The Heston model, a popular stochastic volatility model in financial mathematics, describes the evolution of asset prices by incorporating a stochastic process for the variance (volatility) of the asset. A crucial aspect of working with the Heston model, especially for pricing derivatives, involves changing the probability measure. This change of measure, often from the real-world measure (P) to a risk-neutral measure (Q), is essential for risk-neutral valuation. Filtration plays a fundamental role in this process, ensuring that the information structure is properly handled.
Understanding Filtration
In the context of stochastic processes, a filtration represents the flow of information over time. It is a sequence of σ-algebras (F<sub>t</sub>) indexed by time t, where each F<sub>t</sub> contains all the information available up to time t. Formally, a filtration (F<sub>t</sub>)<sub>t≥0</sub> on a probability space (Ω, F, P) is a collection of σ-algebras F<sub>t</sub> ⊆ F such that:
For all s ≤ t, F<sub>s</sub> ⊆ F<sub>t</sub> (information increases over time).
This means that if an event is measurable with respect to F<sub>s</sub>, it is also measurable with respect to F<sub>t</sub> for any t ≥ s. In simpler terms, if you know something at time s, you still know it at any later time t.
The Heston Model
The Heston model defines the dynamics of the asset price (S<sub>t</sub>) and its variance (v<sub>t</sub>) under the real-world measure P as follows:
dS<sub>t</sub> = μS<sub>t</sub>dt + √v<sub>t</sub>S<sub>t</sub>dW<sup>1</sup><sub>t</sub>
dv<sub>t</sub> = κ(θ - v<sub>t</sub>)dt + σ√v<sub>t</sub>dW<sup>2</sup><sub>t</sub>
where:
μ is the asset's drift.
κ is the mean reversion rate of the variance.
θ is the long-term mean of the variance.
σ is the volatility of the variance.
W<sup>1</sup><sub>t</sub> and W<sup>2</sup><sub>t</sub> are correlated Wiener processes with correlation ρ.
Change of Measure and the Radon-Nikodym Derivative
To price derivatives, we need to transition from the real-world measure P to a risk-neutral measure Q. This is achieved using the Radon-Nikodym derivative (Z<sub>t</sub>), which is defined as:
Z<sub>t</sub> = dQ/dP|<sub>Ft</sub>
This represents the change in probability density between the two measures, conditioned on the information available up to time t (represented by the filtration F<sub>t</sub>).
Girsanov's theorem provides a way to construct the Radon-Nikodym derivative. It allows us to change the drift of a Wiener process. In the Heston model, we typically change the drifts of both W<sup>1</sup><sub>t</sub> and W<sup>2</sup><sub>t</sub>.
Let's define the changes of measure:
dW<sup>1Q</sup><sub>t</sub> = dW<sup>1</sup><sub>t</sub> + λ<sub>1</sub>(t)dt
dW<sup>2Q</sup><sub>t</sub> = dW<sup>2</sup><sub>t</sub> + λ<sub>2</sub>(t)dt
where λ<sub>1</sub>(t) and λ<sub>2</sub>(t) are the market price of risk for the asset and the variance, respectively. These are F<sub>t</sub>-adapted processes, meaning their value at time t is known based on the information available up to time t.
The Radon-Nikodym derivative for this change of measure is given by:
Z<sub>t</sub> = exp(-∫<sub>0</sub><sup>t</sup> λ<sub>1</sub>(s)dW<sup>1</sup><sub>s</sub> - ½∫<sub>0</sub><sup>t</sup> λ<sub>1</sub>(s)<sup>2</sup>ds - ∫<sub>0</sub><sup>t</sup> λ<sub>2</sub>(s)dW<sup>2</sup><sub>s</sub> - ½∫<sub>0</sub><sup>t</sup> λ<sub>2</sub>(s)<sup>2</sup>ds)
The Role of Filtration
Filtration is crucial in several aspects of this process:
Well-Defined Radon-Nikodym Derivative: The Radon-Nikodym derivative Z<sub>t</sub> must be a martingale under the original measure P. This condition is essential for the change of measure to be valid. The filtration F<sub>t</sub> ensures that the stochastic integrals in the definition of Z<sub>t</sub> are well-defined and that Z<sub>t</sub> is indeed a martingale. The adaptedness of λ<sub>1</sub>(t) and λ<sub>2</sub>(t) to the filtration is essential for this.
Information Consistency: The change of measure must preserve the information structure. The filtration F<sub>t</sub> ensures that events that are known at time t under the real-world measure P are also known at time t under the risk-neutral measure Q. This is vital for consistent pricing and hedging.
Risk-Neutral Dynamics: Under the risk-neutral measure Q, the dynamics of the asset price and variance change. These changes are defined using the transformed Wiener processes W<sup>1Q</sup><sub>t</sub> and W<sup>2Q</sup><sub>t</sub>. The filtration ensures that these transformed processes are indeed Wiener processes under Q.
The dynamics of the Heston model under the risk-neutral measure Q become:
dS<sub>t</sub> = rS<sub>t</sub>dt + √v<sub>t</sub>S<sub>t</sub>dW<sup>1Q</sup><sub>t</sub>
dv<sub>t</sub> = (κ(θ - v<sub>t</sub>) - σλ<sub>2</sub>(t))dt + σ√v<sub>t</sub>dW<sup>2Q</sup><sub>t</sub>
where r is the risk-free interest rate.
Market Price of Risk
A critical point is the choice of the market price of risk, λ<sub>1</sub>(t) and λ<sub>2</sub>(t). In a complete market, these would be uniquely determined. However, the Heston model represents an incomplete market, meaning there are infinitely many equivalent martingale measures. The choice of λ<sub>2</sub>(t) (the market price of variance risk) thus becomes a modeling choice. Often, a simple form like λ<sub>2</sub>(t) = λ√v<sub>t</sub> is used, where λ is a constant.
Conclusion
Filtration is a fundamental concept in stochastic calculus and plays a vital role in the change of measure for the Heston model. It ensures the mathematical validity of the change of measure, maintains information consistency between measures, and defines the dynamics of the model under the risk-neutral measure. Without a proper understanding of filtration, working with the Heston model and pricing derivatives accurately would be impossible. The adaptedness of the market price of risk to the filtration is particularly important for ensuring the Radon-Nikodym derivative is well-behaved.