Variance Swap for Heston Model¶
+1. Model Setting¶
1.1 Heston model¶
$$ +\left\{\begin{matrix} + dS_t=\mu S_t dt+\sqrt{v_t}S_t dB_t^S\\ + dv_t=\kappa(\theta-v_t)dt+\sigma_V\sqrt{v_t}dB_t^V + +\end{matrix}\right. +$$
+$(dB_t^S,dB_t^V)=\rho dt$
+Accordding to Heston (1993), these can be transfered to a risk-neutral probability measure: +$$ +\left\{\begin{matrix} + dS_t=\mu S_t dt+\sqrt{v_t}S_t d\tilde B_t^S\\ + dv_t=\kappa^*(\theta^*-v_t)dt+\sigma_V\sqrt{v_t}d\tilde B_t^V + +\end{matrix}\right. +$$
+$\kappa^*=\kappa+\lambda,\quad \theta^*=\frac{\kappa\theta}{\kappa+\lambda}$ +where $\lambda$ is the premium of volatility risk
+1.2 Variance Swap¶
the value of a variance swap at expiry: +$$ +V_T=(\sigma_R^2-K_{VAR})\times L +$$
+-
+
- $\sigma^2:$ annualized realized variance +
- $K_{var}:$ annualized delivery price +
- $L:$ notional amount of swap +
in risk neutral world: $V_t=E_t^Q[e^{-r(T-t)}(\sigma_R^2-K_{var})L]$, therefore fair variance delivery price: $K_{var}=E_0^Q[\sigma_R^2]$
+2. Method Existed in PyFENG¶
Method existed in PyFeng using log-return to calculate realized variance +$$ +K_{var}:=\frac{1}{T}\sum_{i=1}^N E[(ln\frac{S_{t_{i+1}}}{S_{t_{t}}})^2] +$$
+-
+
strike_var_swap_analytic()function in heston.HestonABC()-
+
- refers: Analytic fair strike of variance swap. Eq (11) in Bernard & Cui (2014) +
+return_var_realized()function in heston_mc.HestonMcABC()
+
3. Method Added¶
Refer to Bernard C, Cui Z (2014) Prices and Asymptotics for Discrete Variance Swaps. Applied Mathematical Finance 21:140–173. https://doi.org/10.1080/1350486X.2013.820524 +and +Song-Ping Zhu, Guang-Hua Lian (2011) A Closed-form Exact Solution for Pricing Variance Swaps with Stochastic Volatility
+where the realized variance is caculated by: +$$ +K_{var}:=\frac{1}{T}\sum_{i=1}^N E[(\frac{S_{t_{i+1}}}{S_{t_{t}}}-1)^2] +$$
+We implemented
+-
+
- (eq 2.34) in Zhu & Lian (2011) +
- (Proposition3.2) in Bernard & Cui (2014) +
- percentage change return from MC method +
3.1 (eq 2.34) in Zhu & Lian (2011)¶
The fair strike price for the variance swap is given as: +$$K_{var}=E_0^Q[\sigma_R^2]=\frac{e^{r\Delta t}}{T}[f(v_0)+\sum_{i=2}^Nf_i(v_0)]$$ +where r is interest rate, $\Delta t$ is defined as $\frac{T}{N}$ and N is the number of obersavation days.
+Function $f(v_0)$ and $f_i(v_0)$ are defined as: +$$f(v)=e^{\tilde C(\Delta t)+\tilde D(\Delta t)v}+e^{-r\Delta t}-2$$ +$$\begin{aligned}f_{i}(v_{0})&=e^{\tilde{C}(\Delta t)+\frac{c_{i}e^{-\kappa^{*}t_{i-1}}}{c_{i}-\tilde{D}(\Delta t)}\tilde{D}(\Delta t)v_{0}}(\frac{c_{i}}{c_{i}-\tilde{D}(\Delta t)})^{\frac{2\kappa^{*}\theta^{*}}{\sigma_{V}^{2}}}+e^{-r\Delta t}-2\end{aligned}$$ +where $t_i = i\Delta t$, and $\kappa^{*}, \theta^{*}, \sigma_{V}$ are the parameters of Heston model under risk-neutral probability measure.
+Function $\tilde{C}(\Delta t)$ and $\tilde{D}(\Delta t)$ are derived from the generalized Fourier transform method which is used to solving the PDE of payoff. $\tilde{C}(\Delta t)$ and $\tilde{D}(\Delta t)$ are defined as: +$$\left\{\begin{array}{ll}\widetilde C(\tau)=r\tau+\frac{\kappa^*\theta^*}{\sigma_V^2}[(\widetilde a+\widetilde b)\tau-2\ln(\frac{1-\widetilde ge^{\widetilde b\tau}}{1-\widetilde g})]\\\\\widetilde D(\tau)=\frac{\widetilde a+\widetilde b}{\sigma_V^2}(\frac{1-e^{\widetilde b\tau}}{1-\widetilde ge^{\widetilde b\tau}})\\\\\widetilde a=\kappa^*-2\rho\sigma_V,\quad\widetilde b=\sqrt{\widetilde a^2-2\sigma_V^2},\quad\widetilde g=(\frac{\widetilde a}{\sigma_V})^2-1+(\frac{\widetilde a}{\sigma_V})\sqrt{(\frac{\widetilde a}{\sigma_V})^2-2}\\\end{array}\right.$$
+3.2 (Proposition3.2) in Bernard & Cui (2014)¶
$t_i=\frac{iT}{n},i=0,1,...,n$, $\Delta = T/n$, $\alpha=2\kappa\theta/\gamma^2-1\geq 0$,
+Payoff $\frac{1}{T}\sum_{i=0}^{n-1}(\frac{S_{t_{i+1}-S_{t_i}}}{S_{t_i}})^2$ equals to +$$ +K_{var}(n) = \frac{1}{T}(a_0+\sum_{i=1}^{n-1}a_i)+\frac{n-2ne^{r\Delta}}{T} +$$
+$a_i=\frac{e^{2r\Delta}}{S_0^2}M(2,\Delta)e^{q(2)V_0(\frac{\eta(t_i)e^{-\kappa t_i}}{\eta(t_i)-q(2)}01)}(\frac{\eta(t_i)}{\eta(t_i)-q(2)})^{\alpha+1}$
+where:
+$M(u,t)=E[e^{uX_t}]=S_0^u e^{\frac{\kappa\theta}{\gamma^2}((\kappa-\gamma\rho u-d(u))t-2 ln(\frac{1-g(u)e^{-d(u)t}}{1-g(u)}))}e^{V_0\frac{\kappa-\gamma\rho u-d(u)t}{\gamma}\frac{1-e^{-d(u)t}}{1-g(u)e^{-d(u)t}}}$
+$d(u)=\sqrt{(\kappa-\gamma\rho u)^2+\gamma^2(u-u^2)}$
+$g(u)=\frac{\kappa-\gamma\rho u-d(u)}{\kappa-\gamma\rho u+d(u)}$
+$q(u)=\frac{\kappa-\gamma\rho u -d(u)}{\gamma}\frac{1-e^{-d(u)\Delta}}{1-g(u)e^{-d(u)\Delta}}$
+$\eta(u)=\frac{2\kappa}{\gamma^2}(1-e^{-\kappa u})^{-1}$
+3.3 MC¶
-
+
- added
return_pctvar_realized()function inheston_mc.pyto calcuate annualized variance +
+ - take $v_t=max(0,v_t)$ +
4. Experiments¶
+import sys
+sys.path.insert(sys.path.index('')+1, r'C:\Users\LXY\Desktop\研究生课程\ASP\PyFENG')
+import pyfeng as pf
+import time
+import numpy as np
+import pandas as pd
+# model parameters
+kwargs = {'sigma': 0.04, 'vov': 0.618, 'rho': -0.64, 'mr': 11.35, 'theta': 0.022, 'intr': 0.1}
+Model = pf.HestonMcGlassermanKim2011(**kwargs)
+CMCModel = pf.HestonMcAndersen2008(**kwargs)
+BasicModel = pf.heston_mc.HestonMC(**kwargs)
+Result in Zhu & Lian¶
table for $\sigma_R^2=\frac{1}{T}\sum_{i=1}^N(\frac{S_{t_i}}{S_{t_{i-1}}}-1)^2*100^2$
+
+
times_zhu = []
+times_bern = []
+times_exactmc = []
+times_cmc = []
+times_mc = []
+analytic_zhu = []
+analytic_bern = []
+exactmc_mean = []
+cmc_mean = []
+mc_mean = []
+exactmc_std = []
+cmc_std = []
+mc_std = []
+
+for dt in [1/4, 1/12, 1/26, 1/52]:
+ print('starting simulation for dt = ', dt)
+
+ # analytic variance of Zhu's formula
+ start_time = time.time()
+ analytic_Zhu = Model.strike_var_swap_analytic_ZhuLian(texp=1, dt=dt)
+ end_time = time.time()
+ time_Zhu = end_time-start_time
+
+ # analytic variance of Bernard Cui's formula
+ start_time = time.time()
+ analytic_Bern = Model.strike_var_swap_analytic_pctreturn(texp=1, dt=dt)
+ end_time = time.time()
+ time_Bern = end_time-start_time
+
+ # get mc realized variance and time the simulation
+ CMCModel.set_num_params(dt=dt,n_path=200000)
+ start_time = time.time()
+ vars = CMCModel.return_pctvar_realized(texp=1)
+ mean_cmc = np.mean(vars)
+ std_cmc = np.std(vars)
+ end_time = time.time()
+ time_cmc = end_time-start_time
+
+ Model.set_num_params(dt=dt,n_path=200000)
+ start_time = time.time()
+ vars = Model.return_pctvar_realized(texp=1)
+ mean_exactmc = np.mean(vars)
+ std_exactmc = np.std(vars)
+ end_time = time.time()
+ time_exactmc = end_time-start_time
+
+ BasicModel.set_num_params(dt=dt,n_path=200000)
+ start_time = time.time()
+ vars = BasicModel.return_pctvar_realized(texp=1)
+ mean_mc = np.mean(vars)
+ std_mc = np.std(vars)
+ end_time = time.time()
+ time_mc = end_time-start_time
+
+
+
+ # print results
+ print('Zhu Lian\'s formula: ', analytic_Zhu)
+ print('Bernard Cui\'s formula: ', analytic_Bern)
+ print('MC realized variance: ', mean_mc)
+ print('CMC realized variance: ', mean_cmc)
+ print('ExactMC realized variance: ', mean_exactmc)
+ print('MC realized variance std: ', std_mc)
+ print('CMC realized variance std: ', std_cmc)
+ print('ExactMC realized variance std: ', std_exactmc)
+ print('time for Zhu Lian\'s formula: ', time_Zhu)
+ print('time for Bernard Cui\'s formula: ', time_Bern)
+ print('time for MC simulation: ', time_mc)
+ print('time for CMC simulation: ', time_cmc)
+ print('time for ExactMC simulation: ', time_exactmc)
+ print('----------------------------------')
+
+ analytic_zhu.append(analytic_Zhu)
+ analytic_bern.append(analytic_Bern)
+ exactmc_mean.append(mean_exactmc)
+ exactmc_std.append(std_exactmc)
+ mc_mean.append(mean_mc)
+ mc_std.append(std_mc)
+ cmc_mean.append(mean_cmc)
+ cmc_std.append(std_cmc)
+ times_zhu.append(time_Zhu)
+ times_bern.append(time_Bern)
+ times_cmc.append(time_cmc)
+ times_exactmc.append(time_exactmc)
+ times_mc.append(time_mc)
+
+ time.sleep(1)
+
+res = pd.DataFrame({'dt': [1/4, 1/12, 1/26, 1/52], 'analytic_zhulian': analytic_zhu, 'analytic_bern': analytic_bern,\
+ 'mc_mean': mc_mean, 'cmc_mean': cmc_mean,'exactmc_mean': exactmc_mean, 'mc_std': mc_std, 'cmc_std': cmc_std, 'exactmc_std': exactmc_std, \
+ 'time_ZhuLian': times_zhu, 'time_Bern': times_bern,'time_mc': times_mc, 'time_cmc': times_cmc, 'time_exactmc': times_exactmc})
+starting simulation for dt = 0.25 +Zhu Lian's formula: 0.02632143084055675 +Bernard Cui's formula: 0.02632143084055727 +MC realized variance: 0.03487467662551381 +CMC realized variance: 0.026499679482084662 +ExactMC realized variance: 0.026408175790440937 +MC realized variance std: 0.0340573875616237 +CMC realized variance std: 0.019607639030466525 +ExactMC realized variance std: 0.019721473127171013 +time for Zhu Lian's formula: 0.0 +time for Bernard Cui's formula: 0.0 +time for MC simulation: 0.04999995231628418 +time for CMC simulation: 0.10651612281799316 +time for ExactMC simulation: 0.5356192588806152 +---------------------------------- +starting simulation for dt = 0.08333333333333333 +Zhu Lian's formula: 0.024272086521686692 +Bernard Cui's formula: 0.024272086521682112 +MC realized variance: 0.027859771492387844 +CMC realized variance: 0.024006852994623754 +ExactMC realized variance: 0.024261866608035636 +MC realized variance std: 0.016591855073948374 +CMC realized variance std: 0.013544965155115338 +ExactMC realized variance std: 0.013922716579114067 +time for Zhu Lian's formula: 0.0 +time for Bernard Cui's formula: 0.0 +time for MC simulation: 0.16114234924316406 +time for CMC simulation: 0.30265140533447266 +time for ExactMC simulation: 2.113616704940796 +---------------------------------- +starting simulation for dt = 0.038461538461538464 +Zhu Lian's formula: 0.023858347847937673 +Bernard Cui's formula: 0.02385834784793417 +MC realized variance: 0.025531523731592523 +CMC realized variance: 0.02376758593632581 +ExactMC realized variance: 0.023829697436999395 +MC realized variance std: 0.011737187050216128 +CMC realized variance std: 0.011328117771315259 +ExactMC realized variance std: 0.011355118099651793 +time for Zhu Lian's formula: 0.0 +time for Bernard Cui's formula: 0.002004384994506836 +time for MC simulation: 0.32060742378234863 +time for CMC simulation: 0.6337699890136719 +time for ExactMC simulation: 5.688959836959839 +---------------------------------- +starting simulation for dt = 0.019230769230769232 +Zhu Lian's formula: 0.023710973687990612 +Bernard Cui's formula: 0.02371097368799724 +MC realized variance: 0.024499030904000817 +CMC realized variance: 0.023652189836766483 +ExactMC realized variance: 0.023470922083224946 +MC realized variance std: 0.00974403768485489 +CMC realized variance std: 0.009863110682746281 +ExactMC realized variance std: 0.00949484562057157 +time for Zhu Lian's formula: 0.0005068778991699219 +time for Bernard Cui's formula: 0.002004384994506836 +time for MC simulation: 0.7216520309448242 +time for CMC simulation: 1.268726110458374 +time for ExactMC simulation: 13.508999109268188 +---------------------------------- ++
# good format for printing dataframe res
+pd.options.display.float_format = '{:.6f}'.format
+print(res)
+dt analytic_zhulian analytic_bern mc_mean cmc_mean exactmc_mean \ +0 0.250000 0.026321 0.026321 0.034875 0.026500 0.026408 +1 0.083333 0.024272 0.024272 0.027860 0.024007 0.024262 +2 0.038462 0.023858 0.023858 0.025532 0.023768 0.023830 +3 0.019231 0.023711 0.023711 0.024499 0.023652 0.023471 + + mc_std cmc_std exactmc_std time_ZhuLian time_Bern time_mc time_cmc \ +0 0.034057 0.019608 0.019721 0.000000 0.000000 0.050000 0.106516 +1 0.016592 0.013545 0.013923 0.000000 0.000000 0.161142 0.302651 +2 0.011737 0.011328 0.011355 0.000000 0.002004 0.320607 0.633770 +3 0.009744 0.009863 0.009495 0.000507 0.002004 0.721652 1.268726 + + time_exactmc +0 0.535619 +1 2.113617 +2 5.688960 +3 13.508999 ++