Classical Modelling for numerical data simulation in finance
1. Classical stochastic market models (eg. Heston, SABR) and autoregressive models.
- advantages : tractability and more straightforward suitability to currently prevalent risk-management frameworks.
- disadvantages : relative inflexibility
Stochastic Volatility Model
stochastic volatility model is a financial model that assumes the volatility of the option price is a random variable itself. This is the key difference with the Black-Scholes Model.
(i) Heston Model
\(dS_{t} = rS_{t}dt + \sqrt{V_{t}} S_{t} dW_{1t}\) \(dV_{t} = k(\theta - V_{t})dt + \delta \sqrt{V_{t}}dW_{2t}\) where,<br> $S_{t}$ is stock price at time t.<br> $r =$ risk-free interest rate - theoretical risk-free interest rate<br> $\sqrt{V_{t}}$ is the volatality (standard deviation) of the asset price<br> $\detla$ is the volatality of $\sqrt{V_{t}}$<br> $\theta$ is the long-term price variance<br> $k$ is rate of reversion to $\theta$<br> $dt$ is indefininte small time n=increment<br> $W_{1t}$ is the Browninan motion of asset price<br> $W_{2t}$ is the Browninan motion of asset’s price volatility<br>
– Properties of Heston Model –
- It factors in a possible correlation between a stock’s price and its volatility.
- It conveys volatility as reverting to the mean.
- It gives a closed-form solution, meaning that the answer is derived from an accepted set of mathematical operations.
- It does not require that stock prices follow a lognormal probability distribution.
the Chen model, and the GARCH model.