Modeling and estimating the dynamics of stock prices

What do the models need to model?
What is the purpose of the model?
What did the Black - Scholes model model?

Transcription

1 Modeling and estimating the dynamics of stock prices José Figueroa-López 1 1 Department of Statistics Purdue University VIGRE Seminar Purdue University Sep. 3, 2008

2 A quick glance of mathematical finance Outline 1 A quick glance of mathematical finance 2 Modeling stock prices Stylized statistical features The Black-Scholes model Jump-based models 3 Nonparametric estimation of jump-diffusion models The problem An example 4 Where did we go from here? Market Microstructure Feasibility and robustness

3 A quick glance of mathematical finance First concepts What is mathematical finance? 1 Field concerned with the pricing of financial assets such that the market is free of arbitrage opportunity. 2 What are the assets? 1 Bonds 2 Stocks 3 Derivatives: A contract between two parties where one party is obligated to deliver a payoff at a specified future time called the maturity. The final payoff is contingent to the value of a given stock, called the underlying.

9 A quick glance of mathematical finance First concepts A simple example A put option on a two-state economy A 1-year bond pays an interest of 5% per annum Current price per share is P(0) = $100 In T = 1 year, price can go up to P(1) = $110 with probability 1/4 or go down to P(1) = $95 with probability 3/4 Put Option = Right to sell a share of the stock at T = 1 for $100

14 A quick glance of mathematical finance First concepts Pricing and hedging 1 What is the fair price Π 0 of the derivative? 2 Key idea: Replicating trading strategy. Consider the following trading strategy at time t = 0: (i) Borrow (40)(104) from the bond market. (3)(21) (ii) Buy 2 shares of stock 3 The net necessary investment at time t = 0 is V 0 2 (100) The resulting wealth at time t = 1 is V 1 = { (1.05) = 10 if stock goes up (1.05) = 0 if stock goes down 3 Conclusion: Fair price Π 0 = V 0 =

21 A quick glance of mathematical finance First concepts Risk-neutral valuation A fundamental question: Is the previous arbitrage-free price consistent with the expected discounted payoff" of the option? Risk-neutral valuation 1 The arbitrage-free price Π 0 = is such that ; 2 The probabilities weights q u = 2 3 and q d = 1 3 are such that q u q d =.05 Expected return on the stock = risk-free return

25 A quick glance of mathematical finance First concepts Risk-neutral valuation A fundamental question: Is the previous arbitrage-free price consistent with the expected discounted payoff" of the option? Risk-neutral valuation 1 The arbitrage-free price Π 0 = is such that q u q d = ; with q u = 2 3, q d = The probabilities weights q u = 2 3 and q d = 1 3 are such that q u q d =.05 Expected return on the stock = risk-free return

26 A quick glance of mathematical finance First concepts Risk-neutral valuation A fundamental question: Is the previous arbitrage-free price consistent with the expected discounted payoff" of the option? Risk-neutral valuation 1 The arbitrage-free price Π 0 = is such that q u q d = ; with q u = 2 3, q d = The probabilities weights q u = 2 3 and q d = 1 3 are such that q u q d =.05 Expected return on the stock = risk-free return

27 Modeling stock prices Outline 1 A quick glance of mathematical finance 2 Modeling stock prices Stylized statistical features The Black-Scholes model Jump-based models 3 Nonparametric estimation of jump-diffusion models The problem An example 4 Where did we go from here? Market Microstructure Feasibility and robustness

28 Modeling stock prices Stylized statistical features Stylized features of stock price evolutions 1 A Brownian motion"-like behavior 2 Sudden big" changes in price ( Jumps") 3 Short-term returns exhibit heavy-tails 4 Volatility" clustering (intermittency) 5 Leverage phenomenon

29 Modeling stock prices The Black-Scholes model Black-Scholes for stock prices 1 The log return of the stock price P during [s, t] is given by 2 Properties: R(s, t) := log {P t /P s }. Pt Ps R(s, t) P s, the simple return, if P s P t R(s, t) + R(t, u) = R(s, u), s < t < u 3 Principles of the Black-Scholes model: Log returns on disjoint time periods are independent from one another: Corr (R(s, t), R(t, u)) = 0, s < t < u Log returns on time periods of equal size have the similar statistical features: R(s, t) D = R(t, u), t s = u t The process evolves continuously: P s P t, if s t

36 Modeling stock prices The Black-Scholes model The log price process It is convenient to express P in terms of the log price process X t : X t := log {P t /P 0 } P t = P 0 e X t Clearly, X inherits the following properties: (i) Corr (X t X s, X u X t ) = 0, s < t < u D (ii) X t X s = Xu X t, t s = u t (iii) X s X t, if s t A process X satisfying (i)-(iii) is said to be a (drifted) Brownian motion Striking theoretical property: The histogram of the increments of X (hence, the log returns of P) in consecutive time intervals of equal time span is bell shaped.

41 Modeling stock prices The Black-Scholes model Black-Scholes fundamental results 1 The market consisting of a stock following the Black-Scholes model borrowing and lending of money at a constant interest rate all" derivatives of the stock is arbitrage-free. 2 There exists a trading strategy whose final wealth replicates the payoff of any given derivative 3 The arbitrage-free price of a derivative equals its expected discounted payoff with certain probability weights Q function 4 The probability weight function Q is such that the log return of the stock is equal to the risk-free interest rate

50 Modeling stock prices Jump-based models Need for price jumps 1 Drawback of the Black-Scholes model for stock prices: Distributions with much lighter tails Absence of jumps Don t exhibit volatility intermittence 2 Introduce jumps via a Compound Poisson Process Z t : The arrivals" of jumps are independent from one another Jumps don t occur simultaneously Jumps arrive homogeneously" across time at an expected average rate of λ jumps per unit time. The size of the jumps has the same distribution with density f. 3 Jump-Diffusion model: X t = log {P t /P 0 } = B }{{} t + Z }{{} t Brownian Motion Compound Poisson

58 Nonparametric estimation of jump-diffusion models Outline 1 A quick glance of mathematical finance 2 Modeling stock prices Stylized statistical features The Black-Scholes model Jump-based models 3 Nonparametric estimation of jump-diffusion models The problem An example 4 Where did we go from here? Market Microstructure Feasibility and robustness

59 Nonparametric estimation of jump-diffusion models The problem The problem of nonparametric estimation 1 Set-up: {X t } t 0 is a jump-diffusion process with density of jumps f and intensity of jumps λ. The process is discretely sampled at 0 = t 0 < < t n = T. 2 s(x) = λf (x) is called the Lévy density of the process. 3 Problem: Estimate the function s(x). 4 Why is estimation hard? Times and sizes of the jumps are latent unobservable variables. 5 A natural solution": max i {t i t i 1 } 0 = Recover jumps from {X ti X ti 1 } i. T = t n = Increase of relevant sample size.

65 Nonparametric estimation of jump-diffusion models The problem Histogram type estimators 1 Building blocks: ˆβ(ϕ) := 1 n ϕ ( ) X tk X tk 1 t n k=1 } {{ } "Realized ϕ variation per unit time" 2 Histogram estimators: Fix an estimation window [a, b]. Divide the window in m equal-sized classes: (x 0, x 1 ],..., (x m 1, x m ]. Construct the function estimator: ŝ(x) = ˆβ(ϕ 1 )ϕ 1 (x) + + ˆβ(ϕ m )ϕ m (x), 1 where ϕ i (x) = 1 xi x (xi 1,x i i 1 ].

71 Nonparametric estimation of jump-diffusion models An example An example: Gamma Lévy process 1 Model: Pure-jump process s(x) = α x e x/β 1 {x>ε}. 2 Histogram estimators:

74 Nonparametric estimation of jump-diffusion models An example Performance Maximum-Likelihood estimators: ˆα MLE = 1.01 and ˆβ MLE = Non-parametric least-squares estimators: ˆα LSE = 0.93 and ˆβ LSE = Obtained from fitting the model α x e x/β (using least-squares) to the histogram estimator. Sampling mean and standard errors based on 1000 repetitions. t Histogram-LSF MLE (0.06) 1.40 (0.50) (0.01) 0.99 (0.05) (0.08) 1.12 (0.31) (0.07) 0.99 (0.08) (0.08) 1.13 (0.34) (0.07) 0.99 (0.08)

78 Nonparametric estimation of jump-diffusion models An example Example: One-sided tempered stable distribution 1 Model: s(x) = α x υ+1 e x/β 1 {x>ε}, α = β = 1 and υ =.1. 2 Sampling means and standard errors based on 100 repetitions. t Histogram - LSF Gamma MLE ˆα NP ˆβNP ˆυ ˆα MLE ˆβMLE (.15).97 (.14).09 (.0002) 1.2 (.08) 0.89 (.079) Table: Sampling mean and standard errors (sample size=100) 3 Message: Better to apply a low-performance method to a well specified model than a highly efficient method to a mis-specified model.

82 Where did we go from here? Outline 1 A quick glance of mathematical finance 2 Modeling stock prices Stylized statistical features The Black-Scholes model Jump-based models 3 Nonparametric estimation of jump-diffusion models The problem An example 4 Where did we go from here? Market Microstructure Feasibility and robustness

83 Where did we go from here? Market Microstructure Market microstructure features 1 High-frequency estimation depend heavily on an accurate account of the stock price evolution at a very small-time scale. 2 Real stock prices exhibit several features inherited from the way trading takes place in the market: (i) Nontrading effects (ii) Quotes in discrete units (ii) Clustering noise e.g. Prices tend to fall more often on whole-dollar multiples than on half-dollar multiples, etc. (iii) Bid/ask bounce effect Recorded stock prices can be at the bid or at the ask prices. Bid/ask price bouncing creates spurious correlation in returns.

90 Where did we go from here? Feasibility and robustness of high-frequency methods Estimation under microstructure noise" 1 Ultra high-frequency sampling will eventually recover the tick-by-tick data. 2 How frequently to sample? The higher sampling frequency, the smaller the theoretical standard error of the estimation methods (under absence of noise), but the higher the microstructure noise. 3 Need to analyze the performance of the methods towards microstructure noise (or other kind of noise).

94 Where did we go from here? Feasibility and robustness of high-frequency methods Low-frequency methods 1 Recently, there have been several estimation methods that does not require high-frequency data. 2 These methods are built on three facts: Log returns on equal-sized periods are independent with common distribution, say, f ( s), which depends on the Lévy density s. The distribution f has tractable Fourier transform": ψ(w s) = e iuw f (u s)du. We can estimate ψ using data; let ˆψ be this estimate. Find the Lévy density that approximate the closest the estimated ˆψ: ŝ = inf ψ( s) ˆψ. s

Modeling and estimating the dynamics of stock prices

What do the models need to model?

What is the purpose of the model?

What did the Black - Scholes model model?