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Graphical Representation of Option Price and Sensitivities
Robert Merton was the founder of the Merton model for pricing
an option on dividend-paying stocks, it was one of the extension
and generalization of the Black-Scholes differential equation
It tries to evaluate a fair value of an option, and if
it behaves well then the option's market price will equal the
theoretical fair value.
The mathematics of their derivation is
quite complex. Interested readers can find it in the original
paper Merton (1973), and the books by Hull (1993).
The Merton model was developed to value European-style options
on shares of dividend-paying stocks. It is crucial to remember
that the Merton model is based on a number of assumptions.
The distribution of terminal stock prices (returns) is lognormal.
The underlying asset pays dividends during the life of the option.
There are no arbitrage possibilities.
Transactions cost and taxes are zero.
The risk-free interest rates, the dividend yields, and the asset volatility are known functions of time over the life of the option.
There are no penalties for short sales of stock.
The market operates continuously and the share prices follows a continuous Ito process.