

The one input you can't see: implied volatility

The first five inputs are facts, but volatility is an estimate of the future that no one can observe directly. In practice you work backward: take the price the option trades at in the market, plug it into the formula and solve for the σ that makes the formula's price equal the market price. That is implied volatility (IV).

For example, with the stock at $100, if a 30-day at-the-money call trades at $4.20, solving backward gives an IV of 35.3%. That price is higher than the base case (IV 30%, $3.591), and the extra is there because the market expects bigger moves. That's why traders often quote an option as “35 vol” rather than $4.20: IV puts options with different stock prices and expirations on the same yardstick. Chapter 6 covers how to judge whether IV is high or low.

3.2What Black-Scholes calculates

There are two ways to read the Black-Scholes price: as the expected payoff at expiration, discounted; or as the cost of replicating the option with stock and cash. Both readings give the same number.

Reading 1: the expected payoff, discounted

At expiration, a call pays max(S_T − K, 0). If you know the distribution of the stock price at expiration, average this payoff and discount it back to today at the interest rate, and you have the option price. The two terms of the formula do exactly that:

• Second term Ke^−rTN(d₂): if the option finishes in the money, you pay the strike K. N(d₂) is the probability of finishing in the money; multiply by it and discount. In the base case this is 99.672 × 0.498 = 49.646.
• First term SN(d₁): if the option finishes in the money, you get one share of stock. Whenever it finishes in the money, the stock is above 100, averaging about $107.23; weighted by the probability of finishing in the money and discounted, that comes to 100 × 0.532 = 53.237.
• Subtract one from the other: 53.237 − 49.646 = 3.591. Taking the expectation directly over the lognormal distribution by numerical integration and then discounting also gives 3.591.

The “probability” and “expectation” here come with a condition: they are calculated under the risk-neutral distribution, which assumes the stock's expected return is exactly the risk-free rate of 4%. Why are we allowed to assume that? The second reading explains.

Reading 2: the replicating portfolio

Take the simplest possible example. The stock is at $100, and in one month it can do only one of two things: rise to 110 or fall to 90. A call with a 100 strike then pays 10 or 0.

Now buy 0.5 shares and at the same time borrow $44.85 (repaying 45, with interest, in one month). In one month: if the stock is at 110, the 0.5 shares are worth 55, and after repaying 45 you have 10 left; if it is at 90, the 0.5 shares are worth 45, and after repaying 45 you have 0. In both cases this portfolio pays exactly what the call pays, and it costs 50 − 44.85 = $5.15. So the call is worth $5.15.

Black-Scholes breaks this step into countless tiny steps, readjusting the number of shares at each one to the ratio at that moment. That ratio is N(d₁), which is Delta. That is why Delta appears in the price formula and is also the hedge ratio (Chapter 4).

This has an important consequence for trading: an option's price contains no view on direction, only a view on how big the moves will be. Whether you're bullish or bearish is expressed through whether you buy or sell and which strike you choose; whether an option is expensive depends only on whether its implied volatility is high or low compared with the actual volatility you expect.

3.3The distribution of prices at expiration

Black-Scholes assumes that the stock's log returns are normally distributed, so the stock price at expiration follows a right-skewed lognormal distribution: it can fall no lower than 0, and it has no upper limit.

This assumption has three direct consequences:

1. The median is below the mean. The mean is Se^rT and the median is Se^{(r−σ²/2)T}. At 30 days and IV 30%, they are 100.33 and 99.96, nearly the same. At one year and IV 60%, they become 104.08 and 86.94: the probability that the stock is lower than today in a year is 59.2%, yet the average is higher than today. High-volatility stocks that “fall most of the time and occasionally soar” have exactly this shape.
2. The ±1 standard deviation range is asymmetric. At 30 days and IV 30%, the range containing 68% of the probability is [91.72, 108.94]: +8.94 on the upside and −8.28 on the downside.
3. Symmetric distances, asymmetric probabilities. For the same $10 away from the current price, the probability of finishing above 110 is 13.3%, and the probability of finishing below 90 is 11.1%.

This chart also shows why options are priced in “log” terms. A move from 100 to 200 is +100%, and a move from 100 to 50 is −50%, but on a log scale the two are the same distance (both ln 2). The lognormal distribution handles these “multiplicative” moves naturally and guarantees that the stock price always stays positive.

3.4Limits of the normal assumption

Real stock returns have two features that the lognormal distribution can't describe: extreme moves happen much more often than the model says (fat tails), and big drops tend to be more common and more violent than big rallies (negative skew).

First, see how “optimistic” the normal distribution is about extreme days. Assume daily returns are normally distributed, with 252 trading days a year:

The t distribution column is only an illustration: it is scaled to the same standard deviation as the normal distribution, so the only difference is how thick the tails are. Tail thickness in real markets changes from period to period, and there is no single fixed number. But the direction is certain: measured against long-run average daily volatility, US stock indexes have had several one-day drops of more than 5 standard deviations over the past few decades. Under a normal distribution, a day like that wouldn't come around even once in ten thousand years.

What this means for option prices

• The market charges extra for far out-of-the-money puts. Black-Scholes prices every strike with the same σ. If the tails are fatter, far out-of-the-money puts, which protect against big drops, should cost more. The market does exactly that: for the same expiration, out-of-the-money puts usually carry higher implied volatility than at-the-money options. This is called volatility skew (Chapter 6). For example, a 30-day 90 put is worth $0.40 at IV 30%; if the market quotes it at 36%, it is $0.74, more than 80% more expensive.
• Volatility itself changes. The model assumes σ is constant; in reality, volatility jumps during sharp drops and drifts slowly back down when things are calm. That is exactly the problem Vega deals with in Chapter 5.
• Stocks gap. Earnings, mergers and regulatory news can make a stock jump past many price levels in a single step. The replication argument in Section 3.2 requires continuous rebalancing, which is impossible across a gap, so option sellers carry more risk in a gap than the model calculates.

3.5Put–call parity

The prices of a call and a put with the same strike and expiration are locked together by one equation: call − put = stock − present value of the strike.

Why it must hold

Compare two portfolios. Portfolio A: buy 1 call and deposit Ke^−rT in cash. Portfolio B: buy 1 put and hold 1 share of stock. Here is what each is worth at expiration (strike 100):

Wherever the stock ends up, both portfolios are worth max(S_T, K) at expiration. Two portfolios with exactly the same value at expiration must also cost the same today: C + Ke^−rT = P + S. Rearrange it and you have the parity formula.

Check it with the base-case numbers: 3.591 − 3.263 = 0.328, 100 − 99.672 = 0.328. Note that the whole derivation never used volatility, a distribution or Black-Scholes. As long as you can freely buy, sell, borrow and lend, parity holds. This is also why the change in IV in Section 3.1 raised the call and the put by the same 0.570.

What happens when parity is violated

Suppose the call is quoted at 3.80 while the put is still 3.263. Sell the call, buy the put and buy the stock (a combination called a conversion): you pay out 100 + 3.263 − 3.80 = 99.463. At expiration, wherever the stock is, you sell it at 100: if it has risen, it is called away through the call; if it has fallen, you sell it with the put. You pay 99.463 today and are certain to get 100 back in a month; after 4% interest costs, you still make an extra $0.21 per share, or $21 per contract. Market makers and arbitrage programs would do this trade immediately, until prices moved back near parity. For individual traders, parity is useful for checking quotes and understanding positions; leave the arbitrage to the market makers.

American options and dividends

• American options: US single-stock options are American-style and can be exercised early, so parity turns from an equation into a range. With no dividends, S − K ≤ C − P ≤ S − Ke^−rT. For 30-day at-the-money options the range is 0 to 0.328, which is very narrow; for 1-year at-the-money options it is 0 to 3.92.
• With dividends: C − P = S − PV(dividends) − Ke^−rT. If a $1 dividend will be paid before expiration, the call is about $1 cheaper relative to the put. This is also why in-the-money calls tend to be exercised early before an ex-dividend date (Chapter 2).
• Hard-to-borrow stocks: when short selling means paying a borrow fee, puts become more expensive relative to calls by that cost, and parity seems to “break.” In fact, the formula has simply picked up an extra cost term.

3.6Synthetic positions

Rearrange the parity equation and you get four pairs of equivalent positions, each “holding the same risk in a different form.”

Synthetic long and synthetic short

Buy one 100 call (3.591) and sell one 100 put (3.263), for a net debit of $0.328 per share. This combination has a Delta of 0.532 + 0.468 = 1.000, and the Gammas cancel exactly (a call and a put at the same strike have the same Gamma): its risk is exactly the same as 100 shares of stock, and so is its profit and loss at expiration. The only difference is the money: buying the stock costs $10,000, while the synthetic long costs only $32.8 plus the margin required for the short put. If you deposit the money you save and earn interest on it, the result is identical to buying the stock; if you use it to add more contracts, you are adding leverage.

A synthetic short is the reverse: sell the call and buy the put. It's often used on stocks that are hard or expensive to borrow; the catch is that by then the market has already marked up the puts (see the previous section).

Covered call = short put

The third row is the one most often overlooked. Owning stock and selling a call (a covered call) has exactly the same payoff at expiration as selling a put at the same strike with the full cash set aside (a cash-secured put): above the strike, both earn a fixed maximum gain; below it, both take the loss from the falling stock. Case 1 compares them point by point.

Protective put = long call

Owning stock and buying a put as insurance is equivalent to keeping most of the money in cash and buying just a call. The two have the same maximum loss on the way down and the same gain on the way up. This also shows that the cost of “buying insurance” can be measured directly by the call's time value.

3.7Reading probabilities

Option prices contain the market's view of the distribution at expiration: Delta approximates the probability of finishing in the money, and the probability of touching the strike before expiration is about twice that.

As Chapter 4 covers, Delta is N(d₁), and the risk-neutral probability of finishing in the money is N(d₂). A call's Delta is slightly higher than its probability of finishing in the money. For a put it's the reverse: |Delta| = N(−d₁) is slightly lower than the probability of finishing in the money, N(−d₂). The table below puts them side by side:

Why the probability of touching is about double

Imagine the stock touches 110 at some moment before expiration. From then on, the remaining paths are about equally likely to go up or down, because drift is small over short periods. So “finishing above 110” is only about half as likely as “having touched 110.” Put the other way, probability of touching ≈ 2 × probability of finishing in the money. In mathematics this idea is called the reflection principle; the exact formula, with a correction for drift, differs from the “double” approximation by less than 0.003 in the table above.

The approximation breaks down in three situations:

• Larger drift: with very long terms or high interest rates, “equally likely up or down” no longer holds.
• Closing prices only: the touch probabilities in the table assume continuous monitoring. If you check only once a day at the close (simulated with 30 observation points), the probability of touching 110 falls from 0.266 to about 0.22, and for 105 from 0.57 to about 0.50. Intraday touches that reverse by the close don't count.
• Real distributions have fat tails and gaps: then both probabilities are underestimated.

Finding “one standard deviation” on the option chain

You'll often hear that “16 Delta is about one standard deviation.” This comes from N(−1) = 0.159 in the standard normal distribution. At 30 days and IV 30%, the strikes one standard deviation away are 100 × e^±0.086, or 108.98 and 91.76. At these two strikes the probability of finishing in the money is indeed about 0.16, but the Deltas are 0.179 for the call and −0.140 for the put, because d₁ and d₂ differ by σ√T. So to find one standard deviation on the chain, look for about 0.18 on the call side and about 0.14 on the put side. The higher the IV and the longer the term, the bigger the gap.

3.8Mental math 1: the expected move

One-standard-deviation expected move ≈ stock price × IV × √(days/365). IV is a full-year standard deviation; scale it by the square root of time to the period you care about.

Why the square root? Daily returns are roughly independent, and the variances of independent variables add: the variance over 30 days is 30 times the variance over 1 day, so the standard deviation is √30 times as large. An IV of 30% corresponds to a daily standard deviation of 30% ÷ √365 = 1.57%, and over 30 days it is 1.57% × √30 = 8.6%.

Example: stock at $200, IV 40%, two weeks (14 days): 200 × 0.40 × √(14/365) = 200 × 0.40 × 0.196 = $15.7. The market sees roughly a 68% chance that the stock will be between 184 and 216 in two weeks.

How accurate is the mental math?

Mental math gives a symmetric ±, while the lognormal range is wider on the upside and narrower on the downside, and the whole range also shifts down by −σ²/2. How far apart the two are depends on the size of σ√T:

When σ√T is within 0.2 (for example, up to three months at IV 30%, or up to one month at IV 60%), the mental-math value falls roughly between the exact upper and lower bounds, with an error of about $1. Above 0.3, be careful: in the last row, the upper bound is actually below the mental-math value, because the whole range has been pulled down by −σ²/2.

3.9Mental math 2: the straddle's 0.8

The price of an at-the-money straddle (buying one call and one put at the same strike) ≈ 0.8 × the one-standard-deviation expected move. The 0.8 comes from the mean absolute value of the normal distribution: √(2/π) ≈ 0.798.

Derivation 1: the average absolute move

At expiration, an at-the-money straddle pays |S_T − K|, the absolute value of the stock's move. If the move is roughly normally distributed with standard deviation M = Sσ√T, the average of its absolute value is:

Derivation 2: starting from Black-Scholes

At the money, S = K. Ignoring interest, d₁ = σ√T/2 and d₂ = −σ√T/2, and the call price becomes

Along the way you get an even handier rule: ATM call ≈ 0.4 × stock price × IV × √(days/365). Base case: 0.4 × 100 × 0.30 × 0.287 = 3.44, versus 3.591 from Black-Scholes. The 0.15 gap comes from interest: parity tells us the call is 0.328 more expensive than the put, and the call gets half of that. Add that half back and you get 3.60. The rule also works for backing out IV as in Section 3.1: (4.20 − 0.16) ÷ (0.4 × 100 × 0.287) ≈ 35.2%, against an exact value of 35.3%.

Checking the error with numbers

The rule always overestimates slightly, and the error grows with σ√T. The reason is that derivation 2 approximates the area under the curve as “width × peak,” while the normal density curves downward on both sides of the peak. A more precise expansion is N(x/2) − N(−x/2) ≈ 0.399x(1 − x²/24), with x = σ√T. At x = 0.3 the correction is under 0.4%; at x = 1 it is about 4%. For the terms and IVs you'll use day to day (σ√T < 0.3), the error is under 1%, so you can rely on the mental math.

Two ways to use it in reverse

• Reading the expected move from the straddle: one standard deviation ≈ straddle ÷ 0.8 = 1.25 × straddle.
• Backing out IV from the straddle: IV ≈ straddle ÷ (0.8 × stock price × √(days/365)).

CasesThree cases

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