Option Strategy Calculator: Greeks and P&L

Legs (premium blank uses the model price today)
Leg 1 choose a type to add it

How to use it
  1. Enter the underlying price and the risk-free rate, and the dividend yield if there is one.
  2. Pick a strategy to fill the legs around today’s price, or enter the legs yourself: choose a leg’s type (call, put or stock) and whether it is long (bought) or short (sold). A new blank leg appears below each one you fill, up to six.
  3. For each option leg enter the number of contracts, the strike, the days to expiration and its implied volatility. Enter stock in shares. Enter the premium you paid or received per share (the price per share for stock), or leave it blank to use the model price today.
  4. The cost to open, the position’s greeks, its profit and loss at the first expiration and the chart appear and update as you type.
  5. To see the grid of P&L at a later date, enter the analysis date in days from today.

To see it filled in, choose “Show an example”. Options are priced as European with Black-Scholes, each at its own implied volatility, and legs that have expired are worth their intrinsic value. Commissions, early exercise and the dividends on stock legs are not counted. For study and comparison; not financial advice.

Result

Enter the underlying price.

Black-Scholes-Merton, European exercise, rates and yield continuously compounded, time as calendar days over 365. Greeks are for the whole position: vega and rho per 1 % change, theta per calendar day.

Methodology, equations and examples

An option strategy combines calls, puts and stock into one position. This calculator values each leg with the Black-Scholes-Merton formula, at its own strike, expiration and implied volatility, and adds the legs up. That gives the position's cost, its greeks, its profit and loss (P&L) at any underlying price and date, and how the P&L responds to a change in volatility.

Valuing the position

Each leg controls a number of shares, n. For an option it is the number of contracts times the contract multiplier (100 for US equity options), positive when long and negative when short. For stock it is the number of shares. The position's value at underlying price S and t days from today is

V(S, t) = Σ n × value of the leg at S, with (days to expiration − t) / 365 years left

An option with time left is valued with Black-Scholes-Merton (see the Black-Scholes calculator for the formula). An option past its expiration is worth its intrinsic value, max(0, S − K) for a call and max(0, K − S) for a put. A share of stock is worth S.

The cost to open is Σ n × premium, where the premium is what was paid or received per share. Left blank, it is the model price today, so the position starts at a P&L of zero. A positive cost is a debit (money paid); a negative one is a credit (money received). The P&L is then

P&L(S, t) = V(S, t) − cost to open

Greeks

The greeks of a position are the sums of the greeks of its legs, each times its n. Delta is the position's exposure in shares: a delta of 39 gains about $39 for a $1 rise in the underlying. Gamma is the change in delta per $1. Vega is the change in value per 1 percentage point rise in every leg's volatility, theta the change per calendar day, and rho the change per 1 percentage point rise in the risk-free rate. A share of stock has a delta of 1 and no other greeks.

Max profit, max loss and breakevens

These are worked out at the first expiration among the option legs, when the nearest leg is worth its intrinsic value and any longer-dated legs are still valued with Black-Scholes. The P&L is evaluated at 3,000 prices from $0 to three times the highest strike or today's price, with every strike included. When all the legs expire together the P&L at expiration is made of straight lines that bend only at the strikes, so the maximum and minimum on this grid are exact. Beyond the grid, the slope of the P&L shows whether the profit or the loss keeps growing without limit, as for a long call or a short call. Breakevens are the prices where the P&L crosses zero, found by bisection between grid points.

Scenario grid

The grid shows the P&L at the analysis date for underlying moves of −20 % to +20 % (rows) and for every leg's volatility shifted by −10 to +10 percentage points (columns). The center cell is the P&L with no move and no change in volatility. With the analysis date left at today it equals the P&L today. A shifted volatility is never taken below 1 %.

Example

The bull spread example in Hull, Options, Futures, and Other Derivatives, buys a three-month call with a $30 strike for $3 and sells a three-month call with a $35 strike for $1. With one contract of 100 shares on each leg, n = +100 for the long call and −100 for the short call.

Cost to open = 100 × $3 − 100 × $1 = $200, a debit.

At expiration the spread pays nothing below $30, S − $30 between the strikes and $5 above $35. Its P&L is therefore −$200 below $30 and 100 × $5 − $200 = $300 above $35, the maximum loss and profit. It breaks even where 100 × (S − $30) = $200, at S = $32.

The calculator's example adds an illustrative stock price of $31, 30 % volatility for both legs and a 5 % rate, which set the value today and the greeks. At those inputs the long call is worth $2.58 and the short call $0.67, so the spread is worth $190.94 against the $200 paid, a P&L of −$9.06 today. Its delta is 39 shares and its theta is −$0.26 a day: the position gains about $39 for a $1 rise in the stock and loses about 26 cents a day if nothing else changes.

Limits

Options are priced as European, so the value of early exercise on American options is not counted. Commissions, margin, the dividends paid on stock legs and the bid-ask spread are left out. Every leg's volatility is held at what was entered as time passes, apart from the shifts in the grid. For study and comparison; not financial advice.

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