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The assumption of constant investment opportunities can be relaxed. This requires a model for how ,, change over time. An interest rate model could be added and would lead to a portfolio containing bonds of different maturities. Some authors have added a stochastic volatility model of stock market returns.
The implied volatility smile refers to the pattern of implied volatilities for options contracts with the same expiration date but different strike prices. The slope of the implied volatility smile reflects the market's expectations for future changes in the stock price, with a steeper slope indicating higher expected volatility.
In finance, the Heston model, named after Steven L. Heston, is a mathematical model that describes the evolution of the volatility of an underlying asset. [1] It is a stochastic volatility model: such a model assumes that the volatility of the asset is not constant, nor even deterministic, but follows a random process.
actual historical volatility which refers to the volatility of a financial instrument over a specified period but with the last observation on a date in the past near synonymous is realized volatility , the square root of the realized variance , in turn calculated using the sum of squared returns divided by the number of observations.
Portfolio return volatility is a function of the correlations ρ ij of the component assets, for all asset pairs (i, j). The volatility gives insight into the risk which is associated with the investment. The higher the volatility, the higher the risk. In general: Expected return:
Volatility risk is the risk of an adverse change of price, due to changes in the volatility of a factor affecting that price. It usually applies to derivative instruments , and their portfolios, where the volatility of the underlying asset is a major influencer of option prices .
The main idea behind these two models is that volatility is dependent upon past realizations of the asset process and related volatility process. This is a more precise formulation of the intuition that asset volatility tends to revert to some mean rather than remaining constant or moving in monotonic fashion over time.
Starting from a constant volatility approach, assume that the derivative's underlying asset price follows a standard model for geometric Brownian motion: = + where is the constant drift (i.e. expected return) of the security price , is the constant volatility, and is a standard Wiener process with zero mean and unit rate of variance.