0. Volatility By A.V. Vedpuriswar
June 12, 2014

1. Basics of volatility Volatility is a huge issue in risk management.
Volatility is the key parameter in modeling market risk.
Volatility is essentially the standard deviation of daily portfolio returns.

2. Estimating Volatility
Calculate daily return u1 = ln Si / Si-1
Variance rate per day
We can simplify this formula by making the following simplifications.
ui = (Si – Si-1) / Si-1 ; ū = 0; m-1 = m
If we want to weight ; 2

3. Estimating Volatility
Exponentially weighted moving average model means weights decrease exponentially as we go back in time.
n 2 = 2n-1+ (1 - ) u2n-1
=  [n-22 + (1- )un-22] + (1- )un-12
= (1- )[un un-22] + 2n-22
= (1-) [un-12 + u2n-2 + 2un-32 ] + 3 2n-3
If we apply GARCH model,
n2 = γVL + un 2n-1
VL = Long run average variance rate, γ +  +  = 1.
If γ = 0,  = 1-, = , it becomes exponentially weighted model.
GARCH incorporates the property of mean reversion. 3

4. Measuring Volatility : The VIX
On March 26, 2004, the first-ever trading in futures on the VIX Index began on CBOE Futures Exchange (CFE).
As of February 24, 2006, it became possible to trade VIX options contracts.
The VIX is calculated and disseminated in real-time by the Chicago Board Options Exchange.
It is a weighted blend of prices for a range of options on the S&P 500 index.

5. What VIX implies
The VIX is quoted in percentage points and translates, roughly, to the expected movement in the S&P 500 index over the next 30-day period, which is then annualized.
For example, if the VIX is 15, this represents an expected annualized change of 15% over the next 30 days.
So the index option markets expect the S&P 500 to move up or down over the next 30-day period.

6. Volatility can mean movement in either direction
VIX is often called the "fear index“.
But a high VIX is not necessarily bearish for stocks.
Instead, the VIX is a measure of fear of volatility in either direction, including the upside.
If investors anticipate large upside volatility, they will not sell upside call stock options unless they receive a large premium.
Option buyers will be willing to pay such high premiums only if similarly anticipating a large upside move.
The general increase in upside stock option call prices raises the VIX.
VIX may also go up if there is a general increase in downside stock put option premiums.
This occurs when option buyers and sellers anticipate a likely sharp move to the downside.

7. Significance of VIX
High VIX means investors see significant risk that the market will move sharply, whether downward or upward.
Only when investors perceive neither significant downside risk nor significant upside potential will the VIX be low.
The new VIX is based on the S&P 500® Index (SPXSM), the core index for U.S. equities.
It estimates volatility by averaging the weighted prices of SPX puts and calls over a wide range of strike prices.

8. VIX components
The components of VIX are near- and next-term put and call options, usually in the first and second SPX contract months.
“Near-term” options must have at least one week to expiration.
This is to minimize pricing anomalies that might occur close to expiration.
When near-term options have less than a week to expiration, VIX “rolls” to the second and third SPX contract months.
For example, on the second Friday in June, VIX would be calculated using SPX options expiring in June and July.
On the following Monday, July would replace June as the “near-term” and August would replace July as the “next-term.”

9. VIX and portfolio insurance
Volatility technically means unexpected moves up or down.
But over time, the S&P 500 index option market has become dominated by hedgers.
Hedgers buy index puts when they are concerned about a potential drop in the stock market.
The more investors demand, the higher the price of portfolio insurance.
VIX reflects the price of portfolio insurance.

10. VIX movements
Over its entire history, the median daily closing level of VIX has been
50% of time VIX closed between and (a range of points).
75% of the time VIX closed between and (a range of points).
95% of the time VIX closed between and (a range of points).
The widest range experienced was 2008, with VIX closing between and (a range of index points) about 90%.
The second widest range was in 1987.

14. India VIX
India VIX is a volatility index based on the index option prices of NIFTY.
India VIX is computed using the best bid and ask quotes of the out-of-the-money near and mid-month NIFTY option contracts which are traded on the F&O segment of NSE.
India VIX indicates the investor’s perception of the market’s volatility in the near term.
The index depicts the expected market volatility over the next 30 calendar days. i.e. higher the India VIX values, higher the expected volatility and vice-versa.

15. Problem
The current estimate of daily volatility is 1.5%. The closing price of an asset yesterday was $30. The closing price of the asset today is $ Using the EWMA model, with λ = 0.94, calculate the updated estimate of volatility .
Solution
ht = λ σ2t-1 + ( 1 – λ) rt-12
λ = .94
rt = ln [(30.50 )/ 30]
= .0165
ht = (.94) (.015)2 + (1-.94) (.0165)2
Volatility = = % 15

16. Problem
On Tuesday, return on a stock was 4%. Volatility estimate for Tuesday was 1%. Find volatility estimate for Wednesday using  of 0.94.
Solution
Variance estimate for Wednesday=(1-0.94)*(.04)^2 +(0.94)*(.01)^2 = 0.019%.
Std. dev = √(.019%)=1.378%=.01378
Tuesday volatility (Std. Dev.) estimate was 1%.
Actual return on Tuesday was 4%.
Therefore, volatility estimate for Wednesday is estimated upwards than Tuesday i.e % as compared to 1%.

17. Problem
Continuing the previous example, volatility estimate for Wednesday was 1.378%. Assume that actual return on Wednesday was 0%. What is the variance estimate for Thursday?
Solution
Variance estimate for Thursday
= (1-0.94)*(0)^ *(.01378)^2=
Stdev. =0.0134
In very short-term, like daily returns, estimated volatility is the expected return.
Since latest return of 0% was lesser than estimated volatility (and estimated return) of 1.378%, volatility for next day is revised downward from 1.378% to 1.34%

18. Problem
n2 = Y VL + un 2n-1
Beginning price = 1040, Closing price = 1060
Most recent estimate of volatility = 0.01
Y VL = ,  = 0.06, = 0.92
Find new variance estimate.
Solution
Return = 20/1040 =
New variance estimate
= X.01923* X .01*.01 =
New volatility estimate =

19. The average variance rate is
Problem Suppose that the annualised volatility of an asset will be 20% from month 0 to 6, 22% from month 6 to month 12, and 24% from month 12 to 24. What volatility should be used in Black-Scholes to value a 2-year option?
Solution
The average variance rate is
The volatility used should be
or 22.56%

20. Implied volatility Problem Option price = 4; Stock price = 45
Strike price = 50; Interest rate = 8%
Time to maturity = 1 year
Calculate implied volatility.
Solution
We use Deriva Gem
Answer : 25.12%

21. Implied Volatility
Stock price = 51, Strike price = 50, Time = 1, Interest rate = 8%
We use Deriva Gem to work out Implied Volatility.
Option Price
Implied Volatility 5
7.51 6
15.36 7
21.39 8
27.07 9
32.59 10
38.05

22. Problem
Suppose that the result of a major lawsuit affecting Microsoft is due to be announced tomorrow. Microsoft’s stock price is currently $60.
If the ruling is favourable to Microsoft, the stock price is expected to jump to $75.
If it is unfavourable, the stock is expected to fall to $50. What is the risk- neutral probability of a favourable ruling?
Assume that the volatility of Microsoft’s stock will be 25% for 6 months after the ruling if the ruling is favourable and 40% if it is unfavourable.
Calculate the relationship between implied volatility and strike price for 6-month European options on Microsoft today.
Microsoft does not pay dividends. Assume that the 6-month risk-free rate is 6%. Consider call options with strike price of 30, 40, 50, 60, 70 and 80.
Ref : John C Hull, Options, Futures and Other Derivatives

23. Solution
Suppose that P is the probability of a favourable ruling. The expected price of Microsoft tomorrow is
75p + 50 (1-p)
= p
This must be the price of Microsoft today. (We ignore the expected return to an investor over one day) Hence
p = 60
Or p = 0.4, p being the risk neutral probability.

24. If the ruling is favourable, the volatility, σ, will be 25%
If the ruling is favourable, the volatility, σ, will be 25%. Other option parameters are S0 = 75, r = 0.06, and T = 0.5.
If K = 50, the price of a European call option as calculated by Black Scholes is
If the ruling is unfavourable, the volatility, σ will be 40%.
Other option parameters are S0 = 50, r = 0.06, and T = 0.5.
If K = 50, the price of a European call option is
The price today of a European call option with a strike price 50 is the weighted average of and or:
0.4 x x =
The Black Scholes equation can now be used to calculate the implied volatility when the option has this price.
S0=60,K=50, T = 0.5, r = 0.06 and c =
The implied volatility is 47.76%.

25. Call option Price Favourable outcome
These calculations can be repeated for other strike prices. The results are shown in the table below.
Strike
Price
Call option Price Favourable outcome
Call option price Unfavourable outcome
Weighted
Implied
Volatility 30
45.887
21.001
30.955
46.67 40
36.182
12.437
21.935
47.78 50
26.502
6.310
14.387
47.76 60
17.171
2.826
8.564
46.05 70
9.334
1.161
4.430
43.22 80
4.159
0.451
1.934
40.36