OPTIONS Gram-Charlier
This calculator uses the Gram-Charlier expansion of the Black-Scholes model to compute the theoretical value and greeks of European-style call and put options. The model extends Black-Scholes by introducing skewness and excess kurtosis parameters, allowing it to capture non-normal features of asset returns. To generate results, enter the Inputs and click Calculate.
| Output | Call | Put |
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| Theoretical Value |
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| Delta |
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| Gamma |
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| Theta |
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| Vega |
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| Intrinsic Value |
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| Delta 100's |
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| Lambda |
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| Theta (-7 Days) |
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| Rho |
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| Psi |
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| Strike Sensitivity |
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The OptionsGramCharlier function in FinTools XL prices European options using a Gram-Charlier series expansion around the standard normal distribution. By introducing skewness and excess kurtosis parameters, the model captures the asymmetry and fat tails of empirical return distributions and produces a volatility smile/skew that the standard Black-Scholes model cannot reproduce.
Inputs
Underlying Price: The current price of the underlying asset.
Exercise Price: The price at which the asset can be bought (call) or sold (put).
Time: The time to expiration of the option, expressed in years, between the Value Date and the Expiration Date.
Interest Rate: The risk-free interest rate.
Yield Rate: The annual dividend yield expressed as a percentage of the underlying asset price.
Volatility: The annualized volatility of the underlying asset's returns, expressed as a decimal.
Skewness: The skewness of the return distribution. A value of 0 corresponds to a symmetric (normal) distribution. Negative values indicate a left-skewed distribution, typical of equity returns.
Kurtosis: The excess kurtosis of the return distribution. A value of 0 corresponds to a normal distribution; positive values indicate fat tails.
Market Option Price: The traded price of the option in the market for calculating implied volatility.
Outputs
Theoretical Value The calculated theoretical value of the option based on the Gram-Charlier expansion.
Delta Measures the rate of change of the option price with respect to changes in the underlying asset's price.
Gamma Measures the rate of change in Delta with respect to changes in the underlying asset's price.
Theta Measures the sensitivity of the option price to the passage of time (time decay).
Vega Measures the sensitivity of the option price to changes in the volatility of the underlying asset.
Intrinsic Value The intrinsic value of the option.
Delta 100's The delta of the option multiplied by 100.
Lambda Measures the percentage change in option value per percentage change in the underlying asset price.
Theta (-7 Days) Theta calculated over a shorter timeframe of 7 days.
Rho Measures the sensitivity of the option price to changes in the risk-free interest rate.
Psi Measures the sensitivity of the option price to changes in the dividend yield.
Strike Sensitivity Measures how sensitive the option price is to changes in the strike price.
