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The Options Strategy Lab, Explained

After reading this you will be able to read an option payoff curve, price a multi-leg strategy with Black-Scholes, and interpret the Greeks and probability of profit the lab reports, without mistaking any of it for a forecast.

An option gives you the right, but not the duty, to buy (a call) or sell (a put) 100 shares at a fixed strike price on or before a fixed date. That single asymmetry lets you build positions whose value bends at the strikes. The Options Strategy Lab assembles up to six such legs, prices each one, and draws two lines: the payoff at expiry and the theoretical mark today.

Here is the hook. Sell one call at a 105 strike and buy one call at a 110 strike on a stock trading at 100. You collect a net premium, say 1.50 per share, or 150 for the contract. Your maximum profit is that 150. Your maximum loss is the 5.00 strike gap minus the credit, so 3.50, or 350. That is a bear call spread, and the lab draws it as a downward step between the two strikes.

What the lab computes and what it does not

The lab prices each leg with the Black-Scholes-Merton formula on a non-dividend stock. The "today" curve is the sum of the leg values at each possible underlying price, minus the net premium you paid or plus the credit you received at entry. The expiry curve is the same sum with time set to zero, which collapses every option to its intrinsic value and produces the familiar straight-line kinks.

The lab does not model American-style early exercise, dividends, bid-ask spreads, or assignment risk. Real trades cost more than the theoretical mark because you cross the spread on every leg. Treat the numbers as a clean idealization, not a fill you will actually get.

When to use it, and when not

Use the lab to see the shape of a strategy before you trade it: where it makes money, where it loses, how fast time decay erodes it, and how a jump in implied volatility helps or hurts. It is a teaching instrument for the geometry of payoffs and the sign of your Greeks.

Do not use it to predict where a stock will go. The probability of profit it reports assumes prices are lognormal with a constant volatility you chose. Real returns have fat tails, volatility clusters, and earnings gaps break the model. If you want to size a position by dollar risk instead, the Position Size Calculator is the right tool. For plain buy-and-hold growth, use the Investment Growth Calculator.

The Black-Scholes formula and the intuition

A European call worth C on a non-dividend stock is priced as:

C = S \cdot N(d_1) - K e^{-rT} N(d_2)

Here S is the current stock price, K the strike, r the risk-free rate, T the time to expiry in years, and N(\cdot) the standard normal cumulative distribution. The two arguments are:

d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)T}{\sigma \sqrt{T}}, \quad d_2 = d_1 - \sigma \sqrt{T}

The symbol \sigma is the implied volatility, an annualized standard deviation of returns. Read the call price as the discounted expected payoff under a risk-neutral world: N(d_2) is roughly the chance the option finishes in the money, and S \cdot N(d_1) is the expected value of the stock you receive if it does. A put follows from put-call parity: P = C - S + K e^{-rT}.

The reason the "today" curve is smooth while the expiry curve has sharp corners is the time value. With T \gt 0 there is always a probability the option moves further in the money, so the curve rounds off the kinks. As T \to 0 that rounding vanishes.

A worked example: the demo bear call spread

Pricing two call legs at 30 days

Take the default setup: stock S = 100, implied volatility \sigma = 0.30, rate r = 0.04, and 30 days to expiry, so T = 30/365 = 0.0822. Sell one 105 call, buy one 110 call.

  1. For the 105 call: d_1 = \frac{\ln(100/105) + (0.04 + 0.09/2)(0.0822)}{0.30\sqrt{0.0822}}. The numerator is -0.0488 + 0.00699 = -0.0418, the denominator is 0.0860, so d_1 = -0.486 and d_2 = -0.572.
  2. Then N(d_1) = 0.313 and N(d_2) = 0.284, giving C_{105} = 100(0.313) - 105 e^{-0.04(0.0822)}(0.284) = 31.3 - 29.7 = 1.55.
  3. For the 110 call the same steps give roughly C_{110} = 0.55.
  4. You sell the 105 and buy the 110, so the net credit is 1.55 - 0.55 = 1.00 per share, or 100 per contract.

Maximum profit is the 1.00 credit if the stock stays below 105. Maximum loss is the 5.00 width minus the credit, so 4.00, if the stock finishes above 110. The breakeven sits at 105 + 1.00 = 106.

Profit is flat at +1.00 below 105, slopes down through the 106 breakeven, and floors at -4.00 above 110.

Watching time and volatility bend the curve

The single most instructive move in the lab is dragging days to expiry and implied volatility while a curve is on screen. Higher volatility fattens the today curve away from the expiry kinks, because more uncertainty means more time value everywhere. Fewer days pulls the curve down toward the expiry line as time value bleeds out. For a credit spread you sold, that bleed is your friend: the mark converges to your kept premium.

A single long 100-strike call priced at S=100, r=0.04. At 30 days and 30% volatility it is worth about 3.53. Cutting to 5 days drops it to about 1.42. Raising volatility to 50% at 30 days lifts it to about 5.79. Time decay and volatility both move the price of pure time value.

Reading the Greeks and the probability of profit

The lab reports net position Greeks, summed per share across legs with sign. Each answers a different "what if".

Delta
Change in position value for a 1-point move in the stock. A net delta of -0.20 means you lose about 0.20 per share if the stock rises one point.
Gamma
How fast delta itself changes. High gamma near a strike means your directional exposure flips quickly.
Theta
Value bleed per day. A theta of +0.03 on a short spread means the mark gains about 0.03 per share each day, all else equal.
Vega
Sensitivity to a 1-point change in implied volatility. Vega of -0.05 means a jump from 30% to 31% costs you 0.05 per share.

The probability of profit comes from a Monte Carlo run. The lab simulates thousands of terminal prices from a lognormal distribution at your chosen volatility, evaluates each leg at expiry, and counts the fraction where total P/L exceeds zero. For the demo spread with a breakeven at 106, the model estimates roughly a 70% chance of finishing profitable, because the stock only has to stay below 106.

A high probability of profit is not a good deal by itself. The demo spread wins about 70% of the time but risks 4.00 to make 1.00. Multiply outcomes by their payoffs before you judge: 0.70 \times 1.00 - 0.30 \times 4.00 = -0.50 in this rough sketch, which is why edge, not win rate, is what matters.

Common mistakes

Three errors show up again and again.

First, confusing the today curve with the expiry curve. Your position rarely lands exactly on the expiry line before the last day. If you close early, you realize the smooth mark, not the kinked payoff.

Second, reading probability of profit as truth. It is the model's own estimate under its own assumptions. Feed it 20% volatility on a stock that actually swings 40% and the number is optimistic by a wide margin.

Third, ignoring the Greeks' interaction. A position can be delta-neutral yet lose steadily to theta, or profit from a volatility spike (positive vega) while the stock barely moves. Check all four before you decide what bet you are actually making.

Related tools

Options are one slice of investing. To measure realized results, the ROI Calculator handles simple and annualized return, and the CAGR Calculator gives the compound rate between two values. When you sell a winning position, the Capital Gains Tax Calculator estimates the tax. For discounting a series of cash flows, see the NPV Calculator, and for the rate that zeroes them, the IRR Calculator.

Frequently asked questions

Why does the today curve not touch the expiry payoff?

Because time value keeps every option worth more than its intrinsic value while time remains. The gap is largest near the strikes and near the money. Drag days to expiry toward zero and the two curves merge.

What implied volatility should I enter?

Use the volatility the market is quoting for that expiry, which your broker shows per contract. If you enter your own guess, the price and probability shift to match your guess, not the market.

Is the probability of profit the chance I make money?

It is the model's estimate assuming lognormal prices at the volatility you chose. Real distributions have fatter tails and gaps, so treat it as a rough guide, not a guarantee.

Why are my real fills worse than the lab's mark?

The lab prices at the theoretical mid. Every real leg crosses a bid-ask spread, and multi-leg orders can slip further. The wider the spread, the larger the gap.

Does the lab handle dividends or early exercise?

No. It assumes a non-dividend, European-style underlying. Dividends lower call values and raise put values, and American options can be exercised early, neither of which is modelled here.

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