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254 CONTINUOUS-TIME MODELS3. Assume that the price of IBM stock follows the Ito’s processdP t = µP t dt + σ P t dw t ,where µ and σ are constant and w t is a standard Brownian motion. Considerthe daily log returns of IBM stock in 1997. The average return and the samplestandard deviation are 0.00131 and 0.02215, respectively. Use the data to estimatethe parameters µ and σ assuming that there were 252 trading days in 1997.4. Suppose that the current price of a stock is $120 per share with volatility σ = 50%per annum. Suppose further that the risk-free interest rate is 7% per annum and thestock pays no dividend. (a) What is the price of a European call option contingenton the stock with a strike price of $125 that will expire in 3 months? (b) What isthe price of a European put option on the same stock with a strike price of $118that will expire in 3 months? If the volatility σ is increased to 80% per annum,then what are the prices of the two options?5. Derive the limiting marginal effects of the five variables K , P t , T − t, σ ,andr ona European put option contingent on a stock.6. A stock price is currently $60 per share and follows the geometric Brownianmotion dP t = µP t dt+σ P t dt. Assume that the expected return µ from the stockis 20% per annum and its volatility is 40% per annum. What is the probabilitydistribution for the stock price in 2 years? Obtain the mean and standard deviationof the distribution and construct a 95% confidence interval for the stock price.7. A stock price is currently $60 per share and follows the geometric Brownianmotion dP t = µP t dt+σ P t dt. Assume that the expected return µ from the stockis 20% per annum and its volatility is 40% per annum. What is the probabilitydistribution for the continuously compounded rate of return of the stock over 2years? Obtain the mean and standard deviation of the distribution.8. Suppose that the current price of Stock A is $70 per share and the price followsthe jump diffusion model in Eq. (6.26). Assume that the risk-free interest rate is8% per annum and the stock volatility is 30% per annum. In addition, the priceon average has about 15 jumps per year with average jump size −2% and jumpvolatility 3%. What is the price of a European call option with strike price $75that will expire in 3 months? What is the price of the corresponding European putoption?REFERENCESAbramowitz, M., and Stegun, I. A. (1972), Handbook of Mathematical Functions, 10th ed.,U.S. National Bureau of Standards.Ait-Sahalia, Y. (1996), “Testing continuous-time models for the spot interest rate,” Review ofFinancial Studies,9,385–426.Ait-Sahalia, Y. (1997), “Maximum likelihood estimation of discretely sampled diffusions: aclosed-form approach,” working paper, Economics Department, Princeton University.
REFERENCES 255Bakshi, G., Cao, C., and Chen, Z. (1997), “Empirical performance of alternative option pricingmodels,” Journal of Finance, 52, 2003–2049.Billingsley, P. (1986), Probability and Measure, 2nd ed., Wiley: New York.Billingsley, P. (1968), Convergence of Probability Measures, Wiley: New York.Black, F. and Scholes, M. (1973), “The pricing of options and corporate liabilities,” Journalof Political Economy, 81, 637–654.Conley, T. G., Hansen, L. P., Luttmer, E. G. J., and Scheinkman, J. A. (1997), “Short-terminterest rates as subordinated diffusions,” Review of Financial Studies, 10, 525–577.Cox, J. C., and Rubinstein, M. (1985), Options Markets, Prentice-Hall: Englewood Cliffs,New Jersey.Donsker, M. (1951), “An invariance principle for certain probability limit theorems,” Mem.American Mathematical Society, No. 6.Duffie, D. (1995), Dynamic Asset Pricing Theory, 2nd ed., Princeton University Press: Princeton,New Jersey.Elerian, O., Chib, S., and Shephard, N. (2001), “Likelihood inference for discretely observednon-linear diffusions,” Econometrica, 69, 959–993.Eraker, B. (2001), “MCMC analysis of diffusion models with application to finance,” Journalof Business & Economic Statistics, 19, 177–191.Gallant, A. R., and Long, J. R. (1997), “Estimating stochastic diffusion equations efficientlyby minimum chi-squared, ” Biometrika, 84, 125–141.Gallant, A. R., and Tauchen, G. (1997), “The relative efficiency of method of moments estimators,”Working paper, Economics Department, University of North Carolina.Hull, J. C. (1997), Options, Futures, and Other Derivatives, 3rd ed. Prentice-Hall: UpperSaddle River, New Jersey.Kessler, M. (1997), “Estimation of an ergodic diffusion from discrete observations,” ScandinavianJournal of Statistics, 24, 1–19.Kou, S. (2000), “A jump diffusion model for option pricing with three properties: Leptokurticfeature, volatility smile, and analytic tractability,” working paper, Columbia University.Lo, A. W. (1988), “Maximum likelihood estimation of generalized Ito’s processes with discretelysampled data,” Econometric Theory,4,231–247.Merton, R. C. (1976), “Option pricing when the underlying stock returns are discontinuous,”Journal of Financial Economics,5,125–144.
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Analysis of FinancialTime SeriesFin
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ContentsPrefacexi1. Financial Time
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CONTENTSix6.6 Black-Scholes Pricing
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PrefaceThis book grew out of an MBA
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Analysis of Financial Time Series.
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ASSET RETURNS 3Multiperiod Simple R
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ASSET RETURNS 5r t [k] =ln(1 + R t
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DISTRIBUTIONAL PROPERTIES OF RETURN
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10 FINANCIAL TIME SERIES AND THEIR
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REFERENCES 21Federal Reserve Bank o
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CORRELATION AND AUTOCORRELATION FUN
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CORRELATION AND AUTOCORRELATION FUN
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WHITE NOISE AND LINEAR TIME SERIES
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SIMPLE AUTOREGRESSIVE MODELS 29whic
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SIMPLE AUTOREGRESSIVE MODELS 31wher
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SIMPLE AUTOREGRESSIVE MODELS 33wher
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SIMPLE AUTOREGRESSIVE MODELS 35expa
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SIMPLE AUTOREGRESSIVE MODELS 37Tabl
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SIMPLE AUTOREGRESSIVE MODELS 39Mode
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SIMPLE AUTOREGRESSIVE MODELS 41The
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SIMPLE MOVING-AVERAGE MODELS 43wher
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SIMPLE MOVING-AVERAGE MODELS 45s-rt
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SIMPLE MOVING-AVERAGE MODELS 47The
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SIMPLE ARMA MODELS 49A time series
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SIMPLE ARMA MODELS 51operator, the
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SIMPLE ARMA MODELS 53Table 2.5. Sam
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SIMPLE ARMA MODELS 55This represent
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UNIT-ROOT NONSTATIONARITY 57ˆp h (
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UNIT-ROOT NONSTATIONARITY 59log-pri
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SEASONAL MODELS 61which is commonly
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SEASONAL MODELS 63Series : xSeries
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SEASONAL MODELS 65models use the sa
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• ••• •••••••
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REGRESSION MODELS WITH TIME SERIES
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REGRESSION MODELS WITH TIME SERIES
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LONG-MEMORY MODELS 73=(k − d −
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APPENDIX A: SOME SCA COMMANDS 75est
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EXERCISES 77relation. Draw your con
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STRUCTURE OF A MODEL 813.2 STRUCTUR
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THE ARCH MODEL 83corrected asset re
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THE ARCH MODEL 85Series : xACF0.0 0
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THE ARCH MODEL 87sample mean from t
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THE ARCH MODEL 89where Ɣ(x) is the
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THE ARCH MODEL 91Series : resiSerie
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THE GARCH MODEL 93other softwares a
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THE GARCH MODEL 95ahead forecast ca
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THE GARCH MODEL 97The fitted AR(3)
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THE GARCH MODEL 99Series : stdatACF
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THE GARCH-M MODEL 101two models. Th
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THE EXPONENTIAL GARCH MODEL 103To b
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THE EXPONENTIAL GARCH MODEL 105eros
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THE CHARMA MODEL 107ˆσ 2 864 (3)
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RANDOM COEFFICIENT AUTOREGRESSIVE M
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THE LONG-MEMORY STOCHASTIC VOLATILI
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AN ALTERNATIVE APPROACH 113where Va
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APPLICATION 115(a) IBMlog-rtn-30 -1
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APPLICATION 117equationThe fitted m
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KURTOSIS OF GARCH MODELS 119where K
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SOME RATS PROGRAMS FOR ESTIMATING V
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EXERCISES 123• Is there evidence
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REFERENCES 125Melino, A., and Turnb
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NONLINEAR TIME SERIES 127To put non
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NONLINEAR MODELS 129Example 4.1. Co
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NONLINEAR MODELS 131jumps when it b
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NONLINEAR MODELS 133the series and
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NONLINEAR MODELS 135els to the mont
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NONLINEAR MODELS 137growth-2 -1 0 1
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NONLINEAR MODELS 139average of y t
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NONLINEAR MODELS 141indicating that
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NONLINEAR MODELS 143s T,2t=1T∑K h
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NONLINEAR MODELS 145f i (.) of Eq.
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NONLINEAR MODELS 1474.1.9.1 Feed-Fo
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NONLINEAR MODELS 1494.1.9.2 Trainin
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NONLINEAR MODELS 151prob0.0 0.2 0.4
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NONLINEARITY TESTS 153idea behind v
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NONLINEARITY TESTS 155C l (δ, T )
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NONLINEARITY TESTS 157and obtain th
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NONLINEARITY TESTS 159• Step 2: C
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FORECASTING 161returns. In summary,
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FORECASTING 163in m 11 and m 22 ind
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APPLICATION 165unem. rate4 6 8 10
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APPLICATION 167Table 4.4. Out-of-Sa
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APPENDIX A 169compute a0 = 0.1, a1
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REFERENCES 171P(s t = 2 | s t−1 =
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REFERENCES 173Fan, J. (1993), “Lo
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NONSYNCHRONOUS TRADING 177t − k t
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BID-ASK SPREAD 179The lag-1 autocov
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EMPIRICAL CHARACTERISTICS 181The ma
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n(trades)0 20406080 1200 1000 2000
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EMPIRICAL CHARACTERISTICS 185after-
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MODELS FOR PRICE CHANGES 187deep un
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MODELS FOR PRICE CHANGES 189σ 2i (
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MODELS FOR PRICE CHANGES 191A i = 1
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MODELS FOR PRICE CHANGES 193( )piln
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DURATION MODELS 195For the IBM data
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DURATION MODELS 197(a) is the avera
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DURATION MODELS 199actions might no
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DURATION MODELS 201Series : xACF0.0
Page 211 and 212: DURATION MODELS 203reduces to that
Page 213 and 214: DURATION MODELS 205Series : xACF0.0
Page 215 and 216: THE PCD MODEL 207where w(α) denote
Page 217 and 218: THE PCD MODEL 209(a) All transactio
Page 219 and 220: THE PCD MODEL 2110 10 20 30 40 500
Page 221 and 222: THE PCD MODEL 213⎧⎪⎨1f (x |
Page 223 and 224: THE PCD MODEL 215Generalized Gamma
Page 225 and 226: THE PCD MODEL 217smpl 2 3534compute
Page 227 and 228: REFERENCES 219(a) Use the data to c
Page 229 and 230: Analysis of Financial Time Series.
Page 231 and 232: STOCHASTIC PROCESSES 223write a con
Page 233 and 234: STOCHASTIC PROCESSES 2253. stationa
Page 235 and 236: ITO’S LEMMA 2276.3.2 Stochastic D
Page 237 and 238: ITO’S LEMMA 229This result shows
Page 239 and 240: DISTRIBUTIONS OF PRICE AND RETURN 2
Page 241 and 242: BLACK-SCHOLES EQUATION 233and G t =
Page 243 and 244: BLACK-SCHOLES FORMULA 235risk prefe
Page 245 and 246: BLACK-SCHOLES FORMULA 237The stock
Page 247 and 248: BLACK-SCHOLES FORMULA 239(a) Call o
Page 249 and 250: AN EXTENSION OF ITO’S LEMMA 2410
Page 251 and 252: STOCHASTIC INTEGRAL 243using the It
Page 253 and 254: JUMP DIFFUSION MODELS 245where w t
Page 255 and 256: JUMP DIFFUSION MODELS 247where it i
Page 257 and 258: JUMP DIFFUSION MODELS 249sudden jum
Page 259 and 260: APPENDIX A 251Pricing formulas for
Page 261: EXERCISES 253which involves the CDF
Page 265 and 266: VALUEATRISK 257log return-0.2 -0.1
Page 267 and 268: RISKMETRICS 259we use log returns r
Page 269 and 270: RISKMETRICS 261investor is$10,000,0
Page 271 and 272: ECONOMETRIC APPROACH TO VAR 263Cons
Page 273 and 274: ECONOMETRIC APPROACH TO VAR 265The
Page 275 and 276: QUANTILE ESTIMATION 267Using the fo
Page 277 and 278: QUANTILE ESTIMATION 269ˆx 0.01 = p
Page 279 and 280: EXTREME VALUE THEORY 271= 1 −= 1
Page 281 and 282: EXTREME VALUE THEORY 273shows that
Page 283 and 284: EXTREME VALUE THEORY 275{ [] }r n(i
Page 285 and 286: EXTREME VALUE THEORY 2777.5.3 Appli
Page 287 and 288: EXTREME VALUE TO VAR 2797.6 AN EXTR
Page 289 and 290: EXTREME VALUE TO VAR 281VaR =−2.5
Page 291 and 292: EXTREME VALUE TO VAR 283stock. The
Page 293 and 294: A NEW APPROACH TO VAR 285series (e.
Page 295 and 296: A NEW APPROACH TO VAR 287In Eq. (7.
Page 297 and 298: A NEW APPROACH TO VAR 289Example 7.
Page 299 and 300: A NEW APPROACH TO VAR 291feature is
Page 301 and 302: A NEW APPROACH TO VAR 293(a) Homoge
Page 303 and 304: A NEW APPROACH TO VAR 295Table 7.4.
Page 305 and 306: REFERENCES 297a Gaussian GARCH(1, 1
Page 307 and 308: Analysis of Financial Time Series.
Page 309 and 310: CROSS-CORRELATION 301correlation co
Page 311 and 312: CROSS-CORRELATION 303where ̂D is t
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CROSS-CORRELATION 305Table 8.1. Sum
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30yr20yr10yr5yr1yr-10 0 510-10 0 51
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VAR MODELS 3098.2 VECTOR AUTOREGRES
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VAR MODELS 311where G = Cov(b t ).
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VAR MODELS 313where φ 0 and a t ar
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VAR MODELS 315To specify the order
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VAR MODELS 317the S&P 500 index. Th
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VECTOR MA MODELS 319[ ] [ ] [ ] [r1
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VECTOR MA MODELS 321turn used to co
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VECTOR ARMA MODELS 323For the VAR(1
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VECTOR ARMA MODELS 325log-rate0.0 0
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VECTOR ARMA MODELS 327interest-rate
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CO-INTEGRATION 329Series : x1ACF0.0
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CO-INTEGRATION 331Stock Exchange; o
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THRESHOLD CO-INTEGRATION 333ously b
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PRINCIPAL COMPONENT ANALYSIS 3358.6
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PRINCIPAL COMPONENT ANALYSIS 337(c
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PRINCIPAL COMPONENT ANALYSIS 339(a)
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FACTOR ANALYSIS 341(a) 5 stock retu
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FACTOR ANALYSIS 343andCov(r, F) = E
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FACTOR ANALYSIS 345matrix P to tran
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FACTOR ANALYSIS 347Example 8.9. In
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APPENDIX A. REVIEW OF VECTORS AND M
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APPENDIX A. REVIEW OF VECTORS AND M
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APPENDIX B. MULTIVARIATE NORMAL DIS
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REFERENCES 355(b) Build a bivariate
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REPARAMETERIZATION 359To illustrate
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REPARAMETERIZATION 361where ⊥ den
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GARCH MODELS FOR BIVARIATE RETURNS
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VECTOR VOLATILITY MODELS 377using t
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VECTOR VOLATILITY MODELS 379large-c
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VECTOR VOLATILITY MODELS 381(a) S&P
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FACTOR-VOLATILITY MODELS 383conside
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APPLICATION 385[ ]σ11,t=σ 22,t⎡
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MULTIVARIATE t DISTRIBUTION 387σ 1
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APPENDIX A. SOME REMARKS ON ESTIMAT
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APPENDIX A. SOME REMARKS ON ESTIMAT
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REFERENCES 393(a) Compute the sampl
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GIBBS SAMPLING 397mean adding auxil
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BAYESIAN INFERENCE 399distributions
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BAYESIAN INFERENCE 401normal with m
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ALTERNATIVE ALGORITHMS 403distribut
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ALTERNATIVE ALGORITHMS 40510.4.2 Me
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REGRESSION WITH SERIAL CORRELATIONS
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REGRESSION WITH SERIAL CORRELATIONS
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MISSING VALUES AND OUTLIERS 411y h
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MISSING VALUES AND OUTLIERS 413Cons
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MISSING VALUES AND OUTLIERS 415we h
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MISSING VALUES AND OUTLIERS 417c3t-
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STOCHASTIC VOLATILITY MODELS 419tha
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STOCHASTIC VOLATILITY MODELS 421whe
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STOCHASTIC VOLATILITY MODELS 423den
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STOCHASTIC VOLATILITY MODELS 425The
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STOCHASTIC VOLATILITY MODELS 427(a)
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•••MARKOV SWITCHING MODELS 42
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MARKOV SWITCHING MODELS 431(a) GARC
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MARKOV SWITCHING MODELS 433β i ∼
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MARKOV SWITCHING MODELS 435(a) mont
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MARKOV SWITCHING MODELS 437α ij ar
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FORECASTING 439garchrtn-sq0 200 400
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EXERCISES 441eter uncertainty in pr
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REFERENCES 443Justel, A., Peña, D.
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446 INDEXData (cont.)equal-weighted
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448 INDEXPoisson process, 244inhomo
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