The pricing of options and related instruments has been a major breakthrough for the use of financial theory in practical application. Since the original papers of Black and Scholes and Mertonthere has been a wealth of practical and theoretical applications. In this chapter we will discuss ways of calculating the price of an option in the setting discussed in these original papers.

The discussion is not complete, it needs to be supplemented by one of the standard textbooks, like Hull Setup Let us start by reviewing the setup. The basic assumption used is about the stochastic process governing the price of the underlying asset the option is written on.

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In the following discussion we will use the standard example of a stock option, but the theory is not only relevant for stock options. The price of the underlying asset,is assumed to follow a geometric Brownian Motion process, conveniently written in either of the shorthand forms.

Using Ito's lemma, the assumption of no arbitrage, and the ability to trade continuously, Black and Scholes showed that the price of any contingent claim written on the underlying must solve the following partial differential equation: We will start by discussing the original example solved by Black, Scholes, Merton: European call and put options.

European call and put options, The Black Scholes analysis. A call put option gives the holder the right, but not the obligation, to buy sell some underlying asset at a given pricecalled the exercise price, on or before some given date.

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If the option is European, it can only be used exercised at the maturity date. If the option is American, it can be used at any date up to and including the maturity date.

We use the following notation: Price of the underlying, eg stock price,: Risk free interest rate, continously compounded ,: Standard deviation of the underlying asset, eg stock,: At maturity, a call option is worth. The Black Scholes formulation involves an assumption of continous time and the possibility of trading continously. The Black Scholes formula can be proven a number of other ways.

One is to assume a representative agent and lognormality as was done in Rubinstein The latter second derivative call option 95 pips daily trading system interesting, as it allows us to link the Black Scholes formula to the binomial, allowing the binomial framework to be used as an approximation.

We will return to this in the next chapter. In trading of options, a number of partial derivatives of the option price formula is important.

The first derivative of the second derivative call option price with respect to the day forex guarantee trading win of the underlying security is called the delta of the option price.

It is the derivative most people will run into, since it is important in hedging of options. We limit the discussion to the partials of call options.

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The remaining derivatives are more seldom used, but all of them are relevant. The gamma is the second derivative of the option price with respect to the price of the underlying security, and calculated as: The theta is the partial with respect to time.

For a call option the following two relations hold: The Vega is the partial with respect to volatility: In calculation of the option pricing formulas, in particular the Black Scholes formula, the only unknown is the standard deviation of the underlying stock.

A common problem in option pricing is to find the implied volatility, given the observed price quoted in the market. For example, giventhe price of a call option, the following equation should be solved for the value of.

European call and put options, The Black Scholes analysis.

Instead of this simple bracketing, which is actually pretty fast, and will almost always find the solution, we can use the Newton-Raphson formula for finding the root of an equation in a single variable. The general description of this method starts with a function for which we want to find a root.

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