Volatility By A.V. Vedpuriswar

Transcription

1 Volatility By A.V. Vedpuriswar June 21, 2018

2 Basics of volatility Volatility is the key parameter in modeling market risk. Volatility is the standard deviation of daily portfolio returns. 1

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

4 Estimating Volatility Exponentially weighted moving average model means weights decrease exponentially as we go back in time. n 2 = 2 n-1+ (1 - ) u 2 n-1 = [ n-22 + (1- )u n-22 ] + (1- )u n-1 2 = (1- )[u n-12 + u n-22 ] + 2 n-2 2 = (1- ) [u n-12 + u 2 n u n-32 ] n-3 If we apply GARCH model, n2 = γv L + u n n-1 V L = Long run average variance rate, γ + + = 1. If γ = 0, = 1-, =, it becomes exponentially weighted model. GARCH incorporates the property of mean reversion. 3

5 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. 4

6 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. 5

7 Volatility can mean movement in either direction This occurs when option buyers and sellers anticipate a likely sharp 6 move to the downside. 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.

8 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. 7

9 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. 8

10 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. 9

11 VIX movements Over its entire history, the median daily closing level of VIX has been % of time VIX closed between and (a range of 9.06 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

12 VIX Chart 11

13 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. 12

14 Problem A stock s daily closing prices (in $) for the past 10 days have been 50, 49, 48, 47, 48, 49, 50, 51, 50, 49. Calculate the daily volatility. If a trader has invested $ 1 million in this stock, what is the 95% daily VAR? t Price Returns Square of Returns Total Mean = /9 = Variance =.00372/9 - (.00224) 2 = Std devn =

15 Infosys Volatility Date Opening price Return 5/23/ /24/ /25/ Std Dev 5/28/ /29/ /30/ /31/ /1/ /4/ /5/ /6/ /7/ /8/ /11/ /12/ /13/ /14/ /15/ /18/ /19/ /20/ /21/ /22/

16 BSE Sensex Volatility Date Opening price Return 23-May May May Std Deviation 28-May May May May Jun Jun Jun Jun Jun Jun Jun Jun Jun Jun Jun Jun Jun Jun Jun Jun

17 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 Variance = λ σ 2 t-1 + ( 1 λ) r t-1 2 λ =.94 r t-1 = ln [(30.50 )/ 30] =.0165 Variance = (.94) (.015) 2 + (1-.94) (.0165) 2 = Volatility = = % 16

18 Problem On Tuesday, return on a stock was 4% and volatility estimate was 1%. Find volatility estimate for Wednesday if = Solution Variance estimate for Wednesday=(1-0.94)*(.04)^2 +(0.94)*(.01)^2 = = = 0.019%. Std. dev = (.00019) = = 1.378% Tuesday volatility (Std. Dev.) estimate was 1%. Actual return on Tuesday was 4%. Therefore, volatility estimate for Wednesday is more than that for Tuesday i.e % as compared to 1%. 17

19 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= Std Dev. = In the 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

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

21 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

22 Problem on 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

23 Problem on 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

24 Problem on Implied Volatility 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 riskneutral 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

25 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

26 If the ruling is favourable, the volatility, σ, will be 25%. parameters are S 0 = 75, r = 0.06, and T = 0.5. Other option If K = 50, the price of a European call option Scholes is as calculated by Black If the ruling is unfavourable, the volatility, σ will be 40%. Other option parameters are S 0 = 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. S 0 =60,K=50, T = 0.5, r = 0.06 and c = The implied volatility is 47.76%. 25

27 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 Price Implied Volatility

28 Volatility: Smile, Term Structure and Surface Item Description Volatility smile It is the relationship between implied volatility and strike price for options with a certain maturity. Volatility term structure The variation of implied volatility with the time to maturity of the option. The volatility term structure tends to be downward sloping when volatility is high and upward sloping when it is low. Volatility Surface The implied volatility as a function of the strike price and time to maturity is known as a volatility surface. 27