1. Baosheng Yuan and Kan Chen
Statistical analysis of high-frequency financial data and modeling of financial time series
Baosheng Yuan and Kan Chen
Department of Computational Science, FoS, National University of Singapore
11/30/2018

2. Outline Introduction Financial assets and stochastic process
Statistical tools
Data analysis
Non-Gaussian stochastic model
Analysis of simulation data
Model test: currency option pricing
Summary

3. Introduction (1) What’s the focus of this talk
Understand stock price dynamics: data analysis
Simulate stock price process: modeling
Option pricing: application of model simulation
General properties of stock price
Future return is unknown/uncertain
No known process can describe it
Larger volatility than Gaussian distribution
Return/volatility is clustered -temporally correlated

4. Introduction (2) Widely used approaches Our approach
Data analysis: Return probability distribution
Modeling: Assume an independent stochastic process
Option pricing: Patch-to-match methodology
To match Black-Scholes option price to market price by adjusting implied volatility
Our approach
Data analysis: Conditional return analysis + etc.
Modeling: assume prices are correlated
Option pricing: Simulate the underlying stock process directly, and evaluate the performance by realized profit.

5. Financial assets & stochastic process
Riskless: dS1(t)=r(t)S1(t)dt
Risky: dS2(t)=μ(S2,t)dt + ν(S2,t)dW
Where dW is an unknown stochastic process. Different assumption on dW leads to different model
Itô process
When {Wt: t ≥ 0} is assumed to be a Brownian motion.
Return of Financial asset
Discounted relative price change
R(t)=[S(t)-S(t-Δt)] D(t)/S(t-Δt)
where D(t) is a discounting factor, e.g. interest rate
Logarithm return:
X(t)= ln S(t) - ln S(t-Δt)

6. Analysis Tools Return density estimators
Histogram
Kernel estimators
Statistics – moments of different orders
Mean, variance, skewness, kurtosis
Hurst function
H(Δt)=E[max t<s<t+Δt{X(s)} – min t<s<t+Δt {X(s)}]
Where E[.] is a mathematical expectation
Conditional return *

7. Data analysis - Outline
Global behavior of stock return
Short-term price trend analysis
Trend number analysis
Clustered return analysis
Conditional return analysis
Return conditioned on previous return
Threshold based clustered return
Hurst exponent analysis

9. Data analysis (1) Global behavior of return Kurtosis: Skewness:
Variance:
Var[X] = E[(X-μ)2]
Mean:
μ=E[X]
where E[.] is mathematical expectation

10. (Statistical) moments of HSI
mean
S.D.( )
Skewness
Kurtosis
Gaussian ~
3.0
Δt=2mins
0.0008
Δt=10mins
0.0023
Δt=30mins
0.0042
8.7850
Δt=60mins
0.0059
7.3655
Δt=120mins
0.0086
6.5963
Δt=240mins
0.0126
5.9744

11. Data analysis(2) Global behavior of return
Mean: indiscriminative measure
Variance: >~ Δt (Gaussian: Var[X] ~ Δt):
 larger volatility than Gaussian
Skewness << 0 (Gaussian, symetric)
 fatter negative tails (asymmetric)
Kurtosis >> 3.0 (Gaussian)
 extreme events have higher probability than Gaussian
=>The price process is highly non-Gaussian

12. Data analysis (3) Short-term price trend analysis
Clustered return distribution
Definition: accumulative return for a sequence of returns of same signs
Clustered return over trend number
Trend number: number of time steps in a sequence in which all the returns are the same signs.
Average return per time step: clustered return over the trend number
Dependence of clustered return on trend number
Gaussian: independent
Real high-frequency data ?

15. Data analysis (4) Conditional return analysis
Return conditioned on previous return
Definition: return distribution conditioned on absolute return in previous period
where: X(t)= ln S(t) - ln S(t-Δt); and X1, X2 are the left and right boundary of previous absolute return.
Plot of distribution

21. Data analysis (5) Conditional return analysis
Conditional return distribution
Highly non-Gaussian for all time steps
SD of conditional return ~ absolute return in previous period
Conditional return distributions collapse into a universal curve;.
Conditional returns with different time steps also collapse into a universal curve – time scale free feature

22. Data analysis (6) Conditional return analysis
Threshold based clustered return
Definition: the length of price swing from a regional bottom to a regional top. The top and the bottom are defined such that the trend reversal between the regional top and bottom is less than R0
Distribution:

24. Data analysis (6) Conditional return analysis
Threshold based clustered return
Real data: power law tail with decaying factor ~ 2.0
Random walk: decaying factor ~ 2.7
=> Real financial data is highly non-Gaussian and temporally correlated

25. Data analysis (7) Hurst exponent analysis Hurst function
where E[.] is a mathematical expectation

28. Data analysis (7) Hurst exponent analysis
Gaussian: Hurst exponent ~ 0.5
Real data: Hurst exponent ~ 0.6 for small time scales and --> 0.5 for larger time scales
=> volatility correlation is time-scale dependent, the shorter the time scale is, the stronger the correlation is.

29. Data analysis - summary
Highly non-Gaussian:
Variance: >~ Δt  larger volatility than Gaussian
Skewness << 0 (Gaussian, symetric)
 fatter negative tails (assymetric)
Kurtosis >> 3.0 (Gaussian)
 extreme events have higher probability than Gaussian
Highly temporally correlated
SD of conditional return ~ absolute return in previous period  the larger the volatility now, the larger the volatility next
Volatility correlation has a decreasing dependence on time scales

30. Non-Gaussian stochastic model (1)
Assumption: returns are correlated
Model dynamics:
X(t+1) = X(t) + r(t) ± δ(t)
where r(t) is the intrinsic growth rate at time t and δ(t)= δ0γn(t) the magnitude of price (return) change due to trend.
n(t+1)=n(t) +1 if price moves in a trend; or
n(t+1)=n(t) -1 if price reverses the trend;

31. Non-Gaussian stochastic model(2)
Interest rate:
Applying a risk-neutral measure in binary tree we have:
r(t)=r0 –ln {P(t) exp( δt)+(1-P(t)) exp(-δr)} if X(t)-X(t-1)>0
r(t)=r0 –ln {P(t) exp( -δt)+(1-P(t)) exp(δr)} else
δt = δ0γn(t)+1 ; δr = δ0γn(t)-1 (1)
Where P(t) is the probability of trend
Volatility reversal:
P(t) =P0 – α (n(t)-n0)/(n(t)+n0) if n(t)> n0, or P0 else;
P0 = 1/(1+ γ2) (2)
where γ is a volatility basis, r0 is risk-free interest rate, α is a constant factor for curtaining the volatility from over-shooting, and n0 is the mean of trend number n(t).

32. Non-Gaussian stochastic model(3)
Return mean-reversal:
δx(t) = β (S(t)-S0)/(S(t)+S0) (3)
Where β is a constant factor for adjusting the magnitude of return, S(t) is the stock price at time t, and S0 is the “mean” of the price
Model with volatility-/mean-reversal features:
X(t+1) = X(t) + r(t) + δ(t) (4)
Where δ(t) is determined by:
+ δ0 (1-δx(t)) γn(t)+1 if X(t) –X(t-1) ≥ 0 and R(t)<P(t)
- δ0 (1+δx(t)) γn(t)-1 if X(t) –X(t-1) ≥ 0 and R(t) ≥P(t)
- δ0 (1+δx(t)) γn(t)+1 if X(t) –X(t-1) <0 and R(t)<P(t)
+ δ0 (1-δx(t)) γn(t)-1 if X(t) –X(t-1) <0 and R(t) ≥ P(t)
where R(t) ∈[0,1) is a random number generator. The model is fully described by Eq. (1)-(4).

33. Analysis of simulation data
Simulated stock price
Global return distribution
Conditional return distribution

38. Model test: currency option pricing (1)
European options
C(K,T)=EQ[e-(r-rf)T (ST-K)+]
P(K,T)=EQ[e-(r-rf)T (K- ST)+]
where K is strike price, ST stock price at maturity T; r and rf are interest rate for domestic and foreign currencies respectively; EQ[ .] is an expectation under risk-neutral measure;
The parameters:
Initial trend number n(0): estimated by simulation
Mean trend number n0: estimated by simulation.
α, β and γ(>1) are determined by experiment
δ0 is proportional to volatility of return to be simulated

39. Model test: currency option pricing (2)
Trading strategy:
buy the option only if the simulation price is higher than the market price
Currency option pricing results:
Profit/price
Our Model
Profit/Price
BLS Model
GBP Call+Put
0.5872
SWF Call+Put
0.1543
DMK Call
0.5499

40. Summary (1) Financial time series:
Characterized by “transient” and “recurrent” dynamics according to Hurst exponent
Returns are more volatile than Gaussian
Variance > Δt (small time scalse): more volatile
Kurtosis >> 3 (for Gaussian): extreme returns have larger probability than Gaussian process
Skewness < 0 (Gaussian): Large positive return moves towards Gaussian faster than large negative return does
Returns are temporally correlated
Future volatility is proportional to the current one (universal curve of conditional return distributions)
Correlation depends on time scale: the shorter the time scale is, the stronger the correlation is

41. Summary (2) Unique features of our model
Assume a non-Gaussian and correlated stochastic process
Incorporate short-time price trend and long-time mean-reversal
Price dynamics is easy to simulate and model structure is simple and intuitive
Capture important statistics observed in real data
Can be used directly in option pricing without using implied volatility approach

42. Thank You!