Empirical Performance of Alternative Futures Covered-Call. Strategies under Stochastic Volatility

Transcription

1 Empirical Performance of Alternative Futures Covered-Call Strategies under Stochastic Volatility CHUNG-GEE LIN, MAX CHEN and CHANG-CHIEH HSIEH 1 This study examines the performance of conventional and dynamic covered-call strategies under constant and stochastic volatility options pricing models in Taiwan. In accord with prior literatures, the covered-call strategy may roughly boost portfolio return in some specific moneyness. The conventional covered-call strategy has an increment of 0.11% monthly return than pure futures buy-and-hold strategy under specific moneyness. The dynamic strategy adjusts the moneyness according to different exercise probabilities under constant volatility and stochastic volatility options pricing models. Finally, this study confirmed that the dynamic strategy under stochastic volatility has obvious advantages over the constant volatility and conventional strategies. Key words:covered-call, dynamic strategy, moneyness, stochastic volatility. 1. Introduction Covered-call (buy-write) option trading strategy, which is the portfolio combined by one unit of long underlying asset and writing one call option, has been studied in previous literatures and assessing its performance. This trading strategy is the simplest concept extended from the Capital Asset Pricing Model (CAPM) uses to hold negative correlation between asset and 1 CHUNG-GEE LIN is a Professor of Finance in the Department of Financial Engineering and Actuarial Mathematics, Soochow University, Taiwan. MAX CHEN is an Assistant Professor in the Department of Finance, Ming Chuan University, Taiwan. CHANG-CHIEH HSIEH is a Ph.D. candidate in the Department of Economics, Soochow University, Taiwan. Corresponding Author: Chung-Gee Lin. Address: 56, Kuei-Yang Street, Section 1, Taipei 100, Taiwan. Tel: ext Fax: cglin@scu.edu.tw 1

2 derivative to improve returns and reduce risks. The trading strategy constructors receive a premium when they sell a call option, the premium can reduce the cost of a single long position of underlying asset. The premium amount depends on the extent of the OTM (out-of-money) conditions, as the option strike price is more close to current underlying asset, the trading strategy constructors will receive more premium. Another important factor for the value of option premium is the underlying asset s volatility, which represents the fluctuation degree of the underlying asset. Therefore, choosing the appropriate moneyness with proper volatility evolution expectation of underlying asset is a key to the performance of building a covered-call portfolio. Exchange traded options market was first built at Chicago Board Options Exchange (CBOE) in After this, the options market began to flourish. Taiwan Futures Exchange (TAIFEX) was established in After years of hard work, the exchange is one of the fastest growing options market in the world (refer to Figure 1). TAIFEX option s volume is ranked the sixth greatest market around the world in 2010 according to the report of the World Federation of Exchanges (WFE). [Figure 1] Che and Fung (2011) used the conventional buy-write (covered-call) strategy and a dynamic buy-write strategy to test the performance on Hang Seng Index (HSI) in Hong Kong. Che and Fung (2011) adopted HSI futures to substitute for HSI in order to reduce the impact of transactional cost and execution problems. They found that both strategies outperform the naked futures position. Although the dynamic strategy uses various risk-adjusted measures, which usually has lower returns than the conventional fixed strike strategy. However, the dynamic strategy outperforms the fixed strategy when the market is moderately volatile and in a sharply rising market. 2

3 Figelman (2008) introduces a simple theoretical framework which allows for the decomposition of the observed historical performance of covered-call strategy into three market components: risk-free rate, equity risk premium, and implied-realized volatility spread. Figelman (2008) pointed out that the covered-call strategy is highly correlated with the equity market index (S&P 500), especially on the bull market. Hill et al. (2006), Feldman and Dhruv (2004) and Whaley (2002) also investigated the covered-call strategies on the S&P500. They did not obtain strong evidences that covered-call strategies provided better performance than naked futures position. Nevertheless, they concluded that covered-call strategies can effectively reduce risks (standard derivation). This study extends the works of Che and Fund (2011) and Figelman (2008) to investigate the performance of covered-call strategy in TAIFEX. We divided the market into different situations: rising (bullish) or falling (bearish); sharply or moderately (refer to Figure 2), and examines the performance of conventional and dynamic covered-call strategies under constant and stochastic volatilities environment in Taiwan. [Figure 2] The rest of this paper is organized as follows. In Section 2, the covered-call option trading strategies, the constant volatility and stochastic volatility futures option pricing model were introduced. Section 3 describes the data and defines the four different market situations. The performance of covered-call option trading strategies under conditions will be analyzed. The last section concludes. 2. The model Covered-call trading strategy, in this paper, is built as to buy index futures and simultaneous short a European call on futures. Therefore, the volatility of the underlying futures and option 3

4 strike price will affect the performance of the covered-call trading strategy. When the volatility of the underlying futures increases, the income from short sell the call options will increase. As the option is close to ATM (at the money) condition, the income raises. Building the covered-call strategy involves a position of short sell of futures call options. It is therefore involving the option pricing models for the moneyness probabilities estimation. Conventional covered-call strategy was build under constant volatility assumption, the short sell of a call option position can be constructed by using Black (1976) model, which is shown as equation 1, c FN( d ) XN( d ) (1) whrer c is the valuation of call option on futures with spot futures price F, 0 d F 0 X /2 /, d F 0 X /2 /, X is the exercise price, T is the time to maturity in years, N (.) is the cumulative probability of standard Normal distribution. 2 In a risk neutral world, under the stochastic volatility environment, as shown in equations 2 and 3, ds rs dt v S dz (2) t t t t 1 2, dv k v dt v dz (3) t t t t where S t is the stock price at time t, r is the risk-free interst rate, v t is the volatility of stock price, θ is the long-run mean of v t, is a mean reversion speed of v t, σ is the volatility of v t 2 Put option can be derived form put call parity. 4

5 Heston (1993) has derived a closed form solution for European call option on stock under stochastic volatility, as in equations 4, c S P Xe P (4) H rt where iuln X u 1 1 exp Re j Pj du, j 1 or 2, 2 iu 0 Re. is the real part of complex number. u is the characteristic function. j If the underlying asset is futures, equations 4 can be modified as equation 5, c FP XP (5) ' H where P can be treated as the probability that call option will be exercised (in the money) under stochastic volatility, which will be used for constructing the short sell of a call option position under stochastic volatility environment. The return from a covered-call is derived as below, Return F S F 0 F 0 c X F 0 F S X, 0 F 0 where F S is the last settlement price, c X is the call option price with exercise price X. In this study, the selected strike price is divided into (1) the fixed ratio model and (2) the dynamic adjustment model. The rewards of fixed ratio models consist of a traditional fixed strike price and futures price in the beginning of the period, and which is studied for the degree of difference of 1% to 6% OTM during four specific ranges. 5

6 In Figure 3, we can see a clear change of the price of the option premium. Dynamic adjustment model exhibit a fixed construct dynamic parts of the compliance probability, this study in addition to the construction of Black's (1976) futures option pricing model for the fixed compliance probability and volatility back stepping specific strike price. We assumed seven distinct probabilities of compliance cases. The result of different moneyness trend was shown in Figure 4. This study also extends the Heston stochastic volatility model (1993) together with the optimization techniques for deriving the in-the-money probability (P2). The moneyness trends was shown in Figure 5. [Figures 3, 4 and 5] The Taiwan index options (TXO) is used as the short position in call options. In this study, to sell the option contracts in the past month and to hold to maturity, as the basis of the performance evaluation. According to the model of Che and Fung (2011), to reduce the problems of dividend, hedging, transaction costs and non-synchronous trading, we use Taiwan s stock market futures (TX) to replace the Taiwan Stock Index. 3. Empirical Tests 3.1 Empirical tests for all conditions According to the models, the fixed implementation price strategy and dynamic adjustment strategy in compliance probability, risk values and descriptive statistics was shown in Tables 1 to 3. On Feb-2004 to Jan-2012 period, the average monthly return for simply buy-and-hold strategy is 0.32%. 6

7 In the covered-call strategies, as the moneyness is more deeply OTM, the short position of call will receive fewer premiums. Generally, the short position of 3 to 6% OTM call options will increase the total monthly return. The risk will be smaller than the naked future s position. Finally, whether by the Sharpe Ratio or Sortino Ratio as the performance indicators, the short position of 6% OTM call option can get the best performance, it and can significantly enhance returns and reduce the risk. [Table 1] Table 2 and Table 3 show the results of dynamic adjustment covered-call under Black (1976) model and under Heston (1993) model, respectively. Both implied volatilities are calculated by using all day-end information except the volume is under 100 contracts. In Heston model, we use the method of exhaustion to obtain the optimal parameters in given lower bound [0.01, 0.01, -1, 0.01, and 0.01] and upper bound [0.6, 0.6, 1, 5, and 0.6]. (Represent the volatility of variance, current variance, rho, kappa, and the long-run mean of the variance respectively.) In Black (1976) model, the rewards in all seven compliance probabilities of 17 to 49% are less than the naked future s position. However, there is still an advantage to choice the exercised probability around 30% in order to reduce the fluctuation from of return. In point of Heston (1993) model, which is significantly improve the performance of the remuneration. Apparently, The dynamic covered-call strategies under Black (1976) perform less than the naked future s position. The Heston (1993) models beat easily the Black (1976) model in most cases. [Tables 2 and 3, and Figure 6] 3.2 Empirical tests for different market conditions 7

8 Financial products in different market conditions tend to behave differently. This study refers to the setting of Che and Fund (2011), dividing Taiwan's market into four conditions (sharply rising, sharply falling, moderately rising and moderately falling). The results are shown in Tables 4 to 6. In general, in the rapid decline market, selling close to ATM options can receive more premiums. In sharply falling market, the dynamic adjustment covered-call (both under Black and Heston models) can effectively reduce losses. In moderately falling market, the dynamic covered-call strategy is better than the conventional strategy under Black model. The performance of Heston model is worse than Black model. However, in the character of sensitive market in Taiwan, there are only four months in moderately falling situation in our studied periods. Even so, Heston model is still better than the naked future s position. In sharply rising market, all the covered-call strategies are worse than the naked futures, however, if we evaluate the performance under Sharpe ratio, the covered-call strategies are better than the naked futures, as they can effectively reduce the risks. We can still point out that Heston model is quite best in comparison with Black model. [Tables 4, 5, 6, 7 and Figures 7, 8, 9 and 10] 4. Conclusions This study examines the performance of conventional and dynamic covered-call strategies under constant and stochastic volatility environment in Taiwan. In overall periods, the conventional covered-call strategy under fixed ratio moneyness has a slight increment of 1 basis point on monthly return than pure futures buy-and-hold strategy. However, the dynamic strategy under Black model is worse than naked future s position and fixed ratio strategy. In the alternative Heston model, it can obviously improve the performance of return in our study. 8

9 Finally, this study points out that the dynamic strategy under Heston model has the obvious advantage than Black model. References 1. Bakshi, Gurdip, Charles Cao, and Zhiwu Chen, 1997, Empirical performance of alternative option pricing models, The Journal of Finance 52, Bates, David S., 1991, The crash of '87: Was it expected? The evidence from options markets, The Journal of Finance 46, Blair, Bevan J., Ser Huang Poon, and Stephen J. Taylor, 2001, Forecasting S&P 100 volatility: The incremental information content of implied volatilities and high frequency index returns, Journal of Econometrics 105, Broadie, Mark, Chernov Mikhail, and Johannes Michael, 2009, Understanding index option returns, Review of Financial Studies 22, Chaput, J. Scott, and Louis H. Ederington, 2005, Volatility trade design, Journal of Futures Markets 25, Che, Sanry Y. S., and Joseph K. W. Fung, 2011, The performance of alternative futures buy write strategies, Journal of Futures Markets 31, Christensen, B. J., and N. R. Prabhala, 1998, The relation between implied and realized volatility, Journal of Financial Economics 50, Coval, Joshua D., and Tyler Shumway, 2001, Expected option returns, The Journal of Finance 56, David S, Bates, 2000, Post '87 crash fears in the S&P 500 futures option market, Journal of Econometrics 94, Engle, Robert F., and Andrew J. Patton, 2001, What good is a volatility model? SSRN elibrary. 9

10 11. Figelman, Ilya, 2008, Expected return and risk of covered call strategies, The Journal of Portfolio Management 34, Figelman, Ilya, 2009, Effect of non normality dynamics on the expected return of options, The Journal of Portfolio Management 35, Fischer, Black, 1976, The pricing of commodity contracts, Journal of Financial Economics 3, Harvey, Campbell R., and Robert E. Whaley, 1992, Market volatility prediction and the efficiency of the s & p 100 index option market, Journal of Financial Economics 31, Jones, Christopher S., 2006, A nonlinear factor analysis of S&P 500 index option returns, The Journal of Finance 61, Koopman, Siem Jan, Borus Jungbacker, and Eugenie Hol, 2005, Forecasting daily variability of the S&P 100 stock index using historical, realised and implied volatility measurements, Journal of Empirical Finance 12, Mugwagwa, Tafadzwa, Vikash Ramiah, and Tony Naughton, 2010, The efficiency of the buy write strategy: Evidence from Australia, SSRN elibrary. 10

11 Figure 1 TAIFEX option s historical volumes during 2001 to According to the record from the World Federation of Exchanges (WFE). Volume Trades (Number of Contracts) Transaction of Billions Year 11

12 Figure 2 TAIFEX Future for the period Feb-2004 to Nov Periods 1 4 and 8 represent sharply falling. Periods 3 5 and 7 represent sharply rising. Period 2 represents moderately rising, and the rest represents moderately falling

13 Figure 3 Call Premium (as a percentage of futures) for the period Feb-2004 to Nov

14 Figure 4 Moneyness of the dynamic portfolios and implied volatility under the Black s model for the period Feb-2004 to Nov

15 Figure 5 Moneyness of the dynamic portfolios and implied volatility under the Modified Heston model for the period Feb-2004 to Nov

16 Figure 6 The tradeoff between Return and SD for Pure-Future, Conventional Covered-Call and Dynamic Covered-Call Strategies 0.45% 0.40% 0.35% Return 0.30% 0.25% 0.20% 0.15% 0.10% 4.00% 4.50% 5.00% 5.50% 6.00% 6.50% 7.00% SD Future Fixed Strike Dynamic (BS) Dynamic (Heston) 16

17 Figure 7 The tradeoff between Return and Semi-SD under Sharply Falling Market Condition 1.00% 1.50% Sharply Falling Return 2.00% 2.50% 3.00% 5.50% 6.00% 6.50% 7.00% 7.50% Semi SD Fixed Strike Dynamic (BS) Dynamic (Heston) 17

18 Figure 8 The tradeoff between Return and Semi-SD under Moderately Falling Market Condition 0.20% 0.40% Moderately Falling Return 0.60% 0.80% 1.00% 1.20% 1.40% 3.00% 3.20% 3.40% 3.60% 3.80% 4.00% 4.20% 4.40% Semi SD Fixed Strike Dynamic (BS) Dynamic (Heston) 18

19 Figure 9 The tradeoff between Return and Semi-SD under Sharply Rising Market Condition 4.00% 3.50% Sharply Rising Return 3.00% 2.50% 2.00% 1.50% 1.40% 1.50% 1.60% 1.70% 1.80% 1.90% 2.00% 2.10% Semi SD Fixed Strike Dynamic (BS) Dynamic (Heston) 19

20 Figure 10 The tradeoff between Return and Semi-SD under Moderately Rising Market Condition 0.80% 0.70% Moderately Rising Return 0.60% 0.50% 0.40% 0.30% 2.20% 2.30% 2.40% 2.50% 2.60% 2.70% 2.80% 2.90% 3.00% Semi SD Fixed Strike Dynamic (BS) Dynamic (Heston) 20

21 Table 1 Overall monthly performance of different fixed moneyness for the period Feb-2004 to Jan-2012 Moneyness 1% 2% 3% 4% 5% 6% Pure Future OTM OTM OTM OTM OTM OTM Covered-Call Return 0.32% 0.19% 0.25% 0.34% 0.37% 0.40% 0.43% Covered-Call SD 6.75% 4.62% 4.89% 5.20% 5.48% 5.72% 5.91% Call Premium Return 2.09% 1.71% 1.35% 1.09% 0.86% 0.69% Coefficient of Variation, CV Median 0.98% 2.09% 2.38% 2.05% 2.05% 1.66% 1.39% Max 15.65% 5.98% 5.98% 7.30% 7.34% 8.77% 8.91% Min % % % % % % % Semi-SD 4.53% 5.13% 5.77% 6.52% 7.30% 8.12% 8.95% Sharpe Ratio 4.78% 4.21% 5.13% 6.53% 6.69% 7.01% 7.30% Sortino Ratio 7.12% 3.80% 4.35% 5.21% 5.02% 4.94% 4.82% 21

22 Table 2 Overall monthly performance of different exercise probabilities under the Black s model for the period Feb-2004 to Jan-2012 N(d2) 49% 42% 36% 30% 25% 20% 17% Covered-Call Return 0.13% 0.16% 0.23% 0.30% 0.24% 0.27% 0.25% Covered-Call SD 4.23% 4.68% 5.06% 5.46% 5.68% 5.92% 6.06% Call Premium Return 2.65% 2.05% 1.64% 1.23% 0.97% 0.71% 0.56% Coefficient of Variation, CV Median 1.70% 2.06% 2.05% 2.05% 1.95% 1.95% 1.55% Max 5.25% 5.98% 8.77% 10.39% 11.31% 10.49% 10.49% Min % % % % % % % Semi-SD 8.14% 7.89% 7.51% 6.96% 6.18% 5.18% 3.73% Sharpe Ratio 3.07% 3.47% 4.58% 5.52% 4.20% 4.52% 4.07% Sortino Ratio 1.59% 2.06% 3.09% 4.33% 3.86% 5.16% 6.60% 22

23 Table 3 Overall monthly performance of different exercise probabilities under the Heston model for the period Feb-2004 to Jan-2012 N(d2) 49% 42% 36% 30% 25% 20% 17% Covered-Call Return 0.26% 0.31% 0.39% 0.35% 0.39% 0.44% 0.40% Covered-Call SD 4.61% 4.94% 5.31% 5.57% 5.85% 6.10% 6.18% Call Premium Return 2.20% 1.73% 1.38% 1.09% 0.82% 0.63% 0.53% Coefficient of Variation, CV Median 2.11% 2.34% 2.05% 2.05% 1.66% 1.55% 1.39% Max 5.98% 7.30% 8.70% 9.79% 10.63% 12.35% 12.15% Min % % % % % % % Semi-SD 8.37% 8.07% 7.64% 7.08% 6.31% 5.28% 3.78% Sharpe Ratio 5.58% 6.18% 7.27% 6.36% 6.62% 7.13% 6.50% Sortino Ratio 3.07% 3.78% 5.05% 5.01% 6.14% 8.25% 10.62% 23

24 Table 4 Overall monthly performance of fixed strike strategy under different market conditions Sharply Falling 1% 2% 3% 4% 5% 6% N=33 Pure Future OTM OTM OTM OTM OTM OTM Mean -3.39% -1.86% -2.07% -2.14% -2.34% -2.45% -2.55% SD 7.54% 6.26% 6.44% 6.73% 6.90% 7.10% 7.20% Coefficient of Variation,CV Median -2.75% 0.11% -0.54% -1.11% -1.11% -1.51% -1.54% Max 11.16% 5.98% 5.98% 7.30% 7.34% 8.77% 8.91% Min % % % % % % % Semi-SD 7.53% 5.99% 6.20% 6.76% 6.58% 6.73% 6.84% Sharpe Ratio % % % % % % % Sortino Ratio % % % % % % % Moderately Falling 1% 2% 3% 4% 5% 6% N=5 Pure Future OTM OTM OTM OTM OTM OTM Mean -1.65% -0.93% -0.77% -0.89% -1.03% -1.18% -1.32% SD 4.83% 4.11% 4.53% 4.81% 4.85% 4.95% 4.92% Coefficient of Variation,CV Midian -0.74% 0.94% 0.94% 0.49% 0.15% -0.17% -0.17% Max 3.76% 2.56% 3.14% 4.08% 4.31% 4.31% 4.11% Min -7.99% -6.62% -7.06% -7.37% -7.37% -7.60% -7.74% Semi-SD 4.24% 3.44% 3.61% 3.82% 3.88% 3.88% 4.10% Sharpe Ratio % % % % % % % Sortino Ratio % % % % % % % Moderately Rising 1% 2% 3% 4% 5% 6% N=23 Pure Future OTM OTM OTM OTM OTM OTM Mean 0.75% 0.46% 0.54% 0.57% 0.63% 0.74% 0.75% SD 4.97% 3.10% 3.43% 3.70% 4.01% 4.30% 4.46% Coefficient of Variation,CV Midian 0.32% 1.85% 1.62% 1.32% 1.07% 0.78% 0.50% Max 8.41% 3.12% 3.76% 4.41% 4.90% 6.02% 6.31% Min % -9.31% -9.84% -9.84% % % % Semi-SD 3.15% 2.45% 2.64% 2.72% 2.84% 2.94% 2.96% Sharpe Ratio 15.07% 14.88% 15.68% 15.49% 15.75% 17.14% 16.89% Sortino Ratio 23.79% 18.82% 20.39% 21.09% 22.22% 25.12% 25.48% Sharply Rising 1% 2% 3% 4% 5% 6% N=34 Pure Future OTM OTM OTM OTM OTM OTM Mean 3.93% 2.18% 2.46% 2.77% 3.02% 3.17% 3.36% SD 5.24% 2.43% 2.64% 2.95% 3.28% 3.53% 3.83% Coefficient of Variation,CV Midian 3.24% 2.49% 3.12% 3.44% 3.72% 3.60% 3.48% Max 15.65% 4.54% 5.39% 5.87% 6.48% 7.51% 7.64% Min -9.60% -8.30% -8.63% -8.92% -9.15% -9.31% -9.31% Semi-SD 2.10% 1.59% 1.64% 1.74% 1.82% 1.85% 1.89% Sharpe Ratio 75.03% 89.37% 93.26% 93.83% 92.00% 89.95% 87.94% Sortino Ratio % % % % % % % 24

25 Table 5 Overall monthly performance of dynamic (Black) strike strategy under different market conditions Sharply Falling 49% 42% 36% 30% 25% 20% 0.17 N=33 Mean -1.62% -1.99% -2.12% -2.33% -2.56% -2.81% -2.90% SD 5.89% 6.34% 6.82% 7.19% 7.34% 7.41% 7.52% Coefficient of Variation,CV Median 0.88% -0.04% -1.11% -1.36% -1.67% -2.24% -2.45% Max 5.25% 5.98% 8.77% 10.39% 11.31% 10.49% 10.49% Min % % % % % % % Semi-SD 5.64% 6.10% 6.44% 6.74% 6.94% 7.14% 7.24% Sharpe Ratio % % % % % % % Sortino Ratio % % % % % % % Moderately Falling N=5 49% 42% 36% 30% 25% 20% 0.17 Mean -0.84% -0.67% -0.49% -0.79% -0.85% -1.10% -1.20% SD 3.64% 4.19% 4.60% 4.77% 4.75% 4.76% 4.71% Coefficient of Variation,CV Midian 1.56% 0.94% 0.94% 0.49% 0.15% 0.15% -0.17% Max 1.93% 3.14% 4.08% 4.31% 4.31% 3.98% 3.98% Min -6.03% -6.62% -6.62% -7.06% -7.06% -7.37% -7.37% Semi-SD 3.36% 3.81% 4.14% 4.34% 4.34% 4.40% 4.38% Sharpe Ratio % % % % % % % Sortino Ratio % % % % % % % Moderately Rising N=23 49% 42% 36% 30% 25% 20% 0.17 Mean 0.46% 0.41% 0.46% 0.55% 0.49% 0.61% 0.56% SD 2.84% 3.04% 3.25% 3.52% 3.72% 3.98% 4.06% Coefficient of Variation,CV Midian 1.60% 1.61% 1.32% 1.62% 1.62% 1.07% 1.07% Max 2.72% 3.12% 3.74% 4.41% 4.90% 5.83% 5.83% Min -8.69% -9.31% -9.84% -9.84% % % % Semi-SD 2.32% 2.45% 2.54% 2.65% 2.75% 2.85% 2.87% Sharpe Ratio 16.29% 13.49% 14.11% 15.72% 13.30% 15.33% 13.85% Sortino Ratio 20.01% 16.77% 18.08% 20.83% 18.00% 21.46% 19.57% Sharply Rising N=34 49% 42% 36% 30% 25% 20% 0.17 Mean 1.74% 2.20% 2.47% 2.85% 2.94% 3.22% 3.30% SD 2.11% 2.45% 2.73% 3.14% 3.45% 3.78% 4.02% Coefficient of Variation,CV Midian 1.98% 2.44% 2.72% 3.18% 3.28% 3.46% 3.46% Max 3.70% 5.39% 5.87% 7.49% 7.51% 8.95% 10.03% Min -7.92% -8.30% -8.30% -8.63% -8.92% -8.92% -9.15% Semi-SD 1.47% 1.58% 1.64% 1.77% 1.91% 1.95% 2.02% Sharpe Ratio 82.78% 89.78% 90.42% 90.56% 85.25% 85.20% 82.02% Sortino Ratio % % % % % % % 25

26 Table 6 Overall monthly performance of dynamic (Heston) strike strategy under different market conditions Sharply Falling 49% 42% 36% 30% 25% 20% 0.17 N=33 Mean -1.84% -2.05% -2.20% -2.36% -2.53% -2.67% -2.78% Coefficient of Variation,CV Median 0.11% -0.54% -1.11% -1.36% -1.96% -2.24% -2.26% Max 5.98% 7.30% 8.70% 9.79% 10.63% 12.35% 12.15% Min % % % % % % % Semi-SD 5.97% 6.23% 6.51% 6.73% 6.95% 7.09% 7.17% Sharpe Ratio % % % % % % % Sortino Ratio % % % % % % % Moderately Falling N=5 49% 42% 36% 30% 25% 20% 0.17 Mean -0.81% -0.68% -0.94% -0.90% -1.04% -1.10% -1.21% SD 4.19% 4.38% 4.56% 4.69% 4.89% 4.83% 4.80% Midian 1.56% 0.94% 0.49% 0.15% -0.17% -0.17% -0.34% Max 2.56% 3.14% 3.14% 4.08% 4.31% 4.31% 4.11% Min -6.62% -6.62% -7.06% -7.06% -7.37% -7.37% -7.37% Semi-SD 3.44% 3.44% 3.70% 3.70% 3.88% 3.88% 3.93% Sharpe Ratio % % % % % % % Sortino Ratio % % % % % % % Moderately Rising N=23 49% 42% 36% 30% 25% 20% 0.17 Mean 0.53% 0.50% 0.65% 0.52% 0.66% 0.63% 0.66% SD 3.03% 3.14% 3.47% 3.68% 3.98% 4.14% 4.33% Coefficient of Variation,CV Max 3.12% 3.12% 4.41% 4.90% 5.83% 5.83% 7.38% Min -9.31% -9.31% -9.84% % % % % Semi-SD 2.42% 2.45% 2.59% 2.74% 2.79% 2.90% 2.93% Sharpe Ratio 17.37% 15.83% 18.63% 14.05% 16.52% 15.11% 15.20% Sortino Ratio 21.77% 20.30% 24.93% 18.89% 23.57% 21.53% 22.43% Sharply Rising N=34 49% 42% 36% 30% 25% 20% 0.17 Mean 2.27% 2.61% 2.92% 3.06% 3.24% 3.55% 3.55% SD 2.38% 2.81% 3.12% 3.37% 3.62% 3.96% 4.04% Coefficient of Variation,CV Midian 2.50% 3.02% 3.18% 3.44% 3.76% 3.60% 3.59% Max 4.54% 6.42% 7.51% 7.51% 8.77% 9.09% 9.71% Min -8.30% -8.63% -8.63% -8.92% -9.15% -9.15% -9.31% Semi-SD 1.60% 1.77% 1.81% 1.91% 1.94% 1.95% 1.97% Sharpe Ratio 95.62% 92.74% 93.67% 90.75% 89.57% 89.66% 87.90% Sortino Ratio % % % % % % % 26

27 Table 7 The performance is under conventional strategy and dynamic strategies (Black and Heston). Total Period Average Pure Future Fixed Moneyness % Black Model % % % % % % % % % Heston Model % % % % % % % % Sharply Falling Average Pure Future Fixed Moneyness % Black Model % % % % % % % % % Heston Model % % % % % % % % Moderately Falling Average Pure Future Fixed Moneyness % Black Model % % % % % % % % % Heston Model % % % % % % % % Moderately Rising Average Pure Future Fixed Moneyness % Black Model % % % % % % % % % Heston Model % % % % % % % % Sharply Rising Average Pure Future Fixed Moneyness % Black Model % % % % % % % % % Heston Model % % % % % % % % 27