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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 (FD) 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
References: Online Note
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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