PowerPoint Presentation
Volatility
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The Measurement of Volatility
Different Kinds of Volatility
Trading Actual Volatility
The VIX
Volatility Smiles
Measuring Volatility
The volatility of a price series is a measure of the deviation of price changes around the mean. More precisely, it is the standard deviation of returns on the stock.
We have assumed a lognormal distribution for stock prices. If this is the case then we have a measure for the volatility of the stock price: the probability distributions standard deviation.
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Measuring Volatility
This means that we can make probability statements about the approximate likely range of the stock price in the future
We can define the volatility number associated with an underlying instrument as a one standard deviation price change, in percent, at the end of a one year period.
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Measuring Volatility
If volatility is 25% then one standard deviation means there is a 66% chance that the stock price will be trading + or – 25% of the price today
If the stock price today is £200, and the volatility of the stock is 25% then there is a 67% chance of the stock being in the range £150 to £250.
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Measuring Volatility
The only problem here is that the volatility number is annualised.
What if we are pricing a one week, month option?
An important characteristic of volatility is that it is proportional to the square root of time (see below). As a result, we can approximate for periods shorter than a year by dividing the annual volatility by the square root of the number of trading periods in a year.
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Measuring Volatility
Suppose we want to know the daily volatility of Shell Transport (traded on LIFFE). First we have to determine the number of daily trading periods in a year. As this is exchange traded, there are 256 trading days in a year
The square root of 256 is 16, so to approximate daily volatility we divide the annual volatility by 16.
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Measuring Volatility
So if Shell Transport calls are trading with an annualised volatility of 20%, the daily vol. is 20%/16 = 1.25%.
Equally we could take the daily volatility, multiply it by
where t = 256, and we would obtain the annual volatility.
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Which Volatility?
So far we have seen how volatility is measured. But which volatility are we measuring?
When traders discuss volatility, even experienced traders may find that they are not always talking about the same thing.
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Future Volatility
Future volatility is what every trader would like to know, the volatility that best describes the future distribution of prices for an underlying contract.
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Future Volatility
In theory it is this number to which we are referring when we speak of the volatility input into a theoretical pricing model.
If a trader knows the future volatility, he knows the right “odds”.
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Historical Volatility
Even though we cannot predict the future (chartists would disagree), if a trader intends to use a theoretical pricing model, he must make an intelligent guess about the future volatility.
A good starting point is historical data
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Historical Volatility
If over the past 10 years the volatility of a contract has never been less that 10% or greater than 30% (volatility cone), a guess for future volatility of 5% or 40% would appear to be wrong – this does not mean that these values could not occur
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Historical Volatility
To calculate historical vol., we need two parameters: the historical period over which vol. is to be calculated, and the time interval between successive price changes
Usually traders have computers that examine a wide variety of time periods. Refer to the previous example for the calculation of historical volatility.
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Implied Volatility
Implied volatility can be thought of as a consensus volatility among all market participants with respect to the expected amount of underlying price fluctuation over the remaining life of the option
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Implied Volatility
Its perhaps easier to think of it as the volatility being implied to the underlying contract through the pricing of the option in the marketplace.
It is essentially a psychological number
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Implied Volatility
If a trader has priced an option on a 12% vol. based on a detailed analysis of historical vol. and he finds that the market price implies a volatility for the underlying of 16%, what should he do??
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Implied Volatility
How does the trader know what volatility the market price of the option is implying?
By using iterative procedures, a trader can “back out” the observed market price: plug different vols into the model until the correct market price is found. This is done because “inverting” the Black-Scholes model is an incredibly complex operation.
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Implied Volatility
How much weight should the trader give the implied vol.?
Typically, a trader might give the implied volatility a weighting somewhere between 25% and 75% in making a volatility forecast.
How much depends on a traders confidence in forecasting volatility based on historical volatility data. It is really a matter of market experience. Forecasting volatility has become an industry in itself where methods are becoming more and more sophisticated.
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Implied Volatility
It is a known fact that volatility is mean reverting (see later) and there are models that use this fact in forecasting vol. Some houses use Garch and Egarch models to try and forecast vol. The volatility smile is another consideration (see later). At the end of the day the best forecast should rely to a great extent on the traders feeling for the underlying market.
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Trading Actual Volatility
Given that we know how to calculate return volatility the question we should be asking is: “can we trade volatility?” In other words can we use options to make trading decisions based on our view of the underlying volatility. The answer is yes we can.
The first type of trade is involves trading the actual volatility in the market or trading gamma, delta neutral. We looked at this trade in the “Greeks” section of the notes but we will go over it again here. These ideas are based on K. Connolly “Buying and Selling Volatility”:
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Long Volatility (Gamma) Delta Neutral
To make a profit, most individual investors and fund managers, are forced to take a view on the direction of the price of assets.
If the instrument is cheap, you buy and if it is expensive you sell. The traditional view taker is looking at only one dimension of a price sequence: direction.
Options can allow investors to completely ignore the direction of the price and to concentrate on the second dimension: the volatility of the price
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Long Volatility (Gamma) Delta Neutral
It is actually possible to construct a portfolio containing a given stock and stock options and be completely indifferent to the direction of the price.
This is known as the long volatility delta neutral trade.
Before looking at how this trade works, it would be useful to see how holding a call is, other things being equal, preferable to holding the stock.
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Long Volatility (Gamma) Delta Neutral
Consider the performance of two fund managers, exposed to 50 shares of a stock. One holds 50 shares at £99 (£4550) the other holds a one year call option (exercisable into 100 shares) on the shares priced at £5.46 Delta is 0.5, so the option holder has exposure to 50 shares.
In market jargon the two portfolios are delta equivalent.
Consider the following diagram, where for small changes in the underlying the portfolios behave identically S(0). If the stock trades up S(1) the option outperforms the stock portfolio. If it trades down S(2) the stock portfolio loses more than the option.
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Underlying
Premium
Option Gain > Stock Portfolio Gain
Stock Portfolio Loss > Option Loss
S(0)
S(1)
S(2)
Long Volatility (Gamma) Delta Neutral
The reason for this is the curvature of the option price curve ( gamma measures this: see “the greeks”).
The reason for the curvature is due to the “kink” in the options expiry profile: which is due to the uncertainty about the stock price at expiry. We can either make unlimited amounts but we can only lose the premium.
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Long Volatility (Gamma) Delta Neutral
So why does anyone ever buy stock?? The catch is time decay. Everyday that passes will cause the option price curve to decay slightly toward the kinked expiry profile.
However, given the curvature of the option price curve, the call option will outperform a delta equivalent portfolio if a significant price change is experienced.
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Long Volatility (Gamma) Delta Neutral
Say we believe that for the foreseeable future, the stock price will fluctuate excessively. But we have no idea as to what direction it will take. Can we set up a trade that will profit whichever way the market moves, up or down.??
We need to start up with a position that is initially market neutral but that gets long if the market rises and short if it falls.
The position that achieves this is long one option and short delta shares.
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Long Volatility (Gamma) Delta Neutral
For small stock moves the respective gains and losses on the long and short position cancel out. This is because we are delta neutral.
If the stock price rises significantly the option component always makes more than is lost by the stock component.
If the stock price falls significantly, the profit on the short stock component always exceeds the loss on the option component. So whichever way the stock price moves , we always make a profit. That is the essence of the long volatility trade.
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Long Volatility (Gamma) Delta Neutral
But how do we lock in volatility profits?? e.g if the market moves up significantly, we make money. But what happens if it goes back down again?? We wipe out the gains.
The most straightforward way would be to close out the position, and take the profit. Closing out the position may not make sense. There may still be more profit to be made out of the position.
The way to lock in volatility profit is to re-hedge the portfolio.
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Underlying
Premium
Long Call
Short Stock (inverted)
Long Gamma Delta Neutral Position
80
100
150
= 0.5, sell 500 shares @ 100
= 0.2, buy 300 shares @ 80
= 0.8, sell 600 shares @ 150
Long Volatility (Gamma) Delta Neutral
Remember that we are short the shares. If the market moves up we need to sell delta more shares ( at a higher price), to remain delta neutral. If the market were to fall again to the original position then all we would do is buy back the same amount of delta shares at a cheaper price, therefore locking in profit.
If we had not re-hedged on the way up there would be no profit.
What if the market fell further. We would simply buy the shares back at a cheaper price and make more money.
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Long Volatility (Gamma) Delta Neutral
The whole point is, that by continuously re-hedging , we are selling in rising markets and buying in falling ones.
What happens if the market just continues to rise?? At some point delta will be 1 and the option value will increase in line with the decreasing stock price value. At this point there is no more curvature left in the price curve.
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Long Volatility (Gamma) Delta Neutral
If the market just continued to fall, then we will reach a point where we have bought back all the short stock. We would have locked in profits on the way down, but there is a maximum.
As mentioned earlier, the catch is that the option is a time decaying asset. Although we are locking in profit from re-hedging, the option is losing time value rapidly. ( see Theta )
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Long Volatility (Gamma) Delta Neutral
Note: for a given stock price move, say X%, the more curved the option curve, the more significant the re-hedging profit.
An alternative way of thinking about this is that a trader could achieve the same re-hedging profit with a smaller stock price move using an option with a greater degree of curvature (gamma).The volatility player is always looking for low priced curvature.
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Short Volatility (Gamma) Delta Neutral
Shorting call options is equivalent to establishing a short position in the underlying shares: if the market rises = large losses.
The classic short volatility position, i.e neutral to direction, is short the call long the underlying.
For small moves in the underlying losses cancel out gains if perfectly delta hedged.
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Short Volatility (Gamma) Delta Neutral
If the stock price rises significantly, the option loses more than is made on the stock. If the stock price falls significantly the losses on the stock are greater than the profit on the call. Essentially it is because we are short gamma. The diagram overleaf illustrates this point:
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Short Call
Long Stock (inverted)
Short Volatility Delta Neutral Position
Gain on stock < loss on option
Loss on stock > gain on option
Short Volatility (Gamma) Delta Neutral
Re-hedging the short volatility portfolio:
Assume we are delta neutral. If the market goes up to remain delta neutral we need to buy more shares, at a higher price. If the market immediately goes down, then to remain delta neutral we are forced to sell the shares at a lower price.
In other words , we are locking in a loss:
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Short Call
Long Stock (inverted)
Underlying
100
150
80
= 0.5, buy 500 shares @ 100
= 0.2, sell 300 shares @ 80
= 0.8, buy 600 shares @ 150
Short Volatility Delta Neutral Trade
Short Volatility (Gamma) Delta Neutral
So why re-hedge at all?? If we do not buy stock on the way up then the potential losses are unlimited.Note that as the underlying stock price rises the slope of the option curve becomes more and more negative. At very high stock prices the slope approaches the limiting value of -1.0 and so being short a deep ITM option is like being short the underlying.
Moreover, if we do not sell stock on the way down we could be left with a long position in the stock with unlimited losses.
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Short Volatility (Gamma) Delta Neutral
So why would anyone short volatility.??
Time is on the side of a short volatility player. Also, if the stock price volatility collapses to 0 immediately after the trade is set up, then the combined effects of time decay and low volatility could result in a significant profit being made
The worst case scenario for a short volatility player is if after implementation of the strategy the stock price swings violently up and down, the position would have to be re-hedged frequently.
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Long Volatility with Put Options
Can put options be used to trade volatility delta neutral as above?
Holders of unhedged puts are, by definition, short the market and will lose if the market rises (limited to the premium).
To hedge a long put one needs to be long the underlying as determined by delta.
Consider the following put option on Zeneca (LIFFE option = 1000 shares), with the position starting off delta neutral when delta = -0.5:
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Underlying
Long Volatility Delta Neutral with Puts
Long Put
Long Stock (inverted)
= 0.5, buy 500 shares @ 100
= 0.8, buy 300 shares @ 80
= 0.2, Sell 600 shares @ 150
150
100
80
Long Volatility with Put Options
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If the underlying price rises the short stock exposure of the put decreases and since the portfolio contains 500 long units the total portfolio gets net long. In order to re-hedge the trader would have to sell stock into a rising market.
If the market falls then the trader would buy stock on the way down.
Long Volatility with Put Options
So, as in the case of calls, the long volatility strategy using puts produces re-hedging profits because of the presence of price curvature or gamma. This is why trading in this way is also called gamma trading.
The interesting point here is that ordinary investors view put and call options as being different instruments. To the volatility player they are identical differentiated only by the nature of the original hedge (long call = short stock, long put = long stock)
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Long Volatility with Put Options
Note: When we are gamma trading as above the source of profit comes from trading the actual volatility in the market by re-balancing the hedge: If the option position has high gamma and daily price changes are large enough to cover time decay, profits will be made.
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Pure Volatility Trades
Another source of volatility profit is from the impact that changes in the implied volatility have on the current price of the option. This process of attempting to make profits from predicting changes in implied volatility is sometimes termed vega or pure volatility trading.
The most common type of vega trading strategy is the straddle:
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Long Call
Long Put
100
Long Straddle (Long Vega)
Long Straddle
We can make money from big swings in the underlying, but these price movements have to be big enough to cover the premiums paid.
Any increase in the implied will increase both premiums and we could close the position out at a profit. This is why this is known as trading vega. We are essentially trading on the options sensitivity to the underlying volatility:
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Long Call
Long Put
100
Long Straddle (Long Vega)
Underlying
Premium
Long Straddle
Note that ATM options are the most sensitive to theta. Therefore, very few traders can afford to maintain these strategies for long periods of time. Most trader trade them for a short period (usually a few days) and unwind them when and if volatility increases.
Often, this is an effective strategy to establish prior to the release of important figures etc.
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Trading Vega
With the long straddle we were buyers of volatility (long volatility/vega). Why? In fact buying a plain vanilla call or any other long option position is a long volatility position.
We can also sell volatility, and the short straddle is the most common, and the most risky, pure short volatility trade. Again, selling or writing any option is, by implication, short volatility:
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Short Put
Short Call
100
Short Straddle (Short Vega)
Short Straddle (Short Vega)
A short straddle is a short vega position, in other words you are selling volatility.
If after setting up the position the market stays at around 100 then the seller will make money from the time decay of the option. This is in fact the principal source of profit to a short straddle. The essence of getting the short straddle right lies in the trader’s prediction of future volatility
Again ATM options are optimal because they suffer the most time decay.
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Short Straddle (Short Vega)
The short straddle has unlimited loss potential on both sides of the strike. The position loses money from an increase in the implied (because we are short two ATM options we have effectively doubled our vega exposure), and major market moves. Also recall from “Exploring the Option Greeks,” that we cannot hedge a short straddle delta neutral without locking in a loss. How could we protect our downside without necessarily losing the benefits of time decay:
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Short ATM Call
Short ATM Put
100
Iron Butterfly Spread
Long OTM Call
Long OTM Put
Short Volatility
Note that we could create the same vega exposure by buying a low strike call (95), selling two mid strike calls (100) and buying a higher strike call (105).
If we do not want the extreme risk of having a naked straddle but we still want to be short volatility with no directional bias, we could sell a slightly less risky version. This is known as a strangle. This involves selling options that are OTM, therefore less sensitive to vega, and we can also absorb more market movement due to the options being OTM:
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90
110
Short OTM Put
Short OTM Call
Short Strangle
Leaning Volatility Trades
With the Call Ratio Backspread you buy two or more higher strike calls and sell one lower strike or any other combination that leaves you delta neutral.
When one buys more options than one is selling one is generally a net buyer of implied volatility.
The call back spread however is leaning towards the upside of the market: Buy two 142.5 calls for $3.50 each and sell one 130 call receiving $9.50, when the market is trading at 130.
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130
142.5
Call Ratio Backspread
Long 2 142.5 Calls
Short 1 130 Call
Call Ratio Back Spread
If the market does drifts with low volatility and ends up at around 142.5 this is where the maximum loss could occur due to the high time decay on the long positions and the exposure to the open short call position.
What the call-back spreader is hoping for is a dramatic increase in the upside of the market and in the implied volatility.
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Put Ratio Spread
A put ratio spread is set up if we sell two or more lower strike put options and buy one higher strike put. The net result is that we pay a premium for the strategy
Sell two 120 puts and buy a 140 put.
Because we have sold more puts than we have bought we are net short volatility and expect the market to remain stable & on balance we would rather the market “leant” towards the upside:
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140
120
Buy 1 140 Put
Sell 2 120 Puts
Put Ratio Spread
Put Ratio Spread
If the market trades between 120 and 140 and volatility remains low we make the most money
If we get it wrong and the market fails to remain stable then we would hope that market tends towards the upside
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Call Ratio Spread
Similar to the put ratio spread, the trader can achieve a selling volatility strategy which is initially delta neutral but has a preference for market movement.
In this case the trader sells two higher strike calls and buys a lower strike call, therefore a net seller of volatility. Essentially limiting the loss on the downside of the market with unlimited loss exposure on the upside:
Consider buying a call with a strike of 130 and selling two calls with a strike of 140.
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130
140
Long 1 130 Call
Short 2 140 Calls
Call Ratio Spread
Call Ratio Spread
If the market drifts slowly up to around 140 with low volatility and stays there this is where the most money can be made due to the highest time decay.The greatest potential loss occurs if the market becomes volatile and bullish. The trader would rather the market leant towards the downside
Remember, there is no law requiring a trader to keep the unlimited loss potential on this trade forever (this applies to any trade): for example if the market falls and it looks as if the market could turn bullish then a trader could buy two OTM calls to cash in on the upside.
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Long Time Spread
Another type of volatility play is the time spread. This is a combination of the leaning and pure volatility trades above in that market direction and implied volatility matter.
The most common type of spread consists of opposing positions in two options of the same type where both options have the same exercise price but different expiry dates.
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Long Time Spread
When the long term option is purchased and the short term option is sold, a trader is long the time spread; when the short term option is purchased and the longer term option sold, a trader is short the time spread.
If we assume that the options making up a time spread are approximately ATM, then time spreads have two important characteristics:
1. A long time spread always wants the underlying market to sit still:
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Long Time Spread
An important characteristic of an ATM option’s theta (time decay) is its tendency to become increasingly large as expiration approaches.
As time passes, a short term ATM option will lose its value at a greater rate than a long term ATM option. This principle has an important effect on the value of a time spread.
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Long Time Spread
Consider two ATM calls , one with 3 months to expiry and the other with six months, with values of 6 and 7 ½ respectively.( In other words if we set up this trade we pay 1 ½ points for the spread. Note that spreads are usually quoted as a net figure.)
If one month goes by and the underlying market is unchanged, both options will lose value. But the short term option with the greater theta will lose more.
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Long Time Spread
If the long term option loses ¼ of a point then the short term option could lose, say, 1 full point. Now the options are worth 5 and 7 ¼ respectively, therefore the spread is worth 2 ¼ points now.
If we were to close out the spread now we could sell it for 2 ¼ points and make ¾ of a point profit. If we break the spread down we can see where the profit comes from: buying back the short ATM option makes 1 point and we lose ¼ point on selling the long ATM option.
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Long Time Spread
If the market continues to do nothing then the spread will continue to increase in value as the following table shows:
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Long Time Spread
But what happens if there is a large swing in the underlying market:
As the underlying market rises and the options move deeply ITM they begin to lose their time value. If the move is large enough it won’t matter that the long term option has three more months to expiry. Both options will eventually lose all their time value.
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Long Time Spread
If the time spread consists of two calls with a strike of 100 and the underlying trades up to 150 then both options will probably trade at parity (intrinsic value), or 50 points. The spread will then go to 0. Even if the long term option retains as much as ¼ point, the spread will still have collapsed to ¼ point.
For a large downswing in the market the situation is identical except for the fact that both options now become worthless (no intrinsic value or time value):
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Loss
Profit
Long 20 June 100 calls
Short 20 March 100 Calls
Long Time Spread Profit/Loss Profile
Long Time Spread
The second characteristic of a long time spread is that it always benefits from an increase in implied volatility.
As time to expiration increases, the vega of an option increases. This means that a long term option is always more sensitive in total points to a change in volatility than a short term option with the same exercise price.
Again assume we are short a 3 month call at 6 and long a 6 month call at 7 ½ (the spread is worth 1 ½ points.). Also assume that the spread is based on a volatility of 20%.
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Long Time Spread
If volatility increases to 25% the value of both options increases. But the 6 month call, because of the higher vega, will increase more in value. As with time decay above an increase in the implied widens the spread. Effectively, we lose on the short call but gain more on the long call.
If volatility collapses to say 12% then we lose more on the long call than we gain on the short.
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Long Time Spread
Therefore, when a trader buys or sells a time spread, he is trying to forecast both the movement in the underlying and changes in the implied volatility.
Ideally, the trader who is long a time spread wants two apparently contradictory conditions in the market place:
First, he wants the market to sit still to benefit from time decay & second he wants everyone to think the market is going to move so that implied volatility increases. A seemingly impossible scenario.
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Long Time Spread
This actually happens quite often in markets: Suppose there is an announcement on Reuters that the G7 finance ministers are going to meet in a month to discuss exchange rates and interest rates. No one knows what the outcome of the meeting will be.
In this case there is unlikely to be much movement in the underlying because no one is willing to punt on this kind of risk (apart from ICMA Centre students). Equally, all traders assume that major changes in exchange and interest rates could result from this meeting, which will result in an increase in the implied.
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Long Time Spread
This kind of scenario is very common in the U.S interest rate option market when policy statements are expected from the Fed.
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Volatility Spreads and the Greeks
We can now return to the table which we first came across in the Greeks section of the course and examine each position in more detail:
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The VIX
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 (standard deviation) of 15% over the next 30 days.
This means 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. That is, index options are priced with the assumption of a 68% likelihood (one standard deviation) that the magnitude of the change in the S&P 500 in 30-days will be 4.33% (up or down).
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Brief History of VIX
The VIX was originally launched in 1993, with a slightly different calculation than the one that is currently employed. The ‘original VIX’ (which is still tracked under the ticker VXO differs from the current VIX in two main respects: it is based on the S&P 100 (OEX) instead of the S&P 500; and it targets at the money options instead of the broad range of strikes utilized by the VIX.
The current VIX was reformulated on September 22, 2003, at which time the original VIX was assigned the VXO ticker. VIX futures began trading on March 26, 2004; VIX options followed on February 24, 2006.
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“Fear Index”!
The CBOE has actively encouraged the use of the VIX as a tool for measuring investor fear in their marketing of the VIX and VIX-related products. As the CBOE puts it, “since volatility often signifies financial turmoil, [the] VIX is often referred to as the ‘investor fear gauge’”. The media has been quick to latch onto the headline value of the VIX as a fear indicator and has helped to reinforce the relationship between the VIX and investor fear.
The following graph shows the VIX on an absolute basis, as reformulated in 2003, but using data reverse engineered going back to 1985:
Look at 1987!!
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VIX
The following graph is the VIX of a week in July 2012:
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Volatility Indices
The VIX is the most widely known of a number of volatility indices. The CBOE alone recognizes nine volatility indices, the most popular of which are the VIX, the VXO, the VXN (for the NASDAQ-100 index), and the RVX (for the Russell 2000 small cap index).
In addition to volatility indices for US equities, there are volatility indices for foreign equities -VDAX (DAX Index), VSTOXX (Euro Stoxx 50), VSMI (Swiss) ,VX1, MVX, VAEX,VBEL, VCAC(NYSE EuroNext CAC 40), etc.) as well as lesser known volatility indices for other asset classes such as oil, gold and currencies.
The following show some graphs of other Indices:
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Crude Oil Vol Index
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Gold Vol Index
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Apple Stock Vol Index
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Simple Hedging with the VIX
The VIX is appropriate as a hedging tool because it has a strong negative correlation to the SPX – and is generally about four times more volatile. For this reason, portfolio managers often find that buying out of the money calls on the VIX to be a relatively inexpensive way to hedge long portfolio positions. Similar hedges can be constructed using VIX futures .
The negative correlation between the VIX and the SPY (Standard and Poors Depositary Receipts – ETF’s which mirror the S&P 500) can be seen in the following diagram:
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VIX Hedge
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VIX Hedge
Why the negative correlation?
As perceived risk in the stock market rises, investors tend to purchase more options, particularly puts, for protection against a market decline. This greater demand for options causes implied volatilities to go up. As implied volatility rises, so does the VIX.
Conversely, as investors perceive markets to be less risky, the demand for protective options goes down, lowering implied volatility. The lower implied volatilities are reflected as a decline in the VIX.
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VIX Hedge
Since the VIX typically rises as investors market fears increase, there is generally an inverse correlation between price movement in the VIX and price movement in the S&P 500. So as the stock market falls, the VIX rises and vice versa.
Notice that each time the VIX reached a level over 30 the SPY tended to find a market bottom. This level of 30 or greater thus equates to a high level of nervousness and pessimism in many market participants, which may be used as a bullish contrary indicator. Conversely a relatively low VIX reading below 20 may signal complacency in the market, signalling a bearish contrarian signal.
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VIX Hedge
A very simple hedge (if we are long the SPX, or an ETF which mirrors the SPX) might be to buy a call or a call spread on the VIX. So in a downward moving market when our portfolio might be losing money, the VIX should be increasing in value causing our call spread to increase in value as well, at least partially offsetting our losses in the stock market.
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VIX Hedge
One interesting characteristic of the VIX is that the prices are not lognormally distributed as they are in stocks and stock indexes.
Lognormal distribution means that the price is just as likely to double as to drop in half. If the VIX were to go to 0 this would imply no expected change in daily value of the SPX, which is impossible.
Conversely, if VIX were to reach an extremely high value and persist that would indicate that the market expected extremely large price movements for an extended period of time, again very unlikely.
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VIX Hedge
But the VIX tends to stay in a range roughly between 10 and 35, so it is basically mean reverting
This information is very useful to know when trying to decide the most advantageous time to place a trade in the VIX. For example, if the VIX is extremely low and we want to hedge against a downward move in the stock market, we may decide to put on a bullish position in the VIX (call spread)instead of a more traditional hedge like a bearish position in an index (put spread).
The reason a trade in the VIX may be more advantageous is that if the market does fall, both the VIX call spread and SPY put spread should perform well. However, if you are wrong and the market rallies, you know for certain that the SPY put spread will lose a lot of money. The VIX index, however, may not fall significantly since it is already at an extremely low level.
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VIX Hedge
Comparing SPY and VIX hedges:
At the market close on Friday February 23, 2007 the S&P 500 Depository Receipts (SPY) was at 145.31 and the CBOE Volatility index (VIX) was at 10.58.
At the time you could have purchased an April 143-141 bear put spread on the SPY for $0.40, or a total cost of $40. At that same time, you could have also purchased the April 12-14 bull call spread on the VIX for a net debit of $0.50, or a total cost of $50. The April options had 57 days of life remaining until expiration at this time.
100
VIX Hedge
By the close on Tuesday, February 27 2007 the market had dropped substantially and the SPY was trading at a price of 139.85, a decline of 3.75%. We then closed both trades. The following table shows the performance of both trades over this period of time.
During this same time period, the VIX moved up to 18.22 (an increase of 72.2%).
The following shows the relative performance of the SPY put spread and the SPX call spread:
101
SPX Hedge
102
VIX Hedge
But what if the market had rallied?
Looking at a price chart of the SPY, a move up of nearly the same magnitude occurred just a couple weeks later between March 16, 2007 and March 21, 2007.
At the close on March 16, the SPY was trading at 138.53 and you could have purchased a May 136-134 bear put spread for a net debit of $0.55, or a total cost of $55. The VIX was trading at 16.79 and you could have purchased the May 18-20 bull call spread for a net debit of $0.40, for a total cost of $40. The May options at this time had 64 days of life remaining until expiration.
103
VIX Hedge
By the close on March 21 the market had moved up and the SPY was now trading at 143.29, an increase of 3.44% from when the trades were originally placed. During this same time period, the VIX down to 12.19 (a decrease of 27.4%). The following table shows the performance of both trades over this period of time:
104
VIX Hedge
105
Another VIX Hedge (source: Options House)
A hedge to protect a portfolio against a market crash.
The investor buys near-term slightly out-of-the-money VIX calls while simultaneously, to reduce the total cost of the hedge, sells slightly out-of-the-money VIX puts of the same expiration month.
This strategy is also known as the reverse collar.
The idea behind this strategy is that, in the event of a stock market decline, it is very likely that the VIX will spike high enough so that the VIX call options gain sufficient value to offset the losses in the portfolio.
106
Reverse Collar
To hedge a portfolio with VIX options, the portfolio must be highly correlated to the S&P 500 index with a beta close to 1.0.
The tricky part is in determining how many VIX calls we need to purchase to protect the portfolio. A simplified example is provided below to show how it is done.
107
Reverse Collar
A fund manager oversees a well diversified portfolio consisting of thirty large cap U.S. stocks. For the past two months, the market has been climbing steadily with the S&P 500 index climbing from 1273 in mid-March to 1426 in mid-May.
At the same time, the VIX has been drifting downwards gradually, hitting a five month low of 16.30 on 17th May. The fund manager thinks that the market is getting too complacent and a correction is imminent.
He decides to hedge his holdings by purchasing slightly out-of-the-money VIX calls expiring in one month’s time. Simultaneously, he sells an equal number of out-of-the-money puts to reduce the cost of implementing the hedge.
108
Reverse Collar
As of 17th May:
For simplicity’s sake, let’s assume the value of his holdings is $1,000,000.
The S&P 500 Index stood at 1423.
The VIX is at 16.30.
June VIX calls, with a strike price of $19, are priced at $0.40 each.
June VIX puts, with a strike price of $12.50, are priced at $0.25 each.
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Reverse Collar on VIX
110
12.5
19
16.3
Reverse Collar VIX Hedge
How many VIX calls should the fund manager buy?
According to the CBOE Website, on average, the VIX rise 16.8% on days when the S&P 500 index drops 3% or more. This means that if the SPX move down by 10%, the VIX can potentially shoot up by 56%.
To play it safe, the fund manager assumes that the VIX will rise by only 40% when the SPX drops by 10%. This means that, in theory, the VIX should rise from 16.3 to 22.8 if the S&P 500 drops 10% from the current level of 1423 to 1280.
111
Reverse Collar VIX Hedge
When the VIX is at 22.8, each June $19 VIX call will be worth $380 ($3.80 x $100 contract multiplier).
10% of the fund manager’s portfolio is worth $100,000.
Number of VIX calls required to protect 10% of the portfolio is therefore: $100,000/$380 = 264
Total cost of purchasing the 264 VIX June $19 calls at $0.40 each = 264 x $0.40 x $100 = $10,560
Premium received for selling 264 June $12.50 VIX puts at $0.25 each = 264 x $0.25 x $100 = $6,600
Total investment required to construct the hedge = $10,560 – $6,600 = $3,960
112
113
Reverse Collar VIX Hedge
As can be seen from the table above, should the market retreat, as represented by the declining S&P 500 index, the negatively correlated VIX move upwards at a much faster rate. The VIX puts sold short will expire worthless while the value of the VIX call options rise to offset the loss of value in the portfolio.
Conversely, should the market appreciate, the rise in his holding’s value is offset by the rise in the value of the VIX put options sold short. Notably, because the VIX has traditionally never gone below 10 for long, the put options sold short should not appreciate too much to cause significant damage to the portfolio. Hence, it is more favourable to implement this hedging strategy when the VIX is low.
114
Volatility Skew
Theoretically, all options for a stock should trade with the same measure of volatility and at the money calls and puts with the same strike and expiration should have the same price. In practice, the demand for individual option contracts can drive up the price of some of the options on a stock and not others.
If call volatilities are higher than put volatilities, this indicates that traders are buying calls. An excess of call buyers reveals a skew that favours the stock price going up. A similar argument can be made for expensive puts.
115
Volatility Skew
There are two types of skew, time skew and strike skew. Time skew is a measure of the disparity of option volatility for option contracts with the same price but different expirations. Strike skew is the measure of the disparity of option volatility for option contracts with different strikes but the same expiration.
If the two are combined then this is known as a volatility surface
116
The Skew
If we look at option prices before and after October 1987, there is a distinct break. Option prices began to reflect an “option risk premium” – a crash premium that comes from the experiences traders had in October 1987.
After the crash the demand for protection rose and that lifted the prices for puts; especially out-the-money puts. To afford protection, investors would sell out-the-money calls.
There is thus an over supply of right hand sided calls and demand for left hand sided puts = the skew
But this is only one explanation…
117
The Skew
Another explanation is that the market does not believe that the dispersion of the underlying follows a lognormal distribution.
The situation which often occurs in real markets is called leptokurtosis, when there are higher probabilities of the market either staying put or moving well beyond 2 standard deviations.
The lognormal distribution does not cater for large unexpected swings in the market. The market is effectively saying that there is a much higher probability of a crash than the lognormal distribution would have us believe. Mark Rubinstein calls this “crash-o-phobia”.
118
The Skew
Many market participants compare the differences between the lognormal probabilities and the implied probabilities coming from the options prices.
If the lognormal curve says there is a 1% chance that the stock will go above X at expiry, and the smile probabilities shows the chance is 10%, then this is a strong statement of market expectations.
The following shows the volatility surface for SPX:
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120
EUR/JPY Skew
121
Applied Exercise
Options Strategist Exercise 3
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Henley_Business_School_Logo_HYBRID
t
d
Time to expiry
Long term option
6 months
5 months
4 months
Short term option
3 months
2 months
1 month
long value
7 1/2
7 1/4
6 ¾
short value
6
5
3
Value of spread
1 1/2
2 1/4
3 ¾
Asset Opening Trade Description Closing Trade Description $Gain %Gain
SPY Buy 1 April 143 put @ $1.35
Sell 1 April 141 put @ $0.95
For Net Debit of $40 Sell 1 April 143 put @ $4.80
Buy 1 April 141 put @ $3.70
For Net Credit $110 $70 175%
VIX Buy 1 April 12 call @ $1.55
Sell 1 April 14 call @ $1.05
For Net Debit of $50 Sell 1 April 12 call @ $3.10
Buy 1 April 14 call @ $1.70
For Net Credit $140 $90 180%
Asset Opening Trade Description Closing Trade Description $Loss %Loss
SPY Buy 1 May 136 put @ $2.30
Sell 1 May 134 put @ $1.75
For Net Debit of $55 Sell 1 May 136 put @ $0.70
Buy 1 may 134 put @ $0.65
For Net Credit $5 $50 91%
VIX Buy 1 May 18 call @ $1.00
Sell 1 May 20 call @ $0.60
For Net Debit of $40 Sell 1 May 18 call @ $0.60
Buy 1 May 20 call @ $0.45
For Net Credit $15 $25 63%
S&P 500
Index
VIX
Call Options
Value
Put Options
Value
Net Premium
Received
Unhedged
Portfolio
Hedged
Portfolio
1210
(-15%)
26.08
(+60%)
$186,912 $0 $182,952 $850,000 $1,032,952
1280
(-10%)
22.80
(+40%)
$100,320 $0 $96,360 $900,000 $996,360
1352
(-5%)
19.56
(+20%)
$14,784 $0 $10,824 $950,000 $960,824
1423 16.30 $0 $0 -$3,960 $1,000,000 $996,040
1494
(+5%)
13.04
(-20%)
$0 $0 -$3,960 $1,050,000 $1,046,040
1565
(+10%)
9.78
(-40%)
$0 $71,808 -$75,768 $1,100,000 $1,024,232
1636
(+15%)
9.78**
(-40%)
$0 $71,808 -$75,768 $1,150,000 $1,074,232
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