Realised volatility and parametric estimation of Heston SDEs

Abstract

We present a detailed analysis of observable moment-based parameter estimators for the Heston SDEs jointly driving the rate of returns $(R_{t})$ and the squared volatilities $(V_{t})$. Since volatilities are not directly observable, our parameter estimators are constructed from empirical moments of realised volatilities $(Y_{t})$, which are of course observable. Realised volatilities are computed over sliding windows of size $\varepsilon$, partitioned into $J(\varepsilon )$ intervals. We establish criteria for the joint selection of $J(\varepsilon )$ and of the subsampling frequency of return rates data.

We obtain explicit bounds for the $L^{q}$ speed of convergence of realised volatilities to true volatilities as $\varepsilon \to 0$. In turn, these bounds provide also $L^{q}$ speeds of convergence of our observable estimators for the parameters of the Heston volatility SDE.

Our theoretical analysis is supplemented by extensive numerical simulations of joint Heston SDEs to investigate the actual performances of our moment-based parameter estimators. Our results provide practical guidelines for adequately fitting Heston SDE parameters to observed stock price series.

Appendix: Polynomial functions of volatilities and Theorem 7.1

Proof of Theorem 7.1

By linearity, we only need to consider the case when $h$ is a monomial in $k$ variables. For $k=1$, the result was proved in (6.4). Proceeding by induction on $k$, assume the result is true for monomials in $k-1$ variables $(x_{2}, \ldots , x_{k})$. Any monomial $h$ in $k$ variables can be written as $h = x_{1}^{m} g(x_{2}, \ldots , x_{k})$. Define

$$ G_{T}= g (V_{u(2)+T}, \ldots , V_{u(k)+T} ) \qquad \text{and} \qquad H_{T}= V_{u(1)+T}^{m} G_{T}. $$

The induction hypothesis provides a polynomial $R$ in $k+1$ variables such that for all $T$,

$$ {\mathbb{E}}_{y} [G_{T}] = R (\nu _{T}, y \nu _{T}, w_{1}, w_{2}, \ldots , w_{k-1} ), $$

where the coefficients of $R$ are determined by $g, \boldsymbol{\theta }$. By the Markov property, we thus get

$$ {\mathbb{E}}[G_{T} | {\mathcal{F}}_{u(1) +T} ] = R (\nu _{T}, V_{u(1) +T} \nu _{T}, w_{1}, w_{2}, \ldots , w_{k-1} ) . $$

Since ${\mathbb{E}}_{y} [H_{T}] = {\mathbb{E}}_{y} [ V_{u(1)+T}^{m} { \mathbb{E}}[G_{T} | {\mathcal{F}}_{u(1) +T}] ]$, we then obtain

$$ {\mathbb{E}}_{y} [H_{T}] = {\mathbb{E}}_{y} [ V_{u(1)+T}^{m} R(\nu _{T}, V_{u(1) +T} \nu _{T}, w_{1}, w_{2}, \ldots , w_{k-1}) ]. $$

Each monomial $M$ of $R$ is of the form $\nu _{T}^{p} (V_{u(1) +T} \nu _{T})^{j} S(w_{1}, w_{2}, \ldots , w_{k-1})$ for some $p$, $j$ and some polynomial $S$. Then on the right-hand side of (7.4), $M$ contributes a term of the form

$$ \Gamma (M) = \nu _{T}^{p+j} S(w_{1}, w_{2}, \ldots , w_{k-1}) { \mathbb{E}}_{y} [ V_{u(1)+T}^{m+j} ]. $$

Due to (6.4) with $q= m+j$, this last conditional expectation is a polynomial in the two variables

$$ \nu _{u(1)+T} = \nu _{T} w_{0} \qquad \text{and}\qquad V_{0} \nu _{u(1)+T} = y \nu _{T} w_{0} $$

with coefficients depending only on $m+j$ and $\boldsymbol{\theta }$. Hence $\Gamma (M)$ is a polynomial in $\nu _{T}$ and $y \nu _{T}$, with coefficients which are polynomials in $(w_{0}, w_{1}, w_{2}, \ldots , w_{k-1})$, fully determined by $m$, $j$, $\boldsymbol{\theta }$. The same property must then hold for the sum ${\mathbb{E}}_{y}[H_{T}]$ of all the $\Gamma (M)$ contributed by the monomials $M$ of $R$. This completes the proof of (7.4) by induction on $k$.

Write ${\mathrm{POL}}$ in (7.4) as a polynomial ${\mathrm{POL}}(z)$ in the $k+2$ variables $z_{i}$. The vector $z(T) = (\nu _{T}, y \nu _{T}, w_{0}, \ldots , w_{k-1})$ tends to $z(\infty ) = (0, 0, w_{0}, \ldots , w_{k-1})$ as $T \to \infty$. The polynomial $Q(z(T)) = {\mathrm{POL}}(z(T)) - {\mathrm{POL}}(z(\infty ))$ can be written, for some integer $p$, as

$$ Q \big(z(T)\big) = \nu _{T} A_{0} + \sum _{s=1}^{p} y^{s} \nu _{T}^{s} A_{s}, $$

with each $A_{s}$ a polynomial in the $k+1$ variables $(\nu _{T}, w_{0}, \ldots , w_{k-1})$ for $s=0, \ldots , p$. Since all these positive $k+1$ variables are inferior to 1, each $|A_{s}|$ remains bounded for all $T \geq 0$ and all $u(0) < u(1) < \cdots < u(k)$. Hence there is a constant $C$ such that

$$ | A_{s} | \leq C \quad \text{and} \quad y^{s} \leq C (1+y^{p}) \qquad \text{for all } s= 0, \ldots , p, T \geq 0, y > 0. $$

For all $s \geq 1$, we have $\nu _{T}^{s} \leq \nu _{T}= e^{-T \kappa }$, and hence the expansion of $Q(z(T))$, provides a new constant $C_{1}$ such that for all $u(0) < u(1) < \cdots < u(k)$,

$$ | {\mathbb{E}}_{y} [H_{T}] - {\mathbb{E}}_{\psi }[H] | = \big| Q \big(z(T)\big) \big| \leq C_{1} (1+y^{p}) e^{-T \kappa } \qquad \text{for all } T \geq 0, y > 0. $$

This proves (7.6).

Let ${\overline{H}}= {\mathbb{E}}_{\psi }[H]$. Expand $\beta (T) = (H_{T} - {\overline{H}})^{q}$ as a linear combination of terms of the form ${\overline{H}}^{q-j} H_{T}^{j}$ for $j= 0, \ldots , q$. Recall that $h$ is a polynomial in $x_{1}, x_{2}, \ldots , x_{k}$. For $j$ fixed, $\sigma _{j} = h^{j}$ is also a polynomial in $x_{1}, x_{2}, \ldots , x_{k}$. By the definition (7.3), we can express both $\Sigma = H^{j}$ and $\Sigma _{T} = H_{T}^{j}$ as

$$ \Sigma = \sigma _{j} (V_{u(1)}, \ldots , V_{u(k)} ) \qquad \text{and} \qquad \Sigma _{T}= \sigma _{j} (V_{u(1)+T}, \ldots , V_{u(k)+T} ). $$

For each $j$, (7.6) applied to the polynomial $\sigma = h^{j}$ provides a constant $C_{j}$ and an integer $p(j)$ such that

$$ | {\mathbb{E}}_{y} [\Sigma _{T}] - {\mathbb{E}}_{\psi }[\Sigma ] | \leq C_{j} (1+y^{p(j)} ) e^{-T \kappa } \qquad \text{for all } T \geq 0, y > 0, $$

and hence there are constants $c_{j}$ such that

$$ | {\mathbb{E}}_{y} [\bar{H}^{q-j}H_{T}] - {\mathbb{E}}_{\psi }[ \bar{H}^{q-j} H_{T}] | \leq c_{j} (1+y^{p(j)} ) e^{- T \kappa } \qquad \text{for all } T \geq 0, y > 0. $$

Applying this to $j= 0, \ldots , q$ and using the Newton binomial formula yields, for some new constant $C$,

$$ {\mathbb{E}}[ | \beta (T) | ] \leq C e^{- T \kappa } \sum _{j= 0}^{q} c_{j} (1+y^{p(j)} ) \qquad \text{for all } T \geq 0, y > 0 $$

which completes the proof of (7.5). □