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Creating a New Stochastic Volatility Model from Scratch (Part 2 of 3)
Introduction to stochastic volatility models (Heston) and the creation of a new stochastic volatility model for volatility clustering using Bitcoin Price data
Introduction
The term ‘stochastic’ is defined as randomness stemming from an underlying probability distribution. Stochastic volatility models have a component wherein variance is randomly distributed. By leveraging the fact that price movements are stochastic, we can introduce a function that itself is inertly stochastic. The equation below is based on geometric Brownian motion where ‘v’ is no longer constant, varying in time.
The main focus for stochastic volatility models is to model variable (v) which is not constant throughout time.
The change in (v) is given as two new functions, expressed in terms of (v): Alpha capture trends in (v) and Beta captures the randomness.
A more comprehensive introduction is given in Part 1.
Editing Creating a New Stochastic Volatility Model from Scratch (Part 1 of 3) — Medium
Heston’s Model
The Heston model is a stochastic volatility model which models volatility as non-constant. It improves upon the Black-Scholes model which treats volatility as constant. The Heston Model takes on the form:
- Ws is the Brownian motion of the asset price
- Wv is the Brownian motion of the asset’s price variance
- St is the price of a specific asset at time t
- √Vt is the volatility of the asset price
- zeta is the volatility of the volatility
- r is the drift
- θ is the long-term price variance
- k is the rate of reversion to the long-term price variance
- dt is the indefinitely small positive time increment
The framework of the Heston’s model allows the modelled volatility to reach zero and remain low for long periods of time. In the equation the variables (k) and (theta) are responsible for this…
