1. Option-Implied Indicators for Market Stress
Mentor: Daniel Linders
Bansi Padalia
Tami Beecroft
Biwen Ling
Haocheng Wu
Yuxin Wei
Yun Xia

2. Agenda Introduction Introduction to Financial Stress
Overview of Financial Stress Indicators
Detailed Explanation of HIX
R Pre-processing
HIX Algorithm and Calculation
Conclusion and Future Research Implication
Questions

3. Financial Stress Definition
Financial stress refers to disruptions to the normal functioning of financial markets
Symptoms
Increased uncertainty about value of asset/behavior of investors
Increased asymmetry of information
Decreased Willingness to hold risky assets
Decreased Willingness to hold illiquid assets

4. Interested Parties Investors Gives idea of where the market is
Helps investors diversify their portfolios and mitigate risk accordingly 
Governments
Financial stress has been seen to have economic spillover onto the market 
Provides early warning to take action.

5. Indicators Volatility Credit
 Equity Valuation
Funding 
Safe Assets
Indicators are used to measure financial stress. Together, they create an index.
The Office of Financial Research (33 indicators), classifies their indicators as such:
OFR     FSI

6. VIX Chicago Board Options Exchange (CBOE), the Volatility Index
Measurement of implied volatility within the market

7. The definition of HIX Herd behavior Index
HIX is defined as the ratio of an option-based estimate of the risk-neutral variance and an option-based estimate of the corresponding variance in case of the extreme single factor market situation.
Assumptions:
    Market is arbitrage free and exists a pricing  measure Q.
     The interest rate r is deterministic and constant.

8. Definition: Comonotonicity
Several characterizations exist for the notion of comonotonicity:
Where
Then we define the comonotonic index:

9. R – Initial Preparation
List of component straddle, stored as data frame
Data frame 1, FB
Data frame 2, GS
Data frame 3, MMM
Number of component: X
List of component: FB, GS, MMM, … Stock(x)
Stock MMM
Stock FB
Stock GS
Capture data from Yahoo finance using rvest package

10. Data Before Cleaning
Source: yahoo finance

11. R Data Preparation
Step 1: Find the interaction of call and put price curve
Step 2: Smooth the empirical cumulative density function (ECDF) curve and make it convex

12. Option implied Interest Rate
Use 4 pairs of option that closed to the interaction point, by minimizing errors generated by
Option price implied interest rate: 3.5% on Dec. 6th, 2018.

13. Algorithm and Calculation
Step 1:
Find market option prices and determine the interest rate.
Build ECDF and option curves for each stock and for Dow Jones Index
Step 2:
Build the comonotonic ECDF
Calculate the comonotonic option price
Step 3:
Calculate HIX
Regular 30-days HIX

14. Algorithm: Numerator
In order to do the calculation of real market data, we do the approximation:
Q[Ki+1]
Q[Ki]
Var[S]

15. Asymptotic Equivalence: Empirical Function
We can not get the prices of call options and put options of any strike K, so we just use empirical function to replace cumulative distribution function：
Hobson et al. (2005) and Chen et al. (2008)

16. Algorithm: Denominator
Comonotonic ECDF
We introduce the comontonic sum based on the empirical marginal distributions:
The cumulative distribution function of 
1. Determine a set A:
2. We consider the value of  
3. For ,

17. Algorithm: Denominator
Comonotonic ECDF

18. Algorithm: Denominator
Comonotonic option prices
Comonotonic Call price at strike K
Parameters:
Hobson et al. (2005) and Chen et al. (2008)

19. Algorithm: Denominator
Comonotonic option prices

20. Algorithm: Denominator
Similar to the process, we do the calculation of denominator:

21. The HIX formula Calculate regular 30 days HIX on 2018-12-11
HIX[10d] = , Exp.Date =
HIX[38d] = , Exp.Date =
HIX[30d] = 

22. Conclusion Introduced financial stress indicators Compared VIX and HIX
Calculated HIX
Theorical formulas
Implementation in R
Limitations
Missing data
Data cleaning conditions

23. Data Before Cleaning
Source: yahoo finance

24. Future Research Implication
Find a better way in data cleaning process
 Investigate the behavior of the HIX during several extreme market conditions
 Investigate patterns of HIX and predict HIX in the future

25. Q&A

26. Thank you!

27. Prove: Area -> Variance

28. List of component straddle, stored as data frame
R Initial Data Sampling
A list of data frames that contains information about each stock
List of component straddle, stored as data frame
Data frame 1, FB
Data frame 2, GS
Data frame 3, MMM
Stock 1, e.g. FB
Last Price of option put or call at strike K
Strike
Volume
Open interest
% change in price
Stock 2, e.g. GS
Stock 3 …
……………

29. R Pre-Processing Important packages: rvest
rvest helps you scrape information from web pages.
Used to capture option prices as tables from Yahoo finance
quantmod
Quantitative financial modeling and trading framework.
Used to pull real-time market data for stocks with tickers specified

30. Data Cleaning
Data cleaning
Source: yahoo finance

31. Algorithm: Denominator
Comonotonic ECDF
Formula:
Example: K=190, try probability
Result:

32. Algorithm: Denominator
Comonotonic option prices
Formula:
Calculation:
Result:

33. HIX & VIX
Daniel Linders. (2015)