Options: time value — details
About this app
The fair option value before expiry (Black-Scholes) as a smooth curve above the payoff — the gap is the time value. Sliders for K, σ, T, r; “time decay” runs T → 0 and the curve collapses onto the hockey stick. Delta as a tangent, gamma/vega/theta live. Debunks “out of the money = worthless”.
Intrinsic value and time value
Before expiry, an option is worth its intrinsic value plus its time value. Intrinsic value is the payoff from exercising now, max(S − K, 0) for a call. Time value exists because the price can still move before expiry; it grows with volatility σ and time to maturity T. That is why an out-of-the-money option still has a positive price. As T → 0 the time value decays (theta) and the value falls back onto the payoff. The slope of the value curve is delta, its curvature gamma.
What the app shows
The app draws the Black-Scholes value of a call or put as a smooth curve above the dashed payoff; the gap between them is the time value. A tangent at the price S has delta as its slope. A values panel shows option value, intrinsic value, time value, delta, gamma, vega per volatility point and theta per day. Sliders control strike K, volatility σ, time to maturity T up to two years, interest rate r and price S.
Things to try
Press “play time decay”: T runs towards zero and the value curve settles onto the hockey-stick payoff. Move the price far out of the money and raise σ: time value grows although intrinsic value stays zero. Switch to a put, set the rate to a few percent and move the price deep into the money: the value curve dips below the payoff and time value turns negative. This is why early exercise of an American put can pay off.
Frequently asked questions
- What is the time value of an option?
- Time value is the option price minus its intrinsic value. It reflects the chance that the price still moves favourably before expiry. It is largest at the money and increases with volatility and time to maturity.
- Why is an out-of-the-money option not worthless?
- An out-of-the-money option has no intrinsic value, but the price can still move past the strike before expiry. That possibility has a price, the time value. Only at expiry does an out-of-the-money option expire worthless.
- What do delta, gamma, vega and theta mean?
- Delta measures how much the option value changes for a small move in the underlying and equals the slope of the value curve. Gamma is the curvature, how quickly delta changes. Vega is the sensitivity to volatility and theta the loss of value per unit of time.
- What does the time decay animation show?
- It runs the time to maturity down to zero. You can watch the smooth Black-Scholes curve collapse onto the payoff as time value disappears, fastest near the strike.
Subject: Financial derivatives
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