臺灣博碩士論文加值系統

The major objective of this paper is to investigate the predictive ability of the one-factor stochastic volatility model by extracting helpful information from the implied volatility surface. This model is compared to the GARCH (1, 1) model with respect to forecasting future realized volatility of the underlying asset. We construct the one-factor stochastic volatility model consistent with the theory of implied tree models, which are suggested by Rubinstein (1994), Dupire (1994) and Derman and Kani (1994), and apply the transformation by Kreisler (1998), in which the implied volatility function is included.
We analyze the Dow Jones Industrial Average (DJIA) index options and find that the implied volatility surfaces may really exist through estimation process. In predicting the future realized volatility of the underlying asset, the evidence shows that the GARCH (1, 1) model performs better than the one-factor stochastic volatility model and the ad hoc model. Nevertheless, for a 1-week forecast horizon, the one-factor stochastic volatility model’s correlation coefficient between the realized and the forecasted volatility is much more significant than the GARCH (1, 1) model and the ad hoc model, indicating that the one-factor stochastic volatility model subsumes additional information that others do not take into account.

Chapter 1 Introduction .......................................................................................................1
1.1 Research Background ..................................................................................................1
1.2 Objectives and Major Findings ...................................................................................1
1.3 Research Framework ...................................................................................................2
Chapter 2 Literature Review ..............................................................................................4
2.1 Can Implied Volatilities Predict Future Volatilities? .................................................. 4
2.2 Why is the deterministic volatility model? ..................................................................7
2.3 What is the Contribution of Our Study? ....................................................................11
Chapter 3 Methodology ....................................................................................................12
3.1 Implied Volatility and the One-Factor Stochastic Volatility Model ..........................12
3.2 Implied Volatility Surface and the Ad Hoc Approach ...............................................16
3.3 The GARCH (1, 1) Model .........................................................................................17
3.4 Datasets .....................................................................................................................18
Chapter 4 Empirical Results ........................................................................................... 20
4.1 Estimations for the Implied Volatility Surfaces .........................................................20
4.2 Local Volatility Forecasts and the GARCH (1, 1) .....................................................21
4.3 Comparisons ..............................................................................................................22
Chapter 5 Conclusions and Suggestions ..........................................................................26
References ..........................................................................................................................28
Appendix A .........................................................................................................................32
Appendix B .........................................................................................................................34

List of Tables
Table 1 Implied Volatility Surface of DJIA Index Options .............................................37
Table 2 Ad Hoc Implied Volatility Surface of DJIA Index Options ................................43
Table 3 Volatility Forecasts from the Implied Volatility Surface ....................................49
Table 4 Volatility Forecasts from the Ad Hoc Implied Volatility Surface ......................51
Table 5 Volatility Forecasts from the GARCH (1, 1) Model ...........................................53
Table 6 Forecast Errors ....................................................................................................55
Table 7 The Correlation Matrix between the Forecasted and the Realized Volatilities ...56
















List of Figures
Figure 1 Implied Volatility Surface and Local Volatility Forecast
Observation Date: 08/30/2006 ...........................................................................57
Figure 2 Implied Volatility Surface and Local Volatility Forecast
Observation Date: 09/06/2006 ...........................................................................58
Figure 3 Implied Volatility Surface and Local Volatility Forecast
Observation Date: 09/13/2006 ...........................................................................59
Figure 4 Implied Volatility Surface and Local Volatility Forecast
Observation Date: 09/20/2006 ...........................................................................60
Figure 5 Implied Volatility Surface and Local Volatility Forecast
Observation Date: 09/27/2006 ...........................................................................61

Ait-Sahalia, Yacine, and Andrew W. Lo, 1998, Nonparametric estimation of state-price densities implicit in financial asset prices, Journal of Finance, 53, 499–547.
Ball, Clifford A., and Antonio Roma, 1994, Stochastic volatility option pricing, Journal of Financial and Quantitative Analysis, 29, 589–607.
Bates, David, 1996, Jumps and stochastic volatility: Exchange rate processes implicit in Deutschemark options, Review of Financial Studies 9, 69–107.
Black, Fischer, and Myron Scholes, 1973, The pricing of options and corporate liabilities, Journal of Political Economy, 81, 637–657.
Bollerslev, Tim, 1986, Generalized autoregressive conditional heteroscedasticity, Journal of Econometrics, 31, 307-327.
Canina, Linda, and Stephen Figlewski, 1993, The informational content of implied volatility, Review of Financial Studies, 6, 659–681.
Christensen, Bent Jesper, and Charlotte Strunk Hansen, 2002, New evidence on the implied-realized volatility relation, European Journal of Finance, 8-2, 187-205.
Christensen, B.J., and N.R. Prabhala, 1998, The relation between implied and realized volatility, Journal of Financial Economics, 50, 125–150.
Cox, John C., and Stephen A. Ross, 1976, The valuation of options for alternative stochastic processes, Journal of Financial Economics, 3, 145–166.
Cox, John C., Stephen A. Ross, and Mark Rubinstein, 1979, Option pricing: A simplified approach, Journal of Financial Economics, 7, 229–263.
Day, T.E., and C.M. Lewis, 1992, Stock market volatility and the information content of stock index options, Journal of Financial Economics, 52, 267-287.
Day, T.E., and C.M. Lewis, 1993, Forecasting futures market volatility, Journal of Derivatives, 33-50.
Derman, Emanuel, and Iraj Kani, 1994, Riding on a smile, Risk, 7, 32–39.
Dumas, Bernard, Jeff Fleming, and Robert E. Whaley, 1998, Implied volatility functions: Empirical tests, Journal of Finance, 53, 2059–2106.
Dupire, Bruno, 1994, Pricing with a smile, Risk, 7, 18–20.
Duque, Joao, and Dean Paxson, 1993, Implied volatility and dynamic hedging, Review of Futures Markets, 13, 381–421.
Engle, Robert F., 1982, Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation, Econometrica, 50, 987-1007.
Fleming, Jeff, 1998, The quality of market volatility forecasts implied by S&P 100 index option prices, Journal of Empirical Finance, 5, 317-345
Harvey, Campbell R., and Robert E. Whaley, 1992a, Market volatility prediction and the efficiency of the S&P 100 index option market, Journal of Financial Economics, 31, 43-73.
Harvey, Campbell R., and Robert E. Whaley, 1992b, Dividends and S&P 100 index option valuation, Journal of Futures Markets, 12, 123–137.
Heston, Steven L., 1993, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies 6, 327–343.

Heynen, Ronald, 1993, An empirical investigation of observed smile patterns, Review of Futures Markets, 13, 317–353.
Hull, John, and Alan White, 1987, The pricing of options on assets with stochastic volatility, Journal of Finance 42, 281–300.
Jackwerth, Jens C., 1997, Generalized binomial trees, Journal of Derivatives, 5, 7–17.
Jorion, Philippe, 1995, Predicting volatility in the foreign exchange market, Journal of Finance, 50, 507–528.
Kim, In Joon, and Gun Youb Park, 2006, An empirical comparison of implied tree models for KOSPI 200 index options, International Review of Economics & Finance, 15, 52-71
Kreisler, Michele A., 1996, Information behind the implied volatility smile: Empirical results, Ph.D. dissertation, University of Pennsylvania.
Lamoureux, Christopher G., and William D. Lastrapes, 1993, Forecasting stock-return variance: Toward an understanding of stochastic implied volatilities, Review of Financial Studies, 6, 293–326.
Lim, Kian Guan, and Da Zhi, 2002, Pricing options using implied trees: Evidence from FTSE-100 options, Journal of Futures Markets, 22, 601–626.
Merton, Robert C., 1973, Theory of rational option pricing, Bell Journal of Economics and Management Science, 4, 141–183.
Merton, Robert C., 1976, Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 3, 125–144.
Noh, J., R.F. Engle, and A. Kane, 1994, Forecasting volatility and option prices of the S&P 500 index, Journal of Derivatives, 1, 17-30
Rubinstein, Mark, 1994, Implied binomial trees, Journal of Finance 49, 771–818.
Scott, Louis O., 1987, Option pricing when the variance changes randomly: Theory, estimation and an application, Journal of Financial and Quantitative Analysis, 22, 419–438.
Sheikh, Aamir M., 1993, The behavior of volatility expectations and their effects on expected returns, Journal of Business, 66, 93–116.
Stein, E., & C. Stein, 1993, Stock price distributions for estimating stochastic volatility: An analytic approach. Review of Financial Studies, 4, 727–752.
Taylor, Stephen J., and Xinzhong Xu, 1993, The magnitude of implied volatility smiles: Theory and empirical evidence for exchange rates, Review of Futures Markets 13, 355–380.
Wiggins, James B., 1987, Option values under stochastic volatility, Journal of Financial Economics, 19, 351–372.
Xu, Xinzhong, and Stephen J. Taylor, 1995, Conditional volatility and the informational efficiency of the PHLX currency options market, Journal of Banking and Finance, 19, 803-821.