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In this course, three methods are presented for pricing an option.

1. 1. The first method is an analytical one whereby the Black Scholes formula is used to price a call or a put. The drawback of the analytical approach is that it only works for European options.
2. 1. The second method presented is the binomial tree, which is illustrated in the pricing of an American option to facilitate early exercise.
3. 1. The third method presented is the Monte Carlo simulation.

Then the assumption of constant volatility is challenged, due to the presence of the volatility smile, which is formally defined and shown to be empirically observed in all derivatives markets. Monte Carlo simulations are run to generate a distribution with kurtosis -- a mixture of normal distributions.

Finally, the Heston Model, which relaxes the assumption of constant volatility is presented.

• • • First, the Heston Model is shown to incorporate kurtosis by allowing volatility.
• • Second, the Heston model includes an additional Brownian motion that allows volatility to mean-revert.
• • Third, these Brownian motions are linked by a correlation.

Sample code is provided to run the Heston model. The corresponding implied volatilities are graphed and shown to replicate the volatility smile.

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