Approximate Option Pricing Parameters for Laplace Log Returns

Summary

The document addresses how the familiar d1 and d2 terms in an option-pricing formula might be adapted when log returns follow a Laplace distribution rather than a normal distribution. It points to research on log-symmetric return models, which gives an approximate expression for these parameters under a Laplace assumption. The expression adjusts the rate term using a logarithm involving volatility, while retaining the usual dependence on spot price, strike, and time to expiry.

The answer presents the result as an approximation from a cited paper, not as a general exact pricing formula. It also flags a notation ambiguity: the paper writes “log,” and the respondent suspects it means the natural logarithm. Readers are advised to consult the paper and verify the formula. The short exchange provides no derivation, numerical example, or discussion of how the approximation performs across market conditions, so it serves mainly as a pointer to a model-specific result.

Key ideas

• A Laplace model for log returns changes the option-pricing inputs relative to the normal-return case.
• Research on log-symmetric distributions gives approximate d1 and d2 expressions for Laplace returns.
• The proposed expression depends on spot, strike, interest rate, volatility, and time to expiry.
• The cited result is approximate, and the paper’s logarithm notation may be ambiguous.
• The discussion gives no derivation or empirical performance assessment.

Tags

# What are $d_1$ and $d_2$ for Laplace?


# What are $d_1$ and $d_2$ for Laplace?












What are the formulae for `d1` & `d2` using a Laplace distribution?

## Answer by Richi Wa (score 2, accepted)

https://quant.stackexchange.com/a/7256

Your question is interesting because I thought that the only chance with Lévy-processes is to use Fourier-transform approaches (see e.g. Cont,Tankov).

But in the paper Option Pricing for Log-Symmetric Distributions of Returns by Fima C. Klebaner· Zinoviy Landsman they consider models, where the log of the price has a symmetric distribution. In Corollary 3.2 they propose an approximate formula if the log of the price follows the Laplace distribution where $$ d_{1,2} = \frac{\ln(S_0/K) + (r \pm log(1-\sigma^2/2))T}{\sigma \sqrt{T}}. $$ They write $\log$ but I guess this is just an $\ln$ as before. But please try yourself and/or read the first pages of the paper to be sure.

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