Statistacal Self-Similarity:Fractional Brownian Motion

Transcription

1 Statistacal Self-Similarity:Fractional Brownian Motion Geofrey Wingi Sikazwe Lappeenranta University of Technology March 10, 2010 G. W. Sikazwe (Time Series Seminar, 2010) Statistacal Self-Similarity:Fractional Brownian Motion March 10, / 22

2 Outline 1 Introduction Hurst s Statistical Phenomenon Statistical Self-Similarity 2 Fractional Brownian Motion(FBM) Background and Definition of FBM Basic Properties of FBM 3 Applications of FBM Option Pricing and Volatility Estimation by FBM G. W. Sikazwe (Time Series Seminar, 2010) Statistacal Self-Similarity:Fractional Brownian Motion March 10, / 22

3 Introduction Hurst s Statistical Phenomenon Hurst s Statistical Phenomenon It was introduced by a British hydrologist H. E. Hurst in Discovery of Hurst s parameter or exponent G. W. Sikazwe (Time Series Seminar, 2010) Statistacal Self-Similarity:Fractional Brownian Motion March 10, / 22

4 Introduction Statistical Self-Similarity Definition of Statistical Self-Similarity Two processes are said to be self-similar if their joint distributions are identical. Typical example of self-similar processes is a well known Brownian motion. This process have independent increments. G. W. Sikazwe (Time Series Seminar, 2010) Statistacal Self-Similarity:Fractional Brownian Motion March 10, / 22

5 Fractional Brownian Motion(FBM) Background and Definition of FBM Background and Definition of FBM Fractional Brownian Motion (FBM) belongs to a class of long-memory Gaussian process FBM is a generalization of the more well-known process of Brownian motion The dependence structure of the increments is modeled by a parameter H (0, 1) G. W. Sikazwe (Time Series Seminar, 2010) Statistacal Self-Similarity:Fractional Brownian Motion March 10, / 22

6 Fractional Brownian Motion(FBM) Background and Definition of FBM A standard Fractional Brownian Motion with Hurst parameter 0 < H 1 is a continuous Gaussian process B = (B t ) 0 with zero mean and a covariance function given cov(b) = EB s B t = 1 2 { s 2H + t 2H t s 2H } G. W. Sikazwe (Time Series Seminar, 2010) Statistacal Self-Similarity:Fractional Brownian Motion March 10, / 22

7 Fractional Brownian Motion(FBM) Basic Properties of FBM Basic Properties of FBM B H (0) = 0 and EB H (t) = 0 stationary increments Self-similar G. W. Sikazwe (Time Series Seminar, 2010) Statistacal Self-Similarity:Fractional Brownian Motion March 10, / 22

8 Antipersistent Fractional Brownian Motion(FBM) Basic Properties of FBM When the parameter 0 < H < 1/2, The increments are said to be antipersistent. They are said to have short-range dependence or memory. G. W. Sikazwe (Time Series Seminar, 2010) Statistacal Self-Similarity:Fractional Brownian Motion March 10, / 22

9 Fractional Brownian Motion(FBM) Basic Properties of FBM Figure: 0 < H < 1/2: Antipersistent G. W. Sikazwe (Time Series Seminar, 2010) Statistacal Self-Similarity:Fractional Brownian Motion March 10, / 22

10 Fractional Brownian Motion(FBM) Independent Increments Basic Properties of FBM When the parameter H = 1/2, the process is said to have independent increments. It corresponds to the standard Brownian motion. G. W. Sikazwe (Time Series Seminar, 2010) Statistacal Self-Similarity:Fractional Brownian Motion March 10, / 22

11 Fractional Brownian Motion(FBM) Basic Properties of FBM Figure: H = 1/2: Independent increments G. W. Sikazwe (Time Series Seminar, 2010) Statistacal Self-Similarity:Fractional Brownian Motion March 10, / 22

12 Persistent Fractional Brownian Motion(FBM) Basic Properties of FBM When the parameter 1/2 < H < 1, The increments are said to be persistent. They are said to have long-range dependence or memory. The increments are positively correlated G. W. Sikazwe (Time Series Seminar, 2010) Statistacal Self-Similarity:Fractional Brownian Motion March 10, / 22

13 Fractional Brownian Motion(FBM) Basic Properties of FBM Figure: 1/2 < H < 1: Persistent. G. W. Sikazwe (Time Series Seminar, 2010) Statistacal Self-Similarity:Fractional Brownian Motion March 10, / 22

14 Fractional Brownian Motion(FBM) Basic Properties of FBM Role of Hurst s Parameter H Summary It determines the sign of the covariance of the future and past increments Existence of long-range dependence G. W. Sikazwe (Time Series Seminar, 2010) Statistacal Self-Similarity:Fractional Brownian Motion March 10, / 22

15 Applications of FBM Option Pricing and Volatility Estimation by FBM Background Model due to Cajueiro and Barbachan (2003) To study the behaviour of Brazilian stock returns in continuous time Price European options using Black-Schole type formula G. W. Sikazwe (Time Series Seminar, 2010) Statistacal Self-Similarity:Fractional Brownian Motion March 10, / 22

16 Applications of FBM Option Pricing and Volatility Estimation by FBM The Market Model We consider the fractional Black-Scholes market The investor is confronted by two investment options G. W. Sikazwe (Time Series Seminar, 2010) Statistacal Self-Similarity:Fractional Brownian Motion March 10, / 22

17 Applications of FBM Option Pricing and Volatility Estimation by FBM Risk-free asset: price given by db(t) = ρb(t)dt [B(0) = 1, 0 < t T, ρ > 0] Risk asset: price given by ds(t) = µs(t)dt + σs(t)db H (t) [S(0) = s 0 ] G. W. Sikazwe (Time Series Seminar, 2010) Statistacal Self-Similarity:Fractional Brownian Motion March 10, / 22

18 Applications of FBM Option Pricing and Volatility Estimation by FBM Figure: Investment in risk-free asset: the wealth grows exponentially G. W. Sikazwe (Time Series Seminar, 2010) Statistacal Self-Similarity:Fractional Brownian Motion March 10, / 22

19 Option Pricing Formula Applications of FBM Option Pricing and Volatility Estimation by FBM We consider the fractional Black-Scholes formula, where the total wealth is given by C = S(t)N(d 1 ) Ke r(t t) N(d 2 ) G. W. Sikazwe (Time Series Seminar, 2010) Statistacal Self-Similarity:Fractional Brownian Motion March 10, / 22

20 Sample results Applications of FBM Option Pricing and Volatility Estimation by FBM 3 D plot for FBS - BS call option, Maturity time and Exercise price G. W. Sikazwe (Time Series Seminar, 2010) Statistacal Self-Similarity:Fractional Brownian Motion March 10, / 22

21 Applications of FBM Option Pricing and Volatility Estimation by FBM Figure: FBS-BS, Maturity and Exercise Price G. W. Sikazwe (Time Series Seminar, 2010) Statistacal Self-Similarity:Fractional Brownian Motion March 10, / 22

22 Applications of FBM Option Pricing and Volatility Estimation by FBM THANK YOU G. W. Sikazwe (Time Series Seminar, 2010) Statistacal Self-Similarity:Fractional Brownian Motion March 10, / 22