Option Pricing!

In quantitative trading, market making, and options risk management, rapid mental estimation of how option prices adjust to moves in the underlying asset is an essential skill. Traders use this mental arithmetic to be able to react to market events quickly. In fact, the FPGAs and low-latency systems also change their pricing using this very simple equation!

The Delta-Gamma Approximation

Let $V(S)$ denote the price of an option as a function of the underlying price $S$. When the underlying price moves from an initial price $S_0$ to a new price $S_1$, the change in underlying price is:

Taking a second-order Taylor expansion of $V(S)$ around $S_0$:

Substituting the first-order Greek Delta ($\Delta = \frac{\partial V}{\partial S}$, a measure of how much the price of an option is expected to change for every $1 change in the price of the underlying asset.) and the second-order Greek Gamma ($\Gamma = \frac{\partial^2 V}{\partial S^2}$, derivative of Delta), we arrive at the standard Delta-Gamma formula for option price change $\Delta V$:

The estimated new option price $V_{\text{new}}$ is therefore:

Dissecting the Two Components

1. Delta Effect (Linear PnL): $\Delta \cdot \Delta S$
   • Measures the first-order directional exposure.
   • For long calls ($\Delta > 0$), an upward move increases the option value, while a downward move decreases it.
   • For long puts ($\Delta < 0$), an upward move decreases the option value, while a downward move increases it.
2. Gamma Effect (Convexity PnL): $\frac{1}{2} \Gamma (\Delta S)^2$
   • Captures the curvature of the option price curve (how fast Delta changes as the stock moves).
   • Because $(\Delta S)^2 \ge 0$ and $\Gamma > 0$ for all long vanilla options, the Gamma effect is strictly positive regardless of whether the stock goes up or down.
   • Positive Gamma acts like a tailwind: as the stock moves in your favour, your position becomes longer (Delta increases); as the stock moves against you, your position becomes less exposed (Delta shrinks).

Mental Math Shortcuts for Trading Interviews

Quantitative trading firms may test Delta-Gamma mental math in interviews. You can simplify the convexity term $\frac{1}{2} \Gamma (\Delta S)^2$ with these mental shortcuts:

• Move of $\pm 1$: $(\Delta S)^2 = 1 \implies \text{Gamma PnL} = 0.5 \times \Gamma$
• Move of $\pm 2$: $(\Delta S)^2 = 4 \implies \text{Gamma PnL} = 2 \times \Gamma$
• Move of $\pm 3$: $(\Delta S)^2 = 9 \implies \text{Gamma PnL} = 4.5 \times \Gamma$
• Move of $\pm 4$: $(\Delta S)^2 = 16 \implies \text{Gamma PnL} = 8 \times \Gamma$

Quick Example

Suppose a call option has base price $V_0 = \textdollar5.00$, $\Delta = 0.50$, and $\Gamma = 0.08$. The stock rises by $\textdollar2.00$ ($\Delta S = +2$):

1. Delta effect: $0.50 \times (+2) = +\textdollar1.00$
2. Gamma effect: $\frac{1}{2} \times 0.08 \times (2^2) = 2 \times 0.08 = +\textdollar0.16$
3. Total change $\Delta V$: $+1.00 + 0.16 = +\textdollar1.16$
4. Estimated new price $V_{\text{new}}$: $\textdollar5.00 + \textdollar1.16 = \textdollar6.16$

If instead the stock had fallen by $\textdollar2.00$ ($\Delta S = -2$):

• Delta effect: $0.50 \times (-2) = -\textdollar1.00$
• Gamma effect: $+0.16$ (still positive!)
• New price: $\textdollar5.00 - 1.00 + 0.16 = \textdollar4.16$

Interactive Practice Trainer

Test your mental calculation speed below. Randomized scenarios are generated with given base prices, underlying moves, Delta, and Gamma. Enter your estimated price in the box to check your answer and view the step-by-step breakdown.