Option Strategy Payoff: Method

Option Strategy Payoff: How It Works

What it does

The page answers one question: if this option position is held until the options expire, what does it gain or lose at each possible stock price? It writes out what each part of the position pays, adds the parts, and finds the largest profit, the largest loss and the stock prices at which the position breaks even. The premiums are the ones you type in; the page fetches no quotes. Advanced mode adds a model estimate of what the position is worth today.

The method

Terms. A call option gives its holder the right to buy a stock at a fixed price, the strike K, until the option expires. A put gives the right to sell at the strike. The buyer pays the option's price, the premium, to the seller (also called the writer), who must deliver the stock (call) or buy it (put) if the holder exercises. One contract covers m_c shares, 100 for a listed U.S. stock option. Each option or stock holding in a position is a leg.

Value of a leg at expiry. With S_T the stock price at expiry, per share:

call = max(S_T − K, 0)     put = max(K − S_T, 0)     stock = S_T

A call is worth exercising only when the stock is above the strike, a put only when it is below; otherwise the option expires worthless.

Profit of a leg. With n contracts and premium (or share price paid) p per share:

bought leg: P = n m_c (value − p)     sold leg: P = n m_c (p − value)

Net premium. The cash moved at opening is −n m_c p for each bought leg and +n m_c p for each sold leg. The sum over the option legs is the net premium: negative when paid (a debit), positive when received (a credit).

Profit of the position. The profits of the legs are added. Each leg's profit is a straight line in S_T that bends only at its strike, so the total is a chain of straight lines that bends only at the strikes. Three consequences give exact answers without searching:

1. The largest and the smallest profit lie at a strike, at S_T = 0, or at no finite price. Above the highest strike the total changes by g = m_c × (bought calls and stock − sold calls and stock, in contracts) for each 1.00 of stock price; puts are worthless there. If g is positive the profit is unlimited; if g is negative the loss is unlimited; if g is zero the total is flat.

2. A break-even price lies where the total changes sign between two neighbouring prices X_a and X_b of that list, with totals Y_a and Y_b. The line between them is straight, so

BE = X_a − Y_a (X_b − X_a) / (Y_b − Y_a)

and above the highest strike, BE = X_a − Y_a / g.

3. The description of the shape follows from the slopes of the first and the last line and from the totals at the strikes.

Before expiry (Advanced). Each option leg is valued by the Black-Scholes formula for a European option on a stock that pays no dividend, from the stock price S₀, the strike, the time to expiry T in years (days / 365), the interest rate r and the volatility σ; the formula is worked step by step in the Option Pricing tool. The position's value is the sum over the legs of ± n m_c × value, and its model profit today is that value less the cash paid to open. Its sensitivities are the same sums of the legs' sensitivities: delta (change in value for a 1.00 rise in the stock), gamma (change in delta for a 1.00 rise in the stock), theta (change in value as one calendar day passes) and vega (change in value for a one-point rise in volatility). The model probability that the stock finishes above a price B is N(d) with

d = [ln(S₀ / B) + (r − σ² / 2) T] / (σ √T)

where N is the standard normal distribution function.

What it assumes

All legs are on the same stock and expire on the same day, and the position is held unchanged to expiry. Premiums are the entered numbers; a real order fills at the bid or the ask, not at one quoted price. Commissions, margin interest, taxes and dividends are left out. A stock leg is bought at the entered price, or at today's price when the field is blank. The Advanced values assume European exercise, no dividends, one constant volatility for every strike, a constant interest rate and a lognormal stock price whose average growth is the interest rate.

When it misleads

Early exercise and assignment. Listed U.S. stock options are American-style: the holder may exercise before expiry. The seller of an option can therefore be assigned early, typically just before a dividend or when the option is deep in the money, which ends that leg and can change the position's risk. The expiry diagram does not show this.

The path does not appear. The diagram gives the result at expiry only. Before expiry the position can show a loss larger than its loss at expiry at the same stock price, or call for more margin, even when the expiry result would be a gain.

"Maximum loss" is for the position as entered. A covered call or a collar still holds stock; a cash-secured put commits the cash to buy it. Their largest loss is reached with the stock at zero, and the table says how large it is. A position with a sold call and nothing against it has no limit to its loss.

Dividends and costs. A dividend paid before expiry adds to the result of a stock leg and is not counted. Four legs mean four commissions and four bid-ask spreads, which can take a visible share of a small maximum profit.

The probabilities are model outputs. They use the entered volatility and assume the stock grows at the interest rate. Actual returns have fatter tails than the lognormal curve, the volatility is not known in advance, and quoted options at different strikes imply different volatilities. A probability of finishing above a break-even is not the probability of a profit unless the position gains everywhere above that price.

The page describes a position. It does not say whether a position suits anyone. It is a calculator for education and analysis, not investment advice.

Worked example

The default case: a bull call spread on a stock at 100. Buy one call with strike 100 for 4.35 and sell one call with strike 110 for 1.10; one contract is 100 shares. The premiums are illustrative.

Net premium: −1 × 100 × 4.35 + 1 × 100 × 1.10 = −435 + 110 = −325, so 325 is paid.

If the stock finishes at 108: the bought call is worth 108 − 100 = 8, profit 100 × (8 − 4.35) = 365; the sold call expires worthless, profit 100 × (1.10 − 0) = 110; total 475.

Totals at the prices that matter: at 0 and at 100 both calls are worthless, total −325; at 110 the bought call is worth 10, total 100 × (10 − 4.35) + 110 = 675. Above 110 each further 1.00 adds 100 on the bought call and takes 100 on the sold call, so g = 0 and the total stays at 675.

Maximum profit 675 at 110 or above; maximum loss 325 at 100 or below.

Break-even between 100 (−325) and 110 (675): BE = 100 − (−325) × (110 − 100) / (675 − (−325)) = 100 + 3.25 = 103.25, the lower strike plus the net premium per share.

Advanced, with 60 days to expiry, 25 % volatility and a 4 % rate: the model values are 4.365 and 1.112 per share, the position is worth 325.28 against 325 paid (model profit 0.28), its delta is 34.12 shares, and the model probability of finishing above 103.25 is 38.16 %.