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## Volatility By A.V. Vedpuriswar

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**VolatilityBy A.V. Vedpuriswar**June 12, 2014**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.**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**Estimating Volatility**• Exponentially weighted moving average model means weights decrease exponentially as we go back in time. n2 = 2n-1+ (1 - ) u2n-1 = [n-22 + (1- )un-22] + (1- )un-12 = (1- )[un-12 + un-22] + 2n-22 = (1-) [un-12 + u2n-2 + 2un-32 ] + 3 2n-3 • If we apply GARCH model, n2 = γVL+ un-12 + 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**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.**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.**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.**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.**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.”**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.**VIX movements**• Over its entire history, the median daily closing level of VIX has been 18.88. • 50% of time VIX closed between 14.60 and 23.66 (a range of 9.06 points). • 75% of the time VIX closed between 12.04 and 29.14 (a range of 17.10 points). • 95% of the time VIX closed between 11.30 and 37.22 (a range of 22.92 points). • The widest range experienced was 2008, with VIX closing between 18.16 and 63.31 (a range of 45.15 index points) about 90%. • The second widest range was in 1987.**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.**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 $30.50. Using the EWMA model, with λ = 0.94, calculate the updated estimate of volatility . • Solution ht = λσ2t-1 + ( 1 – λ) rt-12 • λ = .94 • rt-1 = ln [(30.50 )/ 30] = .0165 • ht = (.94) (.015)2 + (1-.94) (.0165)2 • Volatility = .01509 = 1.509 % 15**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. 1.378% as compared to 1%.**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)^2 + 0.94*(.01378)^2=.0001785 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%**Problem**n2 = Y VL + un-12 + 2n-1 Beginning price = 1040, Closing price = 1060 Most recent estimate of volatility = 0.01 Y VL = 0.000002, = 0.06, = 0.92 Find new variance estimate. Solution Return = 20/1040 = 0.01923 New variance estimate = 0.000002 + 0.06X.01923*.01923 + 0.92X .01*.01 = .0001162 New volatility estimate = 0.01078**ProblemSuppose 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%**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%**Implied Volatility**• Stock price = 51, Strike price = 50, Time = 1, Interest rate = 8% • We use Deriva Gem to work out Implied Volatility.**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**Solution**• Suppose that P is the probability of a favourable ruling. The expected price of Microsoft tomorrow is • 75p + 50 (1-p) • = 50 + 25p • This must be the price of Microsoft today. (We ignore the expected return to an investor over one day) Hence • 50 + 25p = 60 • Or p = 0.4, p being the risk neutral probability.**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 26.502. • 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 6.310. • The price today of a European call option with a strike price 50 is the weighted average of 26.502 and 6.310 or: • 0.4 x 26.502 + 0.6 x 6.310 = 14.387 • 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 = 14.387. • The implied volatility is 47.76%.**These calculations can be repeated for other strike prices.**The results are shown in the table below.