Monte Carlo Simulation in Finance to Calculate European Options Value

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Monte Carlo Simulation in Finance for Calculating European Options Value

1. Introduction
An option is a financial instrument whose value depends on a value of underlying security. Options trade started in 1973 at the Chicago Board Options Exchange (Hull, Fundamentals of futures and options markets 2008). Nowadays, options have become a crucial tool in finance; they have become valuable both for financial institutions and investors. Options are attractive to investors since they have great effect in reducing risk in investment. Throughout this independent study, we will be working within the European option. A European option is a vanilla option. A vanilla option is a standard type of option contract with no special features except the simple expiry date (the date in the contract) and the predetermined strike price (the agreed price at which a particular option price can be exercised). There are two types of European option; European call options and European put options. European call options give the owner the right, but not the obligation, to buy an agreed quantity of underlying securities on a certain time (the expiration date), for a specified price (the strike price). The expiration date is called the day until maturity. The seller of call options is obligated to sell the security should the buyer so decide. Vice versa, European put options give the owner the right, but not the obligation, to sell an agree quantity of an underlying securities on a certain time for a specified price. The seller of put options is obligated to buy the security if the buyer so decides. The buyer pays a fee (called premium) for both of these rights. For each call options or put options, there are three conditions; ITM (In-the-money), OTM (Out-of-the-money), and ATM (At-the-money). Call option is called ITM when the strike

price is below the current underlying security’s price (spot price). In this situation, the holder can get a gain (payoff). On the other hand, call option is called OTM, when the strike price is above the spot price of the underlying security. In this situation the option is expired worthless. And lastly, call option is called ATM, when the strike price is equal to the spot price of the underlying security. This situation is the break even condition. For put options, it is called ITM when the strike price is above the spot price of the underlying security, then it is called OTM, when the strike price is below the spot price of the underlying security, and it is called ATM, when the strike price is equal to the spot price of the underlying security.

2. Black-Scholes Model
In 1973, Fischer Black and Myron Scholes articulated the popularly known Black Scholes Option Pricing Model. The Black-Scholes Model is a mathematical description of financial market and derivative investment instruments (Hull, Options, Futures, and other Derivatives, 7th Edition 2009). The model develops partial differential equations whose solution, the Black-Scholes formula, is widely used in the pricing of European options. The model was based on some assumptions as follows:        The market is assumed to be liquid, to be fair and provide all participants with equal access to the available information. This implies that there are no transaction costs. It is possible to borrow and lend cash at a known constant risk-free interest rate. The stock price follows a geometric Brownian motion with constant drift and volatility. The underlying security does not pay a dividend. All securities are infinitely divisible (i.e., it is possible to buy any fraction of a share). There are no restrictions on short selling. There is no arbitrage opportunity. In practice, however, none of these principles can be perfectly satisfied. Nevertheless, as we mentioned above, Black-Scholes model is an important tool for real market analysis, and Black-Scholes prices provide good approximation to the prices of options.

2.1. Black-Scholes Formula for Option...
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