# Simple Entropic Derivation of a Generalized Black-Scholes Option Pricing Model

## Abstract

**:**

## 1 A Simple Black-Scholes Derivation

^{2}Δt, and we commensurately assume that the riskless rate of return is rΔt.

_{T}~ N(0,T). In a lengthy derivation using martingale theory, Dothan [9, pp.210-214] shows that (9) is the log of the density of the equivalent martingale measure used to compute the Black-Scholes option pricing formula as a risklessly discounted expected present value of a call option’s payoff.

#### 1.1 Entropic Option Pricing Literature Review

## 2 Non-Normality: Reinterpreting the Black Scholes Volatility Parameter

_{imp}that makes the Black Scholes formula value equal to each observed option market price, such implied volatilities are “upwardly biased relative to realized volatility...". This upward bias in stock index volatility was also noticed in a systematic study of S&P 100 index option prices by Fleming [11], whose results are summarized later as providing empirical evidence for the following model.

_{imp}). The result will thus be a reinterpretation of the volatility parameter in the Black Scholes formula, needed to help correct fo rnon-normalities.4

#### 2.1 Empirical Implementation

_{imp}from the realized asset volatilities.

## 3 Conclusions

## References and Notes

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^{1}The martingale restriction (2) may alternatively be written as the constraint that the risklessly discounted, conditional risk neutral expected payo® of the underlying asset at s + Δt must be its current price.^{3}This notation for the Black Scholes formula suppresses the notation for its other variables, i.e. the riskless rate of return r, the time to expiration T, the current price of the underlying security, and the exercise price.^{4}A somewhat related contribution of Duan [10] used GARCH to model the actual underlying return process, assumed that the risk neutral conditional return distribution is still normal (albeit with altered mean), and assumed that its variance would equal the actual conditional variance. HÄardle and Hafner [17] showed that the normal distribution with the smallest entropy relative to the actual normal distribution must have the same variance (albeit with altered mean).^{5}Clearly, more extensive empirical testing should be the subject of another paper.

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**MDPI and ACS Style**

Stutzer, M.J. Simple Entropic Derivation of a Generalized Black-Scholes Option Pricing Model. *Entropy* **2000**, *2*, 70-77.
https://doi.org/10.3390/e2020070

**AMA Style**

Stutzer MJ. Simple Entropic Derivation of a Generalized Black-Scholes Option Pricing Model. *Entropy*. 2000; 2(2):70-77.
https://doi.org/10.3390/e2020070

**Chicago/Turabian Style**

Stutzer, Michael J. 2000. "Simple Entropic Derivation of a Generalized Black-Scholes Option Pricing Model" *Entropy* 2, no. 2: 70-77.
https://doi.org/10.3390/e2020070