Heston Model
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debugInConsole: false # Print debug info in Obsidian console1 Heston Model
References: [[ksendal]]; Quantpie Video.
In mathematical finance the Heston model is a stochastic volatility model describing the evolution the volatility of an underlying asset. Specifically, the asset price process \((S_{t})\) and stochastic volatility process \((V_{t})\) have dynamics given by the Cox-Ingersoll-Ross (CIR) Model under the risk-neutral measure by \[ \begin{cases} dS_{t}=rS_{t}dt+S_{t}\sqrt{V_{t}}dW_{t}^1 \\ dV_{t}=\kappa(\theta-V_{t})dt+\eta \sqrt{ V_{t} }dW_{t}^2 \\ dB_{t}=rB_{t}dt, \end{cases} \] where \(W_{t}^1\) and \(W_{t}^2\) are dependent Wiener processes (under the risk-neutral measure) with \(d\left< W^1, W^2 \right>_{t}=\rho\) with additional parameters: 1. \(V_{0}\) - the initial variance; 2. \(\theta\) - the long variance; 3. \(\kappa\) - the mean reversion rate of \(V_{t}\) to \(\theta\); and 4. \(\eta\) - the volatility of the volatility (i.e. variance of \(V_{t}\)).
If the parameters \(\kappa\), \(\eta\) and \(\theta\) obey Feller’s condition \[ 2\kappa \theta> \eta^2, \] then the process \((V_{t})\) is strictly positive.
The construction of this model was motivated to match observed market behavior, specifically high peak and fat tails and the non-constant volatility. The volatility process is positive and mean reverting and the correlation \(\rho\) is a way of controlling the relationship between stock price and volatility.
The Heston model is popular since under the PDE approach, after log transform, one gets a PDE with constant coefficient, solvable through [[fourier-transform]] or Finite-Difference Methods since no closed form solution is availale. In application this model requires calibration (i.e. empirical estimation of the noted parameters) and simulation.
2 Heston Model Pricing PDE
We see clearly that we need three entries in the value function of the option in order to build up to. PDE characterization. Defining the value function of a call option at time \(t\) with stock price \(S_{t}=s\) and volatility \(V_{t}=v\) as \(C(t,s,v)\). Applying the Itô Formula we obtain \[ \begin{align} dC & =C_{t}dt+C_{s}dS_{t}+C_{v}dV_{t}+ \frac{1}{2}C_{ss}d\left< S,S \right> _{t}+\frac{1}{2}C_{vv}d\left< V,V \right>_{t}+C_{vs}d\left< S,V \right> _{t} \\ & =\left( C_{t}+rS_{t}C_{s}+ \frac{1}{2}V_{t}S_{t}^2 C_{ss}+\kappa(\theta-V_{t})C_{v}+\frac{1}{2}\eta^2V_{t}C_{vv}+\eta V_{t}S_{t}\rho C_{vs} \right) dt \\ & \qquad +C_{s}\sqrt{ V_{t} }S_{t}dW_{t}^1+C_{v}\eta \sqrt{ V_{t} } dW_{t}^2. \end{align} \] By assuming the stochastic integrals are martingales we obtain the PDE characterization that \[ 0 = -rC + C_{t}+rsC_{s}+\frac{1}{2}vs^2 C_{ss}+\kappa s^2 C_{ss}+\kappa(\theta-v)C_{v}+\frac{1}{2}\eta^2v C_{vv}+\eta v s \rho C_{vs}. \]
3 Heston Model Simulation
3.1 Heston Model Simulation - Euler-Maruyama Method
References: SDE Simulation. SDE Simulation
4 Relevant Literature
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