Derivative Pricing: The Black And Scholes Model

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Derivative pricing nowadays has been a dispute among finance researchers, finance workers, and experts. Derivative in finance means a financial asset, which is derived from another asset. Another asset, which usually called underlying entity, can be an index, asset, or interest rate. Derivatives can be used for a number of purposes which one of them is option pricing.

Option in finance has a meaning of a contract giving the buyer a right to buy the underlying asset or to sell it at a specified strike price (can be set by looking at market price) and date; also depending on which option they bought. Option also has some factors, which are underlying price, strike price, expected volatility, time until expiration, interest rate and dividends.
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Black and Scholes model is a mathematical model used to calculate the price of European put and call options. This model has an assumption, which the returns will follow the normal distribution. The problem is in the real analytical way; we will find that the returns are skewed to the left and had two heavier tails, which will not satisfy the normal distribution (leptokurtic features). Thus, in this paper, we will produce a model that will satisfy the normal distribution to replace or modify the Black Scholes model.

Literature Review
[1]In 1973, Fisher Black, Robert Merton, and Myron Scholes developed The Black Scholes model, which is one of the most important concepts in today’s financial works.

The key idea of this model is to help investors to buy and sell the underlying asset in the right time with no risk. This model has some more assumptions aside from follow the normal distribution, they are no dividends are paid out during the life of the option, efficient markets, no commission, risk-free rate and volatility are
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Two majors mistake that has been made by Black Scholes model is the leptokurtic and volatility smile. The assumption said that the volatility should be constant. It means that in the market, there are some options with wide varieties of strike prices and an expiration date, by using Black Scholes’ assumption, the volatility will result the same or constant. In reality, when you use Black Scholes formula, the implied volatility curve will resemble a smile. On the other hand, Black-Scholes model, which based on Brownian method, should also follow normal distribution, but this time, Black Scholes failed again to prove its assumption. By plugging in required factors to this model, it will result a two heavier tails, which is not normal
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