class: center, middle, inverse, title-slide .title[ # Method of Moments Estimation for Stochastic Volatility Models ] .author[ ### 吴燕丰 ] .institute[ ### a joint work with 杨翔宇 and 胡建强 ] .date[ ### 2023-03-29 ] --- ### Stochastic Volatility Models Stock price: `\(s(t)\)` Volatility: `\(v(t)\)`, not observable, latent Heston Model(1993)<sup>*</sup>: `\begin{align} \frac{ds(t)}{s(t)} &= \mu dt + \sqrt{v(t)}dw^s(t),\label{eqn:price-process}\\ dv(t) &= k(\theta - v(t))dt + \sigma_v\sqrt{v(t)}dw^v(t).%\label{eqn:volatility-process} \label{eq.v} \end{align}` .footnote[ [*] Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. _The review of financial studies_, 6(2), 327-343. (_cited by 10,903, Google Scholar_) ] --- ### Parameter Estimation - Literatures The **likelihood** function has no closed-form (very hard). #### MLE (Maximum Likelihood Estimation, 极大似然估计) approximate, then maximize the likelihood - Simulated MLE (SMLE), Durham and Gallant (2002)<sup>1</sup> - MLE, approximation, option prices, Aït-Sahalia and Kimmel (2007)<sup>2</sup> - QMLE, Feunou and Okou (2018) - Gradient-based Simulated MLE, Peng et al. (2014, 2016) .footnote[ [1] Durham, G. B., & Gallant, A. R. (2002). Numerical techniques for maximum likelihood estimation of continuous-time diffusion processes. *Journal of Business & Economic Statistics*, 20(3), 297-338. (_cited by 556, Google Scholar_) [2] Aït-Sahalia, Y., & Kimmel, R. (2007). Maximum likelihood estimation of stochastic volatility models. *Journal of financial economics*, 83(2), 413-452. (_cited by 616, Google Scholar_) ] --- ### Parameter Estimation - Literatures #### MCMC (Markov Chain Monte Carlo, 马尔可夫链蒙特卡洛) time-consuming, computationally prohibitive in some cases - Eraker et al. (2003)<sup>1</sup> - Jacquier et al. (2002)<sup>2</sup> - Roberts et al. (2004)<sup>3</sup> .footnote[ [1] Eraker, B., Johannes, M., & Polson, N. (2003). The impact of jumps in volatility and returns. *The Journal of Finance*, 58(3), 1269-1300. (_cited by 1,752, Google Scholar_) [2] Jacquier, E., Polson, N. G., & Rossi, P. E. (2002). Bayesian analysis of stochastic volatility models. *Journal of Business & Economic Statistics*, 20(1), 69-87. (_cited by 2,349, Google Scholar_) [3] Roberts, G. O., Papaspiliopoulos, O., & Dellaportas, P. (2004). Bayesian inference for non‐Gaussian Ornstein–Uhlenbeck stochastic volatility processes. *Journal of the Royal Statistical Society: Series B*, 66(2), 369-393. (_cited by 167, Google Scholar_) ] --- ### Parameter Estimation - Literatures #### Method of Moments (MM, (近似)矩估计) approximation - Simulated MM (SMM), Duffie and Singleton (1993)<sup>1</sup> - Efficient MM (EMM), Bansal et al. (1994), Gallant and Tauchen (1996)<sup>2</sup> - Generalized MM (GMM), Chacko and Viceira (2003), Jiang and Knight (2002), Singleton (2002), - Conditional MM, high-frequency data, Bollerslev and Zhou (2002)<sup>3</sup> - Exact MM, Yang et al. (2021), Levy-driven OU stochastic volatility models .footnote[ [1] Duffie, D. and Singleton, K. J. (1993). Simulated moments estimation of markov models of asset prices. *Econometrica*, 61(4):929–952. (_cited by 1,294, Google Scholar_) [2] Gallant, A. R., & Tauchen, G. (1996). Which moments to match?. *Econometric theory*, 12(4), 657-681. (_cited by 1,555, Google Scholar_) [3] Bollerslev, T., & Zhou, H. (2002). Estimating stochastic volatility diffusion using conditional moments of integrated volatility. *Journal of Econometrics*, 109(1), 33-65. (_cited by 391, Google Scholar_) ] --- ### Summary of our Method of Moments Analytic formulas for the moments and covariances Advantages or features: - simple and easy to implement - quick, without resorting to simulation - No option price data required - only need relatively low order moments and covariances --- ### Notations and Formulas `\begin{equation} d\log s(t) = (\mu-\frac{1}{2}v(t))dt + \sqrt{v(t)}dw^s(t). \end{equation}` `\begin{equation} y_i \triangleq \log s_i - \log s_{i-1}. \end{equation}` Some notations `\begin{align*} I_{s,t} \triangleq & \int_{s}^{t}\sqrt{v(u)}dw^v(u), & I_{s,t}^* \triangleq & \int_{s}^{t}\sqrt{v_u}dw(u),\\ I\!E_{s,t} \triangleq & \int_{s}^{t}e^{ku}\sqrt{v(u)}dw^v(u), & IV_{s,t} \triangleq & \int_{s}^{t}v(u)du, \end{align*}` `\begin{align*} I_i&\triangleq I_{(i-1)h,ih}, &I_i^*&\triangleq I^*_{(i-1)h,ih}, &I\!E_i&\triangleq I\!E_{(i-1)h,ih}, &IV_i&\triangleq IV_{(i-1)h,ih},\\ I_{i,t}&\triangleq I_{ih,t}, &I^*_{i,t}&\triangleq I^*_{ih,t}, &I\!E_{i,t}&\triangleq I\!E_{ih,t}, &IV_{i,t}&\triangleq IV_{ih,t}. \end{align*}` --- ### Notations and Formulas The `\(i\)`th return `\begin{equation} y_i = \mu h - \frac{1}{2}IV_i + \rho I_i + \sqrt{1-\rho^2} I_i^*. \end{equation}` The `\(i\)`th Integrated Volatility `\begin{equation} IV_i = \theta (h-\tilde{h}) + \tilde{h} v_{i-1} - \frac{\sigma_v}{k}e^{-kih}I\!E_i + \frac{\sigma_v}{k}I_i, \end{equation}` where `\(\tilde{h} \triangleq (1-e^{-kh})/k\)`. --- ### Moments and Covariances of returns `\(y_n\)` `\begin{eqnarray} E[y_n] & = &(\mu-\theta/2)h.\\ var(y_n) & = &\theta h + \left(\frac{\sigma_v^2}{4k^2}-\frac{\rho \sigma_v}{k} \right)\theta (h-\tilde{h}).\\ cov(y_n,y_{n+1}) & = & \theta\tilde{h}^2\left(\frac{\sigma_v^2}{8k} - \frac{\rho\sigma_v}{2}\right). \\ cov(y_n,y_{n+2}) & = & e^{-kh}cov(y_n,y_{n+1}) = e^{-kh} \theta\tilde{h}^2\left(\frac{\sigma_v^2}{8k} - \frac{\rho\sigma_v}{2}\right). \end{eqnarray}` -- `\begin{align} cov(y_n^2,y_{n+1}) &= \frac{\theta \sigma_v^4}{8k^3}\tilde{h}(he^{-kh} - \tilde{h}) + \left( \frac{\theta \sigma_v^2}{4k}\mu h - \frac{\theta^2\sigma_v^2}{8k}h - \frac{\theta\sigma_v^2}{4k} \right)\tilde{h}^2\nonumber\\ &\quad - \frac{\rho\sigma_v}{2}\tilde{h}\bigg[ \left( \frac{3\sigma_v^2}{2k^2} - \frac{2\rho\sigma_v}{k} \right)\theta(he^{-kh} -\tilde{h}) + (2\mu\theta -\theta^2)h\tilde{h} \bigg]. \end{align}` --- ### Moments Estimation `\begin{align*} E[y_n] &\approx \overline{Y} \triangleq \frac{1}{N}\sum_{i=1}^N Y_i , \\ var(y_n) &\approx S^2 \triangleq \frac{1}{N}\sum_{i=1}^N (Y_i - \overline{Y})^2 ,\\ cov(y_n,y_{n+1}) &\approx \hat{cov}(y_n,y_{n+1}) \triangleq \frac{1}{N-1}\sum_{i=1}^{N-1} (Y_i - \overline{Y})(Y_{i+1} - \overline{Y}) ,\\ cov(y_n,y_{n+2}) &\approx \hat{cov}(y_n,y_{n+2}) \triangleq \frac{1}{N-2}\sum_{i=1}^{N-2} (Y_i - \overline{Y})(Y_{i+2} - \overline{Y}), \\ cov(y_n^2,y_{n+1}) &\approx \hat{cov}(y_n^2,y_{n+1}) \triangleq \frac{1}{N-1}\sum_{i=1}^{N-1} (Y_i^2 - \overline{Y^2})(Y_{i+1} - \overline{Y}) , \end{align*}` --- ### Central Limit Theorem .center[ <img src='Theorem4-1-notation.png' width=450> <img src='Theorem4-1.png' width=600> ] --- ### Parameter Estimation `\begin{align} \hat{k} &= \frac{1}{M-1}\sum_{m=2}^M\frac{1}{(m-1)h}\ln \left( \frac{\hat{cov}(y_n,y_{n+1})}{\hat{cov}(y_n,y_{n+m})} \right),\\ \hat{\theta} &= S^2/h - \frac{2(h-\tilde{h}_{\hat{k}})}{h\hat{k}\tilde{h}_{\hat{k}}^2}\hat{cov}(y_n,y_{n+1}),\\ \hat{\mu} &= \overline{Y}/h + \hat{\theta}/2,\\ \hat{\sigma}_v^2 &= \frac{4\hat{k}\overline{Y} + [8\hat{d}_h/(\hat{\theta} \tilde{h}_{\hat{k}}^3)]\hat{cov}(y_n,y_{n+1}) - 2\hat{k}\frac{\hat{cov}(y_n^2,y_{n+1})}{\hat{cov}(y_n,y_{n+1})} }{\hat{\theta}\tilde{h}_{\hat{k}}^2/(2\hat{cov}(y_n,y_{n+1})) - \hat{d}_h/(\hat{k}\tilde{h}_{\hat{k}}) },\\ \hat{\rho} &= \frac{\hat{\sigma}_{v}}{4\hat{k}} - \frac{2}{\hat{\theta}\hat{\sigma}_{v}\tilde{h}_{\hat{k}}^2}\hat{cov}(y_n,y_{n+1}), \end{align}` .footnote[ where `\(2\leq M < N\)` (usually `\(M\)` takes a small value, e.g., `\(M\leq 10\)`), `\(\tilde{h}_{\hat{k}} \triangleq (1-e^{-\hat{k}h})/\hat{k}\)` and `\(\hat{d}_h \triangleq he^{-\hat{k}h} - \tilde{h}_{\hat{k}}\)`. ] --- ### Central Limit Theorem .center[ <img src='Theorem4-2.png' width=600> ] --- ### Numerical Experiments .center[ <img src='table1.png' width=500> ] .footnote[ “mean ± standard deviation” based on 400 replications, with 400K samples for each replication ] --- ### Numerical Experiments asymptotic behavior .center[ <img src='table2.png' width=600> ] .footnote[ “mean ± standard deviation” based on 400 replications ] --- ### Numerical Experiments change the value of time interval unit `\(h\)` .center[ <img src='table3.png' width=600> ] .footnote[ “mean ± standard deviation” based on 400 replications ] --- ### Extensions to other SV models - SV model with jumps in the return process `\begin{align*} d\log s(t) &= (\mu- v(t)/2) dt + \sqrt{v(t)}dw^s(t) + jdN(t),\\ dv(t) &= k(\theta - v(t))dt + \sigma_v \sqrt{v(t)}dw^v(t), \end{align*}` - Multi-factor SV model `\begin{align*} d\log s(t) &= (\mu- v(t)/2) dt + \sqrt{v(t)}dw(t),\\ v(t) &= v_1(t) + v_2(t),\\ dv_1(t) &= k_1(\theta_1 - v_1(t))dt + \sigma_{1v} \sqrt{v_1(t)}dw_2(t),\\ dv_2(t) &= k_2(\theta_2 - v_2(t))dt + \sigma_{2v} \sqrt{v_2(t)}dw_2(t), \end{align*}` --- ### Conclusion We obtain the moments and covariances of the asset price based on which we develop MM estimators. A key of our method is to develop a recursive procedure to calculate the moments and covariances of the asset price. Our MM estimators are simple and easy to implement. --- class: middle, center Questions & Answers Thanks for Listening!