Performance measurement for option portfolios in a stochastic volatility framework

Abstract

Measuring the performance of stock portfolios that include options is challenging due to options' nonlinearity in the underlying, their exposure to volatility risk, and their time decay. Our contribution to the literature is twofold: First, we provide a theoretically rigorous derivation of the time-variable factor loadings in a two-factor model under stochastic volatility according to [Heston, S.L., A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud., 1993, 6, 327–343.]. Within this setting, the portfolio returns are explained by the market and an additional option factor, i.e. a portfolio of standard options exposed to volatility risk. We show that (i) any option factor is suitable to perfectly explain the portfolio behavior if simple returns are considered in instantaneous time and that (ii) the option factor's loading equals the fraction of the volatility elasticities of the portfolio and of the option factor while the option factor's underlying elasticity enters the factor loading of the underlying. Second, we analyze the behavior of option factors in practical applications, where time is discrete and factor loadings are estimated in a single regression over a certain time horizon. We show how the bias of (Jensen) alpha and its sign depend on the skewness of market returns. For several option factors from the literature, we conduct a simulation study to analyze their suitability to reduce this bias. As the results are disappointing, we propose a two-step procedure for choosing an adequate factor.

Keywords:

• Performance measurement
• Option portfolios
• Stochastic volatility
• Heston model
• Option-based factors

JEL Classifications:

• C13
• C60
• C61
• C62
• G10
• G12

Acknowledgments

We thank the participants of the 2018 Annual Meetings of the German Academic Association for Business Research (VHB), of the Financial Management Association (FMA), and of the Southern Finance Association (SFA) for valuable comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Heston (1993) assumes $\lambda (M,v,t)$ to be a linear function in the variance.

2 Generally, the two factors are not orthogonal, so both factors will capture the aggregated portfolio sensitivity to the market.

3 In the proof of Proposition 3.3, we insert the portfolio's time decay, $\frac{\mathrm{\partial }{P}_t}{\mathrm{\partial }t}$, from the Heston PDE (see Equation (A39(A39) $\begin{array}{rl}& E[\frac{\mathrm{\Delta }{P}_t}{P_t}-\widehat{\frac{\mathrm{\Delta }{P}_t}{P_t}}]\\ & =E[-\frac{1}{2}{v}_t\frac{M_t^2}{P_t}\frac{{\mathrm{\partial }}^2{P}_t}{\mathrm{\partial }{M}_t^2}\mathrm{\Delta }t-\rho \eta v_t\frac{M_t}{P_t}\frac{{\mathrm{\partial }}^2{P}_t}{\mathrm{\partial }{M}_t\mathrm{\partial }{v}_t}\mathrm{\Delta }t-\frac{1}{2}{\eta }^2\frac{v_t}{P_t}\\ & \times \frac{{\mathrm{\partial }}^2{P}_t}{\mathrm{\partial }{v}_t^2}\mathrm{\Delta }t-({\mu }_{v,t}-\lambda (M,v,t\left)\right)\frac{\mathrm{\partial }{P}_t}{\mathrm{\partial }{v}_t}\frac{1}{P_t}\mathrm{\Delta }t-\frac{\frac{\mathrm{\partial }{P}_t}{\mathrm{\partial }{v}_t}}{\frac{\mathrm{\partial }{F}_t}{\mathrm{\partial }{v}_t}}\frac{\mathrm{\partial }{F}_t}{\mathrm{\partial }{M}_t}\frac{M_t}{P_t}r\mathrm{\Delta }t]\\ & +E[\frac{1}{2}\frac{{\mathrm{\partial }}^2{P}_t}{\mathrm{\partial }{M}_t^2}\frac{\mathrm{\Delta }{M}_t^2}{P_t}+\frac{\mathrm{\partial }{P}_t}{\mathrm{\partial }{v}_t}\frac{\mathrm{\Delta }{v}_t}{P_t}+\frac{1}{2}\frac{{\mathrm{\partial }}^2{P}_t}{\mathrm{\partial }{v}_t^2}\frac{\mathrm{\Delta }{v}_t^2}{P_t}\\ & +\frac{{\mathrm{\partial }}^2{P}_t}{\mathrm{\partial }{M}_t\mathrm{\partial }{v}_t}\frac{\mathrm{\Delta }{M}_t\mathrm{\Delta }{v}_t}{P_t}+\frac{\frac{\mathrm{\partial }{P}_t}{\mathrm{\partial }{v}_t}}{\frac{\mathrm{\partial }{F}_t}{\mathrm{\partial }{v}_t}}\frac{\mathrm{\partial }{F}_t}{\mathrm{\partial }{M}_t}\frac{\mathrm{\Delta }{M}_t}{P_t}]+E\left[\frac{\frac{\mathrm{\partial }{P}_t}{\mathrm{\partial }{v}_t}}{\frac{\mathrm{\partial }{F}_t}{\mathrm{\partial }{v}_t}}\frac{F_t}{P_t}r\mathrm{\Delta }t\right]\\ & -E\left[\frac{\frac{\mathrm{\partial }{P}_t}{\mathrm{\partial }{v}_t}}{\frac{\mathrm{\partial }{F}_t}{\mathrm{\partial }{v}_t}}\frac{\mathrm{\Delta }{F}_t}{P_t}\right]+h.o.t.\\ & =\frac{1}{2}[\frac{{\mathrm{\partial }}^2{P}_t}{\mathrm{\partial }{M}_t^2}-\frac{\frac{\mathrm{\partial }{P}_t}{\mathrm{\partial }{v}_t}}{\frac{\mathrm{\partial }{F}_t}{\mathrm{\partial }{v}_t}}\frac{{\mathrm{\partial }}^2{F}_t}{\mathrm{\partial }{M}_t^2}]\frac{M_t^2}{P_t}{r}^2\mathrm{\Delta }{t}^2\\ & +\frac{1}{2}[\frac{{\mathrm{\partial }}^2{P}_t}{\mathrm{\partial }{v}_t^2}-\frac{\frac{\mathrm{\partial }{P}_t}{\mathrm{\partial }{v}_t}}{\frac{\mathrm{\partial }{F}_t}{\mathrm{\partial }{v}_t}}\frac{{\mathrm{\partial }}^2{F}_t}{\mathrm{\partial }{v}_t^2}]({\mu }_{v,t}-\lambda (M,v,t){)}^2\frac{1}{P_t}\mathrm{\Delta }{t}^2\\ & +[\frac{{\mathrm{\partial }}^2{P}_t}{\mathrm{\partial }{M}_t\mathrm{\partial }{v}_t}-\frac{\frac{\mathrm{\partial }{P}_t}{\mathrm{\partial }{v}_t}}{\frac{\mathrm{\partial }{F}_t}{\mathrm{\partial }{v}_t}}\frac{{\mathrm{\partial }}^2{F}_t}{\mathrm{\partial }{M}_t\mathrm{\partial }{v}_t}]\frac{M_t}{P_t}r({\mu }_{v,t}-\lambda (M,v,t\left)\right)\mathrm{\Delta }{t}^2\\ & +h.o.t.,\end{array}$(A39) ) and Equation (A35(A35) $\begin{array}{rl}\frac{\mathrm{\Delta }{P}_t}{P_t}& =[r-\frac{1}{2}{v}_t\frac{M_t^2}{P_t}\frac{{\mathrm{\partial }}^2{P}_t}{\mathrm{\partial }{M}_t^2}-\rho \eta v_t\frac{M_t}{P_t}\frac{{\mathrm{\partial }}^2{P}_t}{\mathrm{\partial }{M}_t\mathrm{\partial }{v}_t}-\frac{1}{2}{\eta }^2\frac{v_t}{P_t}\frac{{\mathrm{\partial }}^2{P}_t}{\mathrm{\partial }{v}_t^2}\\ & -r\frac{M_t}{P_t}\frac{\mathrm{\partial }{P}_t}{\mathrm{\partial }{M}_t}-({\mu }_{v,t}-\lambda (M,v,t\left)\right)\frac{\mathrm{\partial }{P}_t}{\mathrm{\partial }{v}_t}\frac{1}{P_t}]\mathrm{\Delta }t\\ & +\frac{\mathrm{\partial }{P}_t}{\mathrm{\partial }{M}_t}\frac{\mathrm{\Delta }{M}_t}{P_t}+\frac{1}{2}\frac{{\mathrm{\partial }}^2{P}_t}{\mathrm{\partial }{M}_t^2}\frac{\mathrm{\Delta }{M}_t^2}{P_t}+\frac{\mathrm{\partial }{P}_t}{\mathrm{\partial }{v}_t}\frac{\mathrm{\Delta }{v}_t}{P_t}\\ & +\frac{1}{2}\frac{{\mathrm{\partial }}^2{P}_t}{\mathrm{\partial }{v}_t^2}\frac{\mathrm{\Delta }{v}_t^2}{P_t}+\frac{{\mathrm{\partial }}^2{P}_t}{\mathrm{\partial }{M}_t\mathrm{\partial }{v}_t}\frac{\mathrm{\Delta }{M}_t\mathrm{\Delta }{v}_t}{P_t}+h.o.t.\end{array}$(A35) ) in the appendix). Thus, the nonlinearity and the time decay can be considered two sides of a coin.

4 In this setup, the representative investor has power utility. The skewness preference is related to the positive third derivative of power utility functions.

5 Christoffersen et al. (2013) showed that the pricing kernel found by Rubinstein (1976) can be extended to the Heston model.

6 For details of the implementation, see, for example, Gatheral (2006).

7 The Heston model was calibrated to market prices of European call and put options on the German market index DAX. The calibration was based on all available options traded at the EUREX with maturities from one week to five years. On average, 933 option prices were available, with a minimum of 710. The calibration was repeated each trading day between May 2014 and December 2015, using daily settlement prices of options with maturities up to 2 years traded on the EUREX. In order to determine a global minimum of the root mean squared error (RMSE) between theoretical and observed option prices, the differential evolution algorithm suggested by Storn and Price (1997) was applied. From all daily calibration results, we chose the parameter set which belong to the median volatility of volatility. We thank David Shkel for providing this data.

8 Actually, the calibrated instantaneous variance was $v_0=0.046$. To obtain the Black–Scholes world in the limit when the volatility of volatility approaches zero, we set $v_0=\kappa$.

9 We also tested rolling window regressions, but as they led to worse results, they are not reported here.

10 As we are operating in a simulation setting with no other risk factors except the market return and its variance, we do not take further equity factors as proposed by Fama and French (1993), Carhart (1997), or Fama and French (2015) for example, into account.

11 The valuation in t of a zero-strike variance swap maturing in T in the Heston model is straightforward: the value is given by $\exp (-r(T-t\left)\right)\left(\frac{t}{T}{V}_d\right(0,t)+\frac{T-t}{T}K(t,T{)}^{\ast })$, where $K(t,T{)}^{\ast }=\theta +\frac{{\nu }_t-\theta }{\kappa (T-t)}(1-\exp (-\kappa (T-t\left)\right))$ is the fair variance strike in t and $V_d(0,t_n)=\frac{1}{(n-1)\mathrm{\Delta }t}\sum _{i=0}^{n-1}(\ln (\frac{M_{t_{i+1}}}{M_{t_i}}){)}^2$ is the realized variance until $t_n=t$ based on daily log returns of the market, see Broadie and Jain (2008) and Jacquier and Slaoui (2010).

12 The mean estimated annualized alpha is calculated by $\hat{\alpha }=\frac{1}{10000}\sum _{k=1}^{10000}{\hat{\alpha }}_k$, where ${\hat{\alpha }}_k$ is the estimated annualized alpha obtained from simulation run k. The average RMSE of the estimated alphas is determined by $\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{\frac{1}{10000}\sum _{k=1}^{10000}{\hat{\alpha }}_k^2}$.

13 These figures are calculated as the average absolute values of all option positions in the portfolio with respect to the total portfolio value. For example, for Portfolio 1, we calculate this fraction as ${\mathrm{C}\mathrm{a}\mathrm{l}\mathrm{l}}_t/(M_t-{\mathrm{C}\mathrm{a}\mathrm{l}\mathrm{l}}_t)$, where ${\mathrm{C}\mathrm{a}\mathrm{l}\mathrm{l}}_t$ is the call value at time t and $M_t$ is the market (index) value. This fraction is averaged over all points in time and simulation runs.

14 We choose the structure of the CPF as the basis for the individual factor (ICPF), as the CPF is weakly correlated to the market factor and includes both types of standard options. The individualization of the ICPF only refers to a varying time to maturity, while the moneyness of the included options stays fixed. This is justified by the fact that the moneyness of the included options strongly influences the correlation of the ICPF and the market factor. Only when the options are set at the money, the correlation can be kept low. Further, our observations exhibit a much larger influence of the time to maturity on the quality of the ICPF than the moneyness.

15 In the case of non-augmented option holdings (not reported here), this observation also holds true for Portfolio 2.

16 The actual average option holdings slightly differ from 25.0%, since we fix this value at each rebalancing day, but it can vary over the following six months.