Modeling Tools for Financial Options
What do we mean by option? An option is the right (but not the obligation) to buy or sell a risky asset at a prespecified fixed price within a specified period. An option is a financial instrument that allows —amongst other things— to make a bet on rising or falling values of an underlying asset. The underlying asset typically is a stock, or a parcel of shares of a company. Other examples of underlyings include stock indices (as the Dow Jones Industrial Average), currencies, or commodities. Since the value of an option depends on the value of the underlying asset, options and other related financial instruments are called derivatives (→ Appendix A1). An option is an agreement between two parties about trading the asset at a certain future time. One party is the writer, often a bank, who fixes the terms of the option contract and sells the option. The other party ist the holder, who purchases the option, paying the market price, which is called premium. How to calculate a fair value of the premium is a central theme of this book. The holder of the option must decide what to do with the rights the option contract grants. The decision will depend on the market situation, and on the type of option. There are numerous different types of options, which are not all of interest to this book. In Chapter 1 we concentrate on standard options, also known as plain-vanilla options. This Section 1.1 introduces important terms.
KeywordsAsset Price Stochastic Differential Equation Modeling Tool Wiener Process Implied Volatility
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- 1.The price K as well as other prices are meant as the price of one unit of an asset, say. in $.Google Scholar
- 2.The symbol ∆t denotes a small increment in t (analogously ∆S, ∆W). In case ∆ would be a number, the product with u would be denoted ∆ • u or u∆.Google Scholar
- 3.For an American option it is not certain that Π t can be reached because the holder may choose early exercise. Hence we have only the inequality Π 0 e rT ≤ Π T.Google Scholar
- 4.This assumption together with the assumption of an immediate reaction of the market to arriving informations are called hypothesis of the efficient market [Bo98].Google Scholar
- 5.The less mathematically oriented reader may like to skip the rest of this subsection.Google Scholar
- 6.For the implied volatility see Exercise 1.5.Google Scholar
- 7.Since S i = S i-1 exp(R i,i-1), the log return is also called continuously compounded return in the ith time interval [Tsay02].Google Scholar
- 8.The thickness is measured by the kurtosis E((X — μ)4)/σ 4. The normal distribution has kurtosis 3. So the excess kurtoris is the difference to 3. Frequently, data of returns are characterized by large values of excess kurtosis.Google Scholar