Volatility, Volatility Smiles, Implied Volatility and VIX

Presented By:

Isha Aggarwal (2503) Varun Kalia(2523) Juhi Lakhotia(2534) Vaidahi Singhal (2547)

Sandeep Singh(2550) Madhur Malik (2557) Mehak Sharma (2558) Saurabh Patel (25 )

Option Pricing Basics

Option premiums have two main ingredients: intrinsic value and time value

Intrinsic value is an option's inherent value, for ex. if you own a $50 call option on a stock that is trading at $60, this means that you can buy the stock at $50 and immediately sell it in the market for $60. The intrinsic value is $10

So, the only factor that influences an option's intrinsic value is the underlying stock's price and the option's strike price

If this option is priced at $14, which is $4 more than its intrinsic value. This is where time value comes into play.

Time value is the additional premium that is priced into an option, which represents the amount of time left until expiration.

The price of the option is also affected by volatility.

Volatility Measure of dispersion of an asset price about its mean level over a fixed time interval. Careful modelling crucial for the valuation of options and of portfolios containing options.

The Effect of Volatility

Option Type: Call Option Future Date: T Strike Price: Rs. 50 Value of Option at Maturity: Max(Spot Price – Strike Price,0)

Stock Price 30 40 50 60 70

Payoff 0 0 0 10 20

Stock Price 40 45 50 55 60

Payoff 0 0 0 5 10

High Volatility Scenario Low Volatility Scenario

Mean Stock Price: Rs. 50 Standard Deviation: 14.14 Average Option Payoff: Rs. 15

Mean Stock Price: Rs. 50 Standard Deviation: 7.07 Average Option Payoff: Rs. 7.5

Effect of Volatility Contd.

The higher the volatility, higher the probability that option payoff will at maturity will be greater than the strike price.

In case the prices are lower than the strike price, payoff to the option holder cannot fall below 0.

Asymmetry of payoffs due to nature of contract implies that volatility is of value to the option holder at the time of maturity.

Historical Volatility

Calculation

Return on an asset (rt) = (Pt-Pt-1)/Pt-1

Average return from period t=1 to period t=T

R = ∑rt / T

Variance = ∑(rt – R)2/T-1 Standard Deviation = (Variance)1/2

Assumptions: No intermediate cash flows Independent returns: today’s return is high or low reveals nothing about

tomorrow’s return

Historical Volatility – The Right Measure?

• While historical volatility provides computational convenience, it may be an inaccurate measure of future volatility expected to prevail over the life of an option.

• Future volatility needs to be modelled to give an estimate of risk associated with options

Models for Measuring Volatility

Volatility Models

Deterministic

GARCHExponentially

Weighted Moving Average

Stochastic

With Closed Form Solution

Without Closed Form Solution

Deterministic models assume that volatility can be perfectly predicted from its history and other observable information.Stochastic volatility implies that the future level of the volatility cannot be perfectly predicted using information available today.

Stochastic Models

Volatility is driven by a random source that is

different from the random source driving the asset

returns process, although the two random sources may be correlated with each other.

An investor in the options market bears the additional

risk of a randomly evolving volatility.

Deterministic Models

An investor can hedge the risk from

the asset price by trading an option and a risk-free asset

based on a risk exposure computed using an option

pricing formula.

The investor incurs only the risk from a randomly evolving

asset price.

Generalized Autoregressive Conditional Heteroscedasticity (GARCH)

Makes use of information on past prices to update the current asset volatility.

The term autoregressive refers to the element of persistence in the modelled volatility and the term conditional heteroscedasticity describes the presumed dependence of current volatility on the level of volatility realized in the past.

GARCH places greater weight on more recent squared returns than on more distant squared returns; consequently, ARCH models are able to capture volatility clustering.

Variance is calculated from a long run average variance rate, VL, as well as past variance and returns.

Equation for GARCH:

σ2n = γVL + αu2

n-1 + βσ2n-1; γ + α + β = 1

For computational purpose we estimate the equation:

σ2n = ψ+ αu2

n-1 + βσ2n-1 where ψ = γVL

γ = 1 – α - β

VL = ψ/ γ

Generalized Autoregressive Conditional Heteroscedasticity (GARCH)

It can be seen that variance calculated from the GARCH model follows a mean reversal property. Over time, the variance tends to get pulled back to the long term average of VL.

Exponentially Weighted Moving Average Method

Weights assigned to past returns decrease exponentially as we move back through time.

This weighing scheme leads to the following formula for updating volatility estimates:

σ2n = λσ2

n-1 + (1 - λ)u2n-1

For large m, the term λmσ2

m-n becomes subsequently small to be ignored with weight assigned to the u2

n-I equal to (1 – λ)λi-1

This approach can be used to track changes in volatility.

Stochastic Volatility

Future level of the volatility cannot be perfectly predicted using information available today.

Popularity grew because distributions of the asset returns exhibit fatter tails than those of the normal distribution.

Consistent with the fat tails of the distribution implying for eg. that out of the money options would be underpriced by the Black Scholes Model.

Risk under this model

Random stochastic model poses two kinds of risk : risk from the asset price and volatility risk ( volatility risk can’t be hedged). Investors thus require “Volatility Risk premium”.

Asset price and its volatility are not correlated under

stochastic models, thus the change in volatility alone can change the option price.

This model has a “regressing to the mean” property.

Stochastic Volatility without Closed Form Solution

Based on the assumption that the risk premium of volatility is zero - that is, the volatility risk is not priced in the options market—and that volatility is uncorrelated with the returns of the underlying asset.

Wiggins, who also assumed a zero-volatility risk premium, found that the estimated option values under stochastic volatility were not significantly different from Black-Scholes values, except for long maturity options.

Although computers allow for such calculations but this model largely remains impractical for determining hedge ratios.

Stochastic Volatility with Closed Form Solution

The model gives closed-form solutions not only for option prices but also for the hedge ratios like the deltas and the vegas of options.

In this model, the asset returns rt and the variance s2 t are assumed to evolve through time as

where e1,t and e2,t are two standard normal random variables that could be correlated with each another, either positively or negatively, with a correlation coefficient, r.

Square-Root Volatility Process

In this model, the variance evolves through time in such a way that its long-run average level is measured by u and the speed with which it is pulled toward this long-run mean is measured by k, also known as the mean-reversion coefficient.

The variable g is a measure of the volatility of variance. If g is zero, the model simplifies to a time-varying deterministic-volatility model.

The particular nature of the process ensures that volatility “reflects” away from zero: if volatility ever becomes zero, then the nonzero k ensures that volatility will become positive

Implied Volatility

The price of time is influenced by time until expiration, stock price, strike price, interest rates, and implied volatility.

Implied volatility represents the expected volatility of a stock over the life of the option, and as expectations change, option premiums react appropriately.

Implied volatility is directly influenced by the supply and demand of the underlying options and by the market's expectation of the share price's direction

As expectations rise, or as the demand for an option increases, implied volatility rise. Options that have high levels of implied volatility will result in high-priced option premiums.

As the market's expectations decrease, or demand for an option diminishes, implied volatility decrease. Options with lower levels of implied volatility result in lower option prices

Option Pricing Basics

Implied Volatility - Introduction

The implied volatility of an option contract is the volatility of the price of the underlying security that is implied by the market price of the option based on an option pricing model.

This is the estimated volatility of a security's price. In general, implied volatility increases when the market is bearish and decreases when the market is bullish. This is due to the common belief that bearish markets are more risky than bullish markets.

Implied volatility is a forward-looking measure, and differs from historical volatility because the latter is calculated from known past returns of a security.

Factors Affecting Implied Volatility

Implied volatility of options is determined by market maker's assessment of public expectations regarding events that might change the value of an option.

In one sided markets, market makers are charged with the obligation to sell options to buyers in order maintain liquidity. They then increase the value of the options through increasing their assessment of implied volatility so as to reap profit. And when market is selling off options, market makers charged with the obligation to buy, lower the price of the options by lowering their assessment of implied volatility.

Many option traders wonder who is buying the options when everyone is selling and who is selling the options when everyone is buying. In such one sided markets, option traders are actually dealing directly with market makers.

Calculation of implied volatility

It is the volatility that, when used in a given pricing model, yields a theoretical value for the option equal to the current market price of that option

The value of an option depends on an estimate of the future realized price volatility, σ, of the underlying.

That value of σ which makes theoretical value of option equal to market price of the option is implied volatility

A European call option struck at $50 of XYZ, and the stock is trading at $51.25 and the current market price of option is $2.

Using Black–Scholes model, the volatility implied by the market price is 18.7%, or if we apply the implied volatility back into the pricing model, we will generate a theoretical value of $2

How Implied Volatility Affects Options

The success of an options trade can be enhanced by being on the right side of implied volatility changes. For example, if you own options when implied volatility increases, the price of these options climbs higher.

Each listed option has a unique sensitivity to implied volatility. For example, short-dated options will be less sensitive to implied volatility, while long-dated options will be more sensitive, because long-dated options have more time value priced into them, while short-dated options have less.

Also, each strike price will respond differently to implied volatility changes. Options with strike prices that are near the money are most sensitive to implied volatility changes, while options that are further in the money or out of the money will be less sensitive to implied volatility changes.

Measure of Implied Volatility - Vega

Vega is the change in the value of an option for a 1% increase in implied volatility. The vega of a long option position (both calls and puts) is always positive.

Vega and time to expiration As time passes, option vega decreases. Time amplifies the effect

of volatility changes. As a result, vega is greater for long-dated options than for short dated options.

Vega and volatility As volatility falls, vega decreases for in-the-money and out-of-the-

money options; vega is unchanged for at-the-money options.

Measure of Implied Volatility - Vega

Vega and stock volatility GE stock will have lower volatility values than Apple Computer because

Apple's stock is much more volatile than GE’s .

Each stock has a unique implied volatility range, these values should not be compared to another stock's volatility range.

Implied volatility should be analyzed on a relative basis. Look at the peaks to determine when implied volatility is relatively high, and troughs when implied volatility is relatively low. Implied volatility moves in cycles. High volatility periods are followed by low volatility periods, and vice versa.

Buy undervalued options and sell overvalued options

Determine if implied volatility is high or low and whether it is rising or falling. As implied volatility reaches extreme highs or lows, it is likely to revert back to its mean. As implied volatility increases, option premiums become more expensive. As implied volatility decreases, options become less expensive.

If an options yields expensive premiums due to high implied volatility, understand the reason. After the event occurs, implied volatility will collapse and revert back to its mean.

When you see options trading with high implied volatility levels, consider selling strategies including covered calls, naked puts, short straddles and credit spreads.

When you discover options that are trading with low implied volatility levels, consider buying strategies including buying calls, puts, long straddles and debit spreads

Bullish On Implied Volatility

These option strategies allows you to profit from an increase in implied volatility without any increase in the stock.

Long Straddle: Buy Call + Buy Put

One should use a long straddle when one is confident of a move in the underlying asset but is uncertain as to which direction it may be. For ex. corporate announcement, court verdict, earnings announcement etc.

Establishing a long straddle simply involves the simultaneous purchase of an at the money call option and a put option . A call option allows you unlimited profit to upside and limited loss to down side and a put option allows you unlimited profit to downside and limited loss to upside. Combine them both and you will have a long straddle which profits in both up and down market.

Bullish On Implied Volatility

Long Strangle: Buy OTM call + Buy OTM put

This is another way of gaining from a move in the underlying asset, but is uncertain as to which direction it may be.

Establishing a strangle simply involves the simultaneous purchase of an out of the money (OTM) call option and an OTM put option on the underlying asset

An out of the money call option allows you unlimited profit to upside when the stock moves higher than the strike price with limited loss to down side. An out of the money put option allows you unlimited profit to downside when the underlying stock moves lower than the strike price with limited loss to upside. Combine them both and you will have a strangle which profits when the underlying stock moves up or down

Bullish On Implied Volatility

Short Condor: Sell One Far ITM + Buy ITM + Buy One OTM + Sell One Far OTM

Short Condor Spread is used when one expects the price of the underlying asset to make a quick break to either upside or downside.

There are two ways to establish a Short Condor Spread - Call Short Condor Spread uses only call options, and Put Short Condor Spread uses only put options.

Either way performs the same as long as the underlying asset breaks above or below the upper or lower breakeven points upon option expiration. The composition of both kinds of Short Condor Spread is the same.

Bullish On Implied Volatility

Short Butterfly Spread One should use a Short Butterfly Spread when one expects the price of the

underlying asset to either rise or ditch greatly. There are 3 option trades to establish for this strategy : 1. Sell To Open X

number of In The Money Call Options. 2. Sell To Open X number of Out Of The Money Call Options. 3. Buy To Open 2X number of At The Money Call Options.

This can also be executed by using Put Options instead of Call Options with the same effects.

Gamma Neutral Hedging This is the construction of options trading positions that are hedged such that

the total gamma value of the position is zero or near zero, resulting in the delta value of the positions remaining stagnant irrespective of underlying movements.

Bearish On Implied Volatility

Short Straddle: Sell Call + Sell Put It is used when one is of the opinion that an underlying stock will stay

sideways until option expiration. Establishing a short straddle simply involves the simultaneous writing of a

call option and a put option on the underlying asset, at the same strike price and expiration date.

A short call option allows you to profit when the underlying asset is sideways or down and a short put option allows you to profit when the underlying asset is sideways or up.

These option strategies allows you to profit from a decrease in implied volatility without any decrease in the stock.

Bearish On Implied Volatility

Short Strangle: Sell OTM Call + Sell OTM Put One should use a short strangle when one is confident that the

underlying asset will stay within a tight trading range or stay stagnant until expiration .

Writing an out of the money call option allows you to profit when the option expires after the underlying stock fails to move above its strike price. Similarly, an out of the money put option allows you to profit when the option expires after underlying stock fails to move lower than its strike price.

Combine them both and you will have a short strangle which profits when the underlying stock neither moves up nor down beyond the strike price of the respective options.

Bearish On Implied Volatility

Long Butterfly Spread

One should use a Long Butterfly Spread when one expects the price of the underlying asset to change very little over the life of the option contract.

There are 3 option trades to establish for this strategy : 1. Buy X number of In The Money Call Options. 2. Buy X number of Out Of The Money Call Options. 3. Sell 2X number of At The Money Call Options.

This can also be executed by using Put Options instead of Call Options with the same effects

Bearish On Implied Volatility

Long Condor: Buy One Far ITM + Sell One ITM + Sell One OTM + Buy One Far OTM

Long Condor Spread is used when one expects the price of the underlying asset to remain stagnant.

There are two ways to establish a Long Condor Spread - Call Short Condor Spread uses only call options, and Long Put Condor Spread uses only put options.

Either way performs the same as long as the underlying asset remains within the profitable price range upon option expiration.

Volatility Index (VIX)

VOLATILITY INDEX

• Volatility Index is a measure of market’s expectation of volatility over the near term.

• Volatility Index is a good indicator of the investors’ perception on how volatile markets are expected to be in the near term.

About VIX VIX is a trademarked ticker symbol for the Chicago Board Options Exchange Market

Volatility Index, a popular measure of the implied volatility of S&P 500 index options.

The idea of a volatility index, and financial instruments based on such an index, was first developed and described in 1986, and first published in "New Financial Instruments for Hedging Changes in Volatility," appearing in the July/August 1989 issue of Financial Analysts Journal.

In 1992, the CBOE commissioned Prof. Robert Whaley to create a stock market volatility index based on index option prices.

Based on the history of index option prices, Prof. Whaley computed daily VIX levels in a data series commencing January 1986, available on the CBOE website. Prof. Whaley's research for the CBOE appeared in the Journal of Derivatives.

The VIX is quoted in percentage points and translates, roughly, to the expected movement in the S&P 500 index over the upcoming 30-day period, which is then annualized.

About VIX (Contd.)

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; thus one can infer that the index option markets expect the S&P 500 to move up or down 15%/√12 = 4.33% over the next 30-day period

Even though the VIX is quoted as a percentage rather than a dollar amount there are a number of VIX-based derivative instruments in existence, including:

VIX Futures contracts which began trading in 2004 Exchange listed VIX options which began trading in February 2006 VIX futures based exchange traded notes and exchange traded funds High VIX readings mean investors see significant risk that the market will move

sharply, whether downward or upward. The highest VIX readings occur when investors anticipate that huge moves in either direction are likely. Only when investors perceive neither significant downside risk nor significant upside potential will the VIX be low.

About VIX

Although the VIX is often called the "fear index", a high VIX is not necessarily bearish for stocks.

Instead, the VIX is a measure of market perceived volatility in either direction, including to the upside. In practical terms, when investors anticipate large upside volatility, they are unwilling to 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 resulting aggregate of increases in upside stock option call prices raises the VIX just as does the aggregate growth in downside stock put option premiums that occurs when option buyers and sellers anticipate a likely sharp move to the downside.

When the market is believed as likely to soar as to plummet, writing any option that will cost the writer in the event of a sudden large move in either direction may look equally risky.

Uses of VIX

Tracking the prices of financial options on VIX, gives a numeric measure of how pessimistic or optimistic market actors at large are. A low number in this index indicates a prevailing optimistic or confident investor outlook for the future, while a high number indicates a pessimistic outlook. By comparing the VIX to the major stock-indexes over longer periods of time, it is evident that peaks in this index generally present good buying opportunities.

Uses – Hedging Help in taking trading positions Spot mispriced options Sector specific hedging

About India VIX India VIX is a volatility index based on the Nifty 50 Index Option prices.

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.

A volatility figure (%) is calculated which indicates the expected market volatility over the next 30 calendar days.

Higher the implied volatility higher the India VIX value and vice versa.

India VIX uses the computation methodology of CBOE, with suitable amendments to adapt to the NIFTY options order book using cubic splines, etc

"VIX" is a registered trademark of the CBOE and Standard & Poor’s has granted a license to NSE, with permission from CBOE, to use such mark in the name of the India VIX and for purposes relating to the India VIX.

Factors considered are as follows:

Time to expiry: Computed in minutes instead of days.

Interest Rate:The relevant tenure NSE MIBOR rate (i.e 30 days or 90 days) as risk-free interest rate.

The forward index level:India VIX is computed using out-of-the-money option contracts. Out-of-the-money option contracts are identified using forward index level. The forward index level helps in determining the at-the-money (ATM) strike which in turn helps in selecting the option contracts which shall be used for computing India VIX. The forward indexlevel is taken as the latest available price of NIFTY future contract for the respective expiry month.

Bid-Ask QuotesBest bid and ask quotes of OTM option contracts are used for computation of India VIX. In respect of strikes for which appropriate quotes are not available, values are arrived through interpolation using a statistical method namely “Natural Cubic Spline” After identification of the quotes, the variance (volatility squared) is computed separately for near and mid month expiry.

Computation of India VIX

Volatility Smiles

Volatility Smile

It is the relationship between implied volatility and strike price for options with a certain maturity

The volatility smile for European call options should be exactly the same as that for European put options

The same is at least approximately true for American options

Why the Volatility Smile is the Same for European Calls and Put

Put-call parity p + S0e−qT = c +K e–rT holds for market prices (pmkt and cmkt) and for Black-Scholes-Merton prices (pbs and cbs)

As a result, pmkt− pbs=cmkt− cbs

When pbs = pmkt, it must be true that cbs = cmkt

It follows that the implied volatility calculated from a European call option should be the same as that calculated from a European put option when both have the same strike price and maturity

The Volatility Smile for Foreign Currency Options

At-the-money options tend to have lower volatilities that in- or out-of-the-money options

ImpliedVolatility

StrikePrice

Implied Distribution for Foreign Currency Options

Lognormal

Implied

Properties of Implied Distribution for Foreign Currency Options

Both tails are heavier than the lognormal distribution

It is also “more peaked” than the lognormal distribution

Possible Causes of Volatility Smile for Foreign Currencies

Exchange rate exhibits jumps rather than continuous changes.

Volatility of exchange rate is stochastic

(One reason for a stochastic volatility in the case of equities is the relationship between volatility and leverage)

Historical Analysis of Exchange Rate Changes

Real World (%) Normal Model (%)

>1 SD 25.04 31.73

>2SD 5.27 4.55

>3SD 1.34 0.27

>4SD 0.29 0.01

>5SD 0.08 0.00

>6SD 0.03 0.00

The Volatility Smile for Equity Options (Figure 19.3, page 414)

ImpliedVolatility

Strike

Price

Implied Distribution for Equity Options

Lognormal

Implied

Properties of Implied Distribution for Equity Options

The left tail is heavier than the lognormal distribution

The right tail is less heavy than the lognormal distribution

Reasons for Smile in Equity Options

Leverage

Crashophobia

Other Volatility Smiles?

What is the volatility smile if:

True distribution has a less heavy left tail and heavier right tail

True distribution has both a less heavy left tail and a less heavy right tail

Ways of Characterizing the Volatility Smiles

Plot implied volatility against K/S0 Plot implied volatility against K/F0

Note: traders frequently define an option as at-the-money when K equals the forward price, F0, not when it equals the spot price S0

Plot implied volatility against delta of the option Note: traders sometimes define at-the money as a call with

a delta of 0.5 or a put with a delta of −0.5. These are referred to as “50-delta options”

Volatility Term Structure

In addition to calculating a volatility smile, traders also calculate a volatility term structure.

This shows 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

Example of a Volatility Surface

K/S0 0.90 0.95 1.00 1.05 1.10

1 mnth 14.2 13.0 12.0 13.1 14.5

3 mnth 14.0 13.0 12.0 13.1 14.2

6 mnth 14.1 13.3 12.5 13.4 14.3

1 year 14.7 14.0 13.5 14.0 14.8

2 year 15.0 14.4 14.0 14.5 15.1

5 year 14.8 14.6 14.4 14.7 15.0

Greek Letters

If the Black-Scholes price, cBS is expressed as a function of the stock price, S, and the implied volatility, simp, the delta of a call is

Is the delta higher or lower than for equities?

S

c

S

c

imp

imp

BSBS

S

c

BS

Volatility Smiles When a Large Jump is Expected

At the money implied volatilities are higher that in-the-money or out-of-the-money options (so that the smile is a frown!)

The Smile Problem

Traders use the Black-Scholes method to price “plain-vanilla” options.

These options are priced with a consistent implied volatility.

Using actual market data, the implied volatilities are actually different.

Maple example

Volatility Smile and Possibility of Default

Company Ticker

Symbol % Short Bond Rating

AverageDefaultRate Definition

Amazon AMZN 8.00% Baa2 0.15%

MinimumInvestment

Grade

Citigroup C NA A3 0.02% High Quality

General Motors GM 17.10% Ca 24.73%Very Poor

Quality

JP Morgan JPM 1.90% Aa3 0.01%Very High

Quality

Microsoft MSFT 1.30% Aaa 0.00%Highest Rating

Available

There appears to be a correlation between the volatility smile, the bond rating for the company, and the % short value, which gives a percentage that the company’s stock price will drop. Good companies with shallow smiles have good bond ratings and low percent shorts and vise versa.

Measuring the Problem - Dollar Error

Measures the error of Black-Scholes with actual market data Select two options with same S, same T, but different K For given volatility Dollar error=|market – BS price|

Record maximum difference

Repeat but alter assumed volatility– measure from 0 to ∞

Measuring the Problem– MiniMax Dollar Error (MDE)

Dollar error gives a set of maximum dollar errors for all volatilities.

The minimum of all of these maxes is the minimax dollar error Comparing the initial two options, the BS formula must

have at least this amount of error for one of the options, irrespective of volatility.

Measuring the Problem– Pricing

MDE scaled to an S of 100 by taking: Scaled MDE = (MDE*100)/concurrent underlying

Negative sign added to scaled MDE if in the option pair, the higher K option has lower implied volatility than the lower K option

Stock Market Crash of 1987

In 1986, when comparing calls that were 9% in-the-money with calls 9% out-of-the-money: MDE < 1% or Scaled MDE about $.04

Crash of 1987 caused the MDE error to increase for out-of-the-money puts as the stock market fell

Span of implied volatilities in the -9% to 9% striking price range jumped from 1.5% in 1986 to 6.5% in 1992