Why Delta-Gamma-Theta Estimates Fail for Large Option Moves
Summary
The document explains why a delta-gamma-theta estimate can differ from an option’s actual price change when the underlying price moves substantially. The approximation combines the first-order delta effect and second-order gamma effect of a stock-price move with a first-order theta effect from time passing. It is a truncated Taylor expansion, so it is most reliable for relatively small changes in the underlying and time.
In the example, the option price rises from 0.046 to 0.065 after the stock moves from 95 to 96 and one day passes, while the Greek-based estimate is larger. The response attributes the gap to omitted higher-order effects and suggests either adding higher-order Greeks, such as speed, or using a smaller underlying move. For a move from 95 to 95.1 over the same time step, the stated approximation matches the example’s repriced value change. The illustration uses a particular calculator setup; it does not establish an error bound or address changes in volatility or other inputs.
Key ideas
- Delta-gamma-theta is a local approximation based on a truncated Taylor expansion.
- Its accuracy declines when the underlying price movement is large.
- Higher-order Greeks can capture additional curvature in option value.
- Smaller price increments can make the approximation more accurate in the example.
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# option price change # option price change I am trying to match change in European Call option price to greeks using the calculator here e.g. for `S=95, K=100, r=0, V=25, t=5 and dividend=0,` I get ``` Theoretical Price 0.046 Delta 0.041 Gamma 0.032 Vega 0.01 Theta -0.024 Rho 0.001 ``` Now I move 1 day forward and change S by 1, So now for `S=96, K=100, r=0, V=25, t=4 and dividend=0,` I get ``` Theoretical Price 0.065 Delta 0.061 Gamma 0.048 Vega 0.012 Theta -0.038 Rho 0.001 ``` However, If i use $dc = \Delta ds + 0.5 \Gamma ds^2 + \theta dt$ I get ``` dc = 0.041 * 1 + 0.5 * 0.032 * 1^2 + (-0.024) *1 = 0.033 ``` compared to `(0.065-0.046) = 0.019` fromt he numbers above. Am I missing something? ## Answer by JejeBelfort (score 3, accepted) https://quant.stackexchange.com/a/41344 Your equation: $$dC = \Delta S + 0.5\Gamma (\Delta S)^2 + \theta \Delta t$$ is actually an approximation of the option price changes (more precisely a "delta-gamma-theta" approximation) which is relevant only for sufficiently small underlying price movements. It basically captures first and second-order moves in the stock price along with first order move in the time-to-maturity. See Taylor Series for more details on this. If you want a better approximation of what will be your call option price the next day given an underlying price move, you should either: - capture higher-order variations of your option price by introducing third, fourth, ..., n-th order greeks (e.g. Speed, equal to $d\Gamma/dS = d^3C/dS^3$) in your approximation equation; - reduce your underlying price move: for instance, with a new underlying price of `95.1`, you get a new option value equal to `0.027` so `0.046 - 0.027 = 0.019`. The delta-gamma-theta approximation yields to `0.019` as well: `(0.041 * 0.1) + (0.5 * 0.032 * 0.1 * 0.1) - 0.024*1 = 0.019`.
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