Where: - logarithmic rate of return for i-th period - price of security (value of the index) at time i Volatility implied from other derivative contracts traded on the market( recheck sentences below carefully please) Under assumption of validity of the option pricing formulas, we can imply what level of the volatility is perceived by the market participants. Implied volatility is the value that calibrates the model (e.g. Black-Scholes), so that it yields current market price of the instrument. Rationale for using implied volatility is that price of volatility should be the same for all traded assets. Using other volatility than the market, means that you are buying (selling) uncertainty on other conditions than the market. This leads to arbitrage opportunities. Remark: There is no exact analytical formula for implied volatility (or covariance). Values are obtained by means of numerical algorithms. Variance is stochastic It is true that volatility is unstable in time, but patterns which volatility follows are still under discussion. One of the findings, which is supported by the majority of market analysts and researchers is called “volatility clustering”. It describes following property: period with high (low) volatility is usually followed by a period with high (low) volatility. For that reason most popular models for stochastic volatility incorporate this autoregressive property. Examples of such models are ARCH and GARCH. Those methods are especially successful in modeling volatility of short term contracts. According to arguments above, GARCH model would be best suited for valuation of the analyzed option, but due to simplicity of the model we did not employ idea of stochastic volatility. Pricing of the option was performed with estimates based on historical data.
Where: - logarithmic rate of return for i-th period - price of security (value of the index) at time i Volatility implied from other derivative contracts traded on the market( recheck sentences below carefully please) Under assumption of validity of the option pricing formulas, we can imply what level of the volatility is perceived by the market participants. Implied volatility is the value that calibrates the model (e.g. Black-Scholes), so that it yields current market price of the instrument. Rationale for using implied volatility is that price of volatility should be the same for all traded assets. Using other volatility than the market, means that you are buying (selling) uncertainty on other conditions than the market. This leads to arbitrage opportunities. Remark: There is no exact analytical formula for implied volatility (or covariance). Values are obtained by means of numerical algorithms. Variance is stochastic It is true that volatility is unstable in time, but patterns which volatility follows are still under discussion. One of the findings, which is supported by the majority of market analysts and researchers is called “volatility clustering”. It describes following property: period with high (low) volatility is usually followed by a period with high (low) volatility. For that reason most popular models for stochastic volatility incorporate this autoregressive property. Examples of such models are ARCH and GARCH. Those methods are especially successful in modeling volatility of short term contracts. According to arguments above, GARCH model would be best suited for valuation of the analyzed option, but due to simplicity of the model we did not employ idea of stochastic volatility. Pricing of the option was performed with estimates based on historical data.
ProjectAFIfinal.docx