Continous time models and realized variance: Simulations

Transcription

1 Continous time models and realized variance: Simulations Asger Lunde Professor Department of Economics and Business Aarhus University September 26, 2016

2 Continuous-time Stochastic Process: SDEs Building blocks There are two basic building blocks: The Wiener process (Brownian motion) [normal increments] The Poisson process (rare event) We will focus on SDEs that build on the Wiener process. Literature: Glasserman, (2004), Monte Carlo Methods in Finanical Engineering, Springer Verlag. 2 / 121

3 Continuous-time Stochastic Process: SDEs The Wiener Process Definition 1 (Wiener Process) A continuous-time stochastic process {W(t)} is a Wiener process if (i) W(0) = 0. (ii) W(t) W(s) N(0, t s) for t > s. (iii) W(t) W(s) W(v) W(u) for t > s v > u. (iv) W(t) is a continuous process, i.e. there are no jumps. 3 / 121

4 Continuous-time Stochastic Process: SDEs The Wiener Process An artimetric Browian motion (or Generalized Weiner process), P(t), is where X(t) X(0) = µt + σw(t) so for this we have X(t) X(s) N(µ(t s), σ 2 (t s)) 4 / 121

5 Continuous-time Stochastic Process: SDEs The Wiener Process: Construction Theorem 2 (Donsker) Let ε i N(0, 1) and t [0, 1], then i=1 W n (t) 1 [nt] d ε i W(t) for n. n So we can simulate an (artimetric) Browian motion on [0, 1] with increments at steps of size 1/N using the following scheme: X j/n = j s=1 ( ) σ µ/n + ε s N =µ j N + σ j N ε s, ε s N(0, 1), s = 1,..., N. s=1 5 / 121

6 Continuous-time Stochastic Process: SDEs Simulations The Brownian Motion. In matlab just write: X = cumsum(randn(n,1)/sqrt(n)); 6 / 121

7 Simulation of SDE Brownian Motion: Path 7 / 121

8 Brownian Motion: Distribution of increments 8 / 121

9 Simulation of SDE Brownian Motion: Path 9 / 121

10 Brownian Motion: Distribution of increments 10 / 121

11 Simulation of SDE Brownian Motion: Path 11 / 121

12 Brownian Motion: Distribution of increments 12 / 121

13 Simulation of SDE Brownian Motion: Path 13 / 121

14 Brownian Motion: Distribution of increments 14 / 121

15 Simulation of SDE Brownian Motion: Path 15 / 121

16 Brownian Motion: Distribution of increments 16 / 121

17 Simulation of SDE Brownian Motion: Path 17 / 121

18 Brownian Motion: Distribution of increments 18 / 121

19 Simulation of SDE Brownian Motion: Path 19 / 121

20 Brownian Motion: Distribution of increments 20 / 121

21 Continuous-time Stochastic Process: SDEs Simulations The Arithmetric Brownian Motion. In matlab write: X = cumsum(mu/n+randn(n,1)*sigma/sqrt(n)); 21 / 121

22 Arithmetric Brownian Motion: Path 22 / 121

23 Arithmetric Brownian Motion: Path 23 / 121

24 Arithmetric Brownian Motion: Path 24 / 121

25 Arithmetric Brownian Motion: Path 25 / 121

26 Arithmetric Brownian Motion: Path 26 / 121

27 Arithmetric Brownian Motion: Path 27 / 121

28 Arithmetric Brownian Motion: Path 28 / 121

29 Continuous-time Stochastic Process: SDEs Diffusion Processes A diffusion process is a stochastic differential equation that takes the following form: dx(t) = a(x(t), t)dt + b(x(t), t)dw(t). It is often written as dx(t) = µ(x(t), t)dt + σ(x(t), t)dw(t), (1) and µ(x(t), t), and σ(x(t), t) are called the instantaneous expected return and volatility functions, respectively. 29 / 121

30 Continuous-time Stochastic Process: SDEs Diffusion Processes The meaning of dw(t) comes from the stochastic integral W(T) = T 0 dw(t) [ adding up random increments over ever smaller intervals ] So what (1) really means is X(T) X(0) = T 0 T µ(x(t), t)dt + σ(x(t), t)dw(t), 0 When µ(, ) and σ(, ) are constants then T T X(T) X(0) = µ dt + σ dw(t) (= µt + σdw(t)). 0 0 Which is why we write dx(t) = µdt + σdw(t) for the aritmetric Brownian motion. 30 / 121

31 Continuous-time Stochastic Process: SDEs The Geometric Brownian Motion Replacing X by its logarithm, ln(x), will ensure that the process always stays positive. When we use then this is equivalent to the SDE d ln X(t) = (µ 1 2 σ2 )dt + σdw(t), dx(t) = µx(t)dt + σx(t)dw(t), which is know as the Geometric Brownian Motion. It has ln(x(t)) ln(x(s)) N((µ 1 2 σ2 )(t s), σ 2 (t s)), so we may simulate this process by { } P j/n = exp (µ 1 2 σ2 ) j N + σ j N ε s, s=1 ε s N(0, 1), s = 1,..., N 31 / 121

32 Continuous-time Stochastic Process: SDEs Simulations The Geometric Brownian Motion. In matlab write: X = exp(-0.5*sigma*(1:1:n) /N +cumsum(randn(n,1)*sigma/sqrt(n))); 32 / 121

33 Geometric Brownian Motion: Path 33 / 121

34 Geometric Brownian Motion: Path 34 / 121

35 Geometric Brownian Motion: Path 35 / 121

36 Geometric Brownian Motion: Path 36 / 121

37 Geometric Brownian Motion: Path 37 / 121

38 Geometric Brownian Motion: Path 38 / 121

39 Geometric Brownian Motion: Path 39 / 121

40 Continuous-time Stochastic Process: SDEs Simulations The Geometric Brownian Motion with drift. In matlab write: X = exp((mu-0.5*sigma)*(1:1:n) /N +cumsum(randn(n,1)*sigma/sqrt(n))); 40 / 121

41 Geometric Brownian Motion with drift: Path 41 / 121

42 Geometric Brownian Motion with drift: Path 42 / 121

43 Geometric Brownian Motion with drift: Path 43 / 121

44 Geometric Brownian Motion with drift: Path 44 / 121

45 Geometric Brownian Motion with drift: Path 45 / 121

46 Geometric Brownian Motion with drift: Path 46 / 121

47 Geometric Brownian Motion with drift: Path 47 / 121

48 Continuous-time Stochastic Process: SDEs The Geometric Brownian Motion, plus jumps We continue considering the Geometric Brownian Motion, dx(t) = µx(t)dt + σx(t)dw(t), but we add a jump term and write the process like this: dx(t) = µdt + σdw(t) + dj(t). (2) X(t ) J is a process that is independent of W that has piecewise constant sample paths. 48 / 121

49 Continuous-time Stochastic Process: SDEs The Geometric Brownian Motion, plus jumps J is given by J(t) = j=1 N(t) (D j 1) where D 1, D 2,... are random variables and N(t) is a counting process. N(t) counts the number of random arrival times 0 < τ 1 < τ 2 < in [0, t]. dj(t) denotes the jump in J at time t. So D j 1 if t = τ j dj(t) = 0 if t = τ j any j X(t ) denotes the value of X just before a potential jump. 49 / 121

50 Continuous-time Stochastic Process: SDEs The Geometric Brownian Motion, plus jumps the solution to dx(t)/x(t ) = µdt + σdw(t) + dj(t) is ( X(t) = X(0) exp (µ 1/2σ 2 )t + σw(t) ) N(t) D j j=1 Before we can simulate we must make some distributional assumptions about J(t)... the last term. Simplest assumptions: N(t) is a Poisson process, N(t) Poi(λ) log D j N(ν, φ 2 ) It is now possible to simulate the process at a fixed set of dates 0 = t 0 < t 1 < < t n. 50 / 121

51 Continuous-time Stochastic Process: SDEs The Geometric Brownian Motion, plus jumps this is because we have the following discretization log X(t i+1 ) = log X(t i ) + (µ 1/2σ 2 )(t i+1 t i ) + σ [W(t i+1 ) W(t i )] + N(t i+1 ) N(t i )+1 log D j so we just have to repeatively draw Z and M and compute: log X(t i+1 ) = log X(t i ) + (µ 1/2σ 2 )(t i+1 t i ) + σ t i+1 t i Z + M 51 / 121

52 Continuous-time Stochastic Process: SDEs The Geometric Brownian Motion, plus jumps General method of simulating the GBM with jumps: 1. generate Z N(0, 1) 2. generate N Poi (λ(t i+1 t i )); if N = 0, set M = 0 and go to generate Z 2 N(0, 1); set M = νn + φ NZ 2 4. set log X(t i+1 ) = log X(t i ) + (µ 1/2σ 2 )(t i+1 t i ) + σ t i+1 t i Z + M 52 / 121

53 Continuous-time Stochastic Process: SDEs The Geometric Brownian Motion So in our previous way of writing, we may simulate this prices process by P j/n = exp { (µ 1 2 σ2 ) j N + σ N j s=1 ε s + j s=1 νn s + φ N s d s } for s = 1,..., N, with ε s N(0, 1), and N s Poi (λ/n), d s N(0, 1). 53 / 121

54 Continuous-time Stochastic Process: SDEs The Geometric Brownian Motion The GBM plus jumps. In matlab write: mu = 0; sigma = 1; lambda = 15; nu = 0; phi = sqrt(0.1); eps = sigma/sqrt(n)*randn(n,1); Np = poissrnd(thels(n)/n,n,1); M = nu*np+phi*sqrt(np).*randn(n,1); X = exp((mu-0.5*sigma)*(1:1:n) /N+cumsum(eps+M)); 54 / 121

55 Simulation of SDE Geometric Brownian Motion plus jumps λ = 15 and 500 obs. 55 / 121

56 Simulation of SDE Geometric Brownian Motion plus jumps λ = 15 and 1000 obs. 56 / 121

57 Simulation of SDE Geometric Brownian Motion plus jumps λ = 15 and 5000 obs. 57 / 121

58 Simulation of SDE Geometric Brownian Motion plus jumps λ = 15 and obs. 58 / 121

59 Simulation of SDE Geometric Brownian Motion plus jumps λ = 10 and obs. 59 / 121

60 Simulation of SDE Geometric Brownian Motion plus jumps λ = 5 and obs. 60 / 121

61 Simulation of SDE Geometric Brownian Motion plus jumps λ = 0.1 and obs. 61 / 121

62 Continuous-time Stochastic Process: SDEs The Ornstein-Uhlenbeck (OU) Process The previous SDEs were non-stationary process that have a random walk property. This is not a plausible model for the volatility of the price of a financial asset. The OU process is a stationay (mean reverting) process defined by dv(t) = α(µ V(t))dt + σdw(t). Note that α is the rate at which the process is pulled back toward the mean parameter µ. The OU process is often used to model the logarithm of volatility. 62 / 121

63 Continuous-time Stochastic Process: SDEs The Ornstein-Uhlenbeck (OU) Process We can generate sample paths from the OU process using the following fact: with φ = exp( α ). V(t + ) V(t) N(µ + φ(v(t) µ), (1 φ 2 ) σ2 2α ), So we can simulate an OU process on [0, 1] with increments of size 1/N using the following scheme (φ = exp( α/n)): V (j+1)/n = µ + φ(v j/n µ) + with ε s N(0, 1) for s = 1,..., N. (1 φ 2 ) σ2 2α ε s,. 63 / 121

64 Continuous-time Stochastic Process: SDEs Simulations The Ornstein-Uhlenbeck (OU) Process. In matlab write: X = zeros(n,1); eps = randn(n,1); phi = exp(-alpha/n); for i=2:n X(i) = phi*x(i-1)... +sqrt((1-phi*phi)*sigma2/(2*alpha))*eps(i); end; 64 / 121

65 OU process: Path 65 / 121

66 OU process: Path 66 / 121

67 OU process: Path 67 / 121

68 OU process: Path 68 / 121

69 OU process: Path 69 / 121

70 Continuous-time Stochastic Process: SDEs Simulations The exponential Ornstein-Uhlenbeck (OU) Process. In matlab write: X = zeros(n,1); eps = randn(n,1); phi = exp(-alpha/n); for i=2:n X(i) = phi*x(i-1)... +sqrt((1-phi*phi)*sigma2/(2*alpha))*eps(i); end; X = exp(x); 70 / 121

71 OU process: Path 71 / 121

72 OU process: Path 72 / 121

73 OU process: Path 73 / 121

74 OU process: Path 74 / 121

75 OU process: Path 75 / 121

76 Continuous-time Stochastic Process: SDEs The Euler Scheme for simulating SDEs In the previous examples, the BM, the GBM, and the OU process, we had an excat solution to use for simulation. In general we can use the so-called Euler approximation. For and a given discretization one can use (writing X τj = X(τ j )): dx(t) = a(x(t), t)dt + b(x(t), t)dw(t). t 0 < τ 0 < τ 1 <... < τ j <... < τ N = T X τj+1 = X τj + a(x τj, τ j )(τ j+1 τ j ) + b(x τj, τ j )(W τj+1 W τj ). Or simply with j = τ j+1 τ j, we can write X τj+1 = X τj + a(x τj, τ j ) j + b(x τj, τ j ) j ε s, ε s N(0, 1), s = 1,..., N. 76 / 121

77 Continuous-time Stochastic Process: SDEs The Euler Scheme for simulating SDEs So for the GBM this is P (j+1)/n = P j/n + µp j/n 1 N + σ2 N P j/n ε j, ε j N(0, 1), j = 1,..., N. Which we can compare to the exact simulation that used to get P j/n = exp ln(x(t)) ln(x(s)) N((µ 1 2 σ2 )(t s), σ 2 (t s)), { } (µ 1 2 σ2 ) j N + σ j N ε s, ε s N(0, 1), s = 1,..., N s=1 77 / 121

78 Some Specific Models Constant Volatility Model In this model we take X = Y + U, where Y is a Brownian motion and U is a Gaussian white noise process with mean zero and variance ω 2. So in the discretized form we set Y 0 = 0 and Y j/n and U j/n iid N(0, ω 2 ). j ε s, where ε s iid N(0, 1/N), j = 1,..., N. s=1 The observed process is then given by: X j/n = Y j/n + U j/n, j = 0,..., N, 78 / 121

79 Some Specific Models OU Stochastic Volatility Model We consider the following SV model for the log price: dy t = µdt + σ t dw t, σ t = exp (β 0 + β 1 τ t ), dτ t = ατ t dt + db t, corr(dw t, db t ) = ρ To simulate the price process we utilize the Euler scheme, and the exact discretization of OU-process. We want to simulate the process from time 0 to time 1 so we divide the [0, 1] period into N subperiods of equal size, = 1/N. 79 / 121

80 Some Specific Models OU Stochastic Volatility Model We get the following iterative scheme Y (j+1)/n = Y j/n + µ + σ j/n ε W j σ j/n = exp[β 0 + β 1 τ j/n ] τ (j+1)/n = e α τ j/n + (1 e 2α )/( 2α)ε B j where εw j ε B j i.i.d. N (( 0 0 ) ( 1, ρ 1 )). 80 / 121

81 Some Specific Models OU Stochastic Volatility Model To ( be comparable to the constant volatility simulations impose ) 1 E 0 σ2 udu = 1. This is done by setting β 0 = β 2 1 /(2α), which follows from the fact that ( τ t τ 0 N e αt τ(0), ) 1 2α (1 eαt ) ( L N 0, ) 1, 2α We utilize this stationary distribution in our simulations to restart the process each day at τ 0 N(0, ( 2α) 1 ). Note that the variance of σ 2 is exp( 2β 2 1 /α) 1. Finally, we again add noise simulated as U j/n i.i.d. N(0, ω 2 ). 81 / 121

82 Realized variance: GBM with constant volatility 82 / 121

83 Volatility Estimation (Constant Vol) 83 / 121

84 Volatility Estimation (Constant Vol) 84 / 121

85 Volatility Estimation (Constant Vol) 85 / 121

86 Volatility Estimation (Constant Vol) 86 / 121

87 Volatility Estimation (Constant Vol) 87 / 121

88 Volatility Estimation (Constant Vol) 88 / 121

89 Volatility Estimation (Constant Vol) 89 / 121

90 Volatility Estimation (Constant Vol) 90 / 121

91 Volatility Estimation (Constant Vol) 91 / 121

92 Realized variance: GBM with OU Volatility 92 / 121

93 Volatility Estimation (OU Vol) 93 / 121

94 Volatility Estimation (OU Vol) 94 / 121

95 Volatility Estimation (OU Vol) 95 / 121

96 Volatility Estimation (OU Vol) 96 / 121

97 Volatility Estimation (OU Vol) 97 / 121

98 Volatility Estimation (OU Vol) 98 / 121

99 Volatility Estimation (OU Vol) 99 / 121

100 Volatility Estimation (OU Vol) 100 / 121

101 Volatility Estimation (OU Vol) 101 / 121

102 Realized variance: GBM with Constant Volatility plus noise 102 / 121

103 Volatility Estimation (Constant Vol and noise) 103 / 121

104 Volatility Estimation (Constant Vol and noise) 104 / 121

105 Volatility Estimation (Constant Vol and noise) 105 / 121

106 Volatility Estimation (Constant Vol and noise) 106 / 121

107 Volatility Estimation (Constant Vol and noise) 107 / 121

108 Volatility Estimation (Constant Vol and noise) 108 / 121

109 Volatility Estimation (Constant Vol and noise) 109 / 121

110 Volatility Estimation (Constant Vol and noise) 110 / 121

111 Volatility Estimation (Constant Vol and noise) 111 / 121

112 Realized Kernel variance: GBM with Constant Volatility plus noise 112 / 121

113 Kernel based Volatility Estimation (Const. Vol and noise) 113 / 121

114 Kernel based Volatility Estimation (Const. Vol and noise) 114 / 121

115 Kernel based Volatility Estimation (Const. Vol and noise) 115 / 121

116 Kernel based Volatility Estimation (Const. Vol and noise) 116 / 121

117 Kernel based Volatility Estimation (Const. Vol and noise) 117 / 121

118 Simulation of SDE Kernel based Volatility Estimation (Const. Vol and noise) 118 / 121

119 Simulation of SDE Kernel based Volatility Estimation (Const. Vol and noise) 119 / 121

120 Simulation of SDE Kernel based Volatility Estimation (Const. Vol and noise) 120 / 121

121 Simulation of SDE Kernel based Volatility Estimation (Const. Vol and noise) 121 / 121