CS计算机代考程序代写 PowerPoint Presentation

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The Greeks: Trading Tools

Where business comes to life

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Delta, Theta, Gamma, Vega, Rho

Risk exposure

Position Delta

Trading Delta & Gamma Neutral

Detailed example of delta-gamma hedging

Position Analysis

Summary of the Greeks

Delta
The Black-Scholes model can be differentiated with respect
to each of its five parameters; S, X, T, r, .The result is a set
of formulas that predict how much an options value will change
if a parameter changes, ceteris paribus.

Delta, measures changes in option premium for a given
movement in the underlying price

Formally,

Which is simply the slope of the option price curve at
any one point

Delta =

Underlying
Premium

= 0.5
+0.5

Option Price Curve

Delta
Example:

Delta for a call option on Shell Transport stock is +0.5.
Strike = 100, stock = 100: ATM. Its premium is 1.98

Delta = 0.5: If the stock moves by 1 point the option will move by half a point. If the underlying price rises by 5 points what is the new option premium?

The new premium is 4.48

Delta
Can also be expressed as where C is the price of the call and U is the underlying.

If we know that a 1 point move in the underlying causes the option to move by 0.5 points, then we must be able to hedge that 1 point move.

Put differently, how many options are required to produce the same price movement as a 1 point change in the assets price ??

Delta
Two options are required: simply divide delta into the assets price. Intuition: For every one point asset price movement, one option creates a half point movement in the option premium (for a delta of 0.5). So two options will create a one point movement in the option premium.

In other words, 2 call options are required to hedge a one unit price change in the underlying.
Understanding delta is fundamental in understanding what you are buying and selling when you trade options. In other words, your exposure to the market
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Delta
It is intuitively easier to think in terms of how many units of the underlying are required to hedge an options position rather than the other way around:
If we are short 1 call option with a delta of 0.6 then we need to buy 0.6 of the underlying to be hedged against a 1 point move in the underlying.

Delta
How many Shell Transport shares do we need to hedge if we are short 1 call option (each option gives the right to buy 1000 shares. Delta = 0.5)?
To be Delta Neutral we need to buy 500 shares

Delta neutral means our position is insensitive to small changes in the underlying.

Delta Neutral= Buy 500 shares
Short 1 Call on Shell, Delta = 0.5

Delta Short
Delta Long

Delta
Delta also tells us what our effective exposure to the underlying market is.
Short 1 call on Shell Transport with Delta = 0.5, means we have short exposure to 500 shares.
If we know this exposure then we know what our delta hedge should be.
Is delta positive or negative for a short call option position?

Delta
The Delta of a call is always positive, but delta exposure can be either positive or negative

Take the short call position in Shell Transport for example:

Short 1 Call on Shell: 1 contract = 1,000 shares
Delta = +0.5, Exposure = -500 Shares

Delta = +0.8, Exposure = -800 Shares

Delta = +0.2, Exposure = -200 Shares

Put Delta
The delta of puts is always negative: if the underlying increases the put premium will decrease by delta

Delta =

= -0.5

Option Price Curve
+1

-0.5
Underlying
Premium
Long Put

Delta Exposure with a Put
Although the delta of a put option is always negative the delta exposure, as with a call option, can be either positive or negative.
Consider the following short put position on Boots:

Short Put on Boots: 1 contract = 1,000 shares

Delta = -0.5, Exposure= +500 shares

Delta = -0.8, Exposure = +800 shares

Delta = -0.2, Exposure = +200 shares

Delta
To re-cap, consider the following:
LIFFE Options
Position delta delta exposure delta hedge
-10 Calls +0.5 -5,000 +5000
+10 Calls +0.5 +5,000 -5,000
-10 Puts -0.5 +5,000 -5,000
+10 Puts -0.5 -5,000 +5,000

Delta
Delta also gives us an approximate probability of being exercised

Consider the following diagram for a short call option position:

Short Call on Hanson, Strike = $7.00
Underlying

$7.00

$4.00

$10.00

= 0

= +0.5

= +1

A
B
C

Delta as the probability of exercise
In the above diagram, Delta ranges from 0 to 1. At point A the option is deep OTM, the probability of exercise is low and the option is therefore insensitive to movements in the underlying. We can be delta neutral with a small amount of the underlying

Delta as the probability of exercise
At point B the option is ATM, and depending on time to expiry, is becoming more and more sensitive to the underlying. The probability of exercise is now approximately 50% .
At point C the option is deep ITM and, again, depending on time to expiry the probability of exercise is now approaching 100%. In fact the nearer we get to expiry the more the option starts behaving like the underlying.

Delta and Time to Expiry
Consider the following call deltas for a stock price of 100.
With a strike of 90 (ITM), and 6 months to expiry, the calls delta is .80 . With a month to go delta is .94. With one day to go delta is 1.
With a strike of 105 (OTM), at 6 months delta is .48, at 1 month .28, and at 1 day 0.

Delta and Time to Expiry
Given the above examples of delta changing through time consider the following diagram:

90 Days

60 Days

Delta and Time to Expiry
30 Days

1 Day

Delta and Time to Expiry
The delta graph shows how the smooth profile for the longer dated options becomes more acute as expiry approaches. With one day to expiry, as mentioned previously, the delta becomes virtually a zero/one variable for small movements in the underlying.

This is due to the one fundamental issue that underpins all options: Whether it will be exercised or not (and, hence, whether it will have any intrinsic value or not)

Delta and Time to Expiry
With a long way to expiry the issue is essentially unresolved, even for options not ATM. As maturity approaches, there is less and less time for any significant move in the underlying price: so it becomes clearer whether the option will expire ITM or OTM.

Risk Exposure and Position Delta
Any hedge with a portfolio of options and the underlying is delta neutral as long as all the deltas in the hedge add up to 0. If they do not then, if we want to remain delta neutral we have to compensate by either buying or selling further assets.

The position delta determines how much a portfolio changes in value if the price of the underlying stock changes by a small amount.
The portfolio might consist of several puts and calls on the same stock, with different strikes and expiry dates, and long and short positions in the stock itself.

Risk Exposure and Position Delta
Regardless of the makeup of the portfolio position delta is calculated as follows:

The portfolio delta is simply the sum of all the deltas in the portfolio.

Position Delta
Example: ( each option = 1 share)

Delta Total Deltas
Long 300 shares +1 +300
Long 40 puts -0.46 -18.4
Short 150 calls +0.80 -120
Long 62 calls +0.28 +17.36
Position Delta +178.96

Position Delta
In order to be delta neutral, the trader would need to sell short 178.96 shares.

This is because the portfolio exposure is “delta long”

Note that you do not necessarily have to trade the underlying to make a portfolio of options delta neutral.

Position Delta
Example:

Suppose you bought 100 puts with a delta of -0.3. ( ITM OR OTM?). How would you create a delta neutral position with ITM puts that have a delta of -0.85?.Should they be bought or sold??.

Position Delta
Delta Neutral =

0=(100)(-0.3) + (n2)(-0.85) = -35.29

To achieve delta neutrality we need to sell 35.29 puts.
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Position Delta
Another example:
An option trader buys a near the money call with a delta of .55. He also buys 2 OTM puts with each delta being -0.35. Later he sells two OTM calls each with a delta of 0.30. Each option = 100 shares.

The position delta:
1(.55)+2(-.35)-2(.30)=-.75.

Position Delta
For a $1 increase in the underlying the portfolio loses $75. We should buy 75 shares to go delta neutral.

Alternatively, we could buy an ITM call with a delta of .75. Why??.

Pin Risk
Pin risk occurs when the market price of the underlying of an option contract at the time of the contract’s expiration is close to the option’s strike price. In this situation, the underlying is said to have pinned.
The problem with this is situation is that the trader could be left with an unwanted (and risky) underlying position after expiry.

Example 1
Short a UK call option with Strike 100. Just before expiry market at 100.
Trader doesn’t know if the market will stay where it is, fall or rise.
Takes the view that the market will rise. Buys 1000 shares to delta hedge the position.
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Pin Risk
Just at the close the underlying market falls to 99.50. The call is OTM, and the trader is left with 1000 shares long the day after expiry and the underlying market trades down again to 99, resulting to a mark to market loss of £1000 on the shares.

Example 2
Short a UK call strike 100. The trader assumes that the option will not be exercised so does not delta hedge the position. Unfortunately the trader won’t know if he has been assigned until the day after expiration
The following day he is assigned for exercise and ends up short 1000 shares at 100
Question: why was an ATM option assigned for exercise?

Pin Risk
Even though the option has no theoretical value, the owner of the option might want to be long the shares.
He has two choices- exercise the option or buy the shares outright
The right of exercise is included in the original transaction cost so will be cheaper than buying the shares outright (brokerage fees etc).
There is no certain solution to pin risk. The practical solution is not to carry open positions until expiry.
Certain options have no pin risk because they are settled in cash and therefore don’t carry the risk of an unwanted underlying position at expiry. Examples of these types of options are stock index options and Eurodollar options

Gamma
Is delta an accurate measure of the relative movement of asset and option prices?

No, because of the convexity, or curvature of the option price curve (refer diagram overleaf)

When the underlying trades up, delta underestimates the change in the call premium.

When the market falls the delta overestimates the decline in the value of the premium.

Underlying
Premium
Delta underestimates change in call premium

Delta overestimates change in call premium

Gamma
Gamma is the rate of change of delta for a unit change in the price of the underlying
The gamma of an option is usually expressed in the amount of delta lost or gained per one unit change in the underlying

Formally:

It is the second derivative of the premium with respect to the underlying

Gamma
If delta is the hedge ratio, then intuitively gamma expresses how much the hedge ratio changes when the underlying asset price changes. The larger the gamma in absolute terms, the more risky the position, especially short ones (see later).

If we are trading delta neutral, then the larger the absolute gamma, the larger will be the associated change in delta: which means that we need more of the underlying to remain delta neutral for any given change in the underlying

Gamma
Example:
Consider a short call option position with a delta of .25. Theoretically we should be delta neutral for small movements in the underlying if we buy .25 of the share.

However, with a gamma of .05, for a 1 point move up in the underlying delta becomes .30, otherwise we would not be delta neutral.

And the more convex the option, the higher the gamma, the more delta will change for any given change in the underlying.

Gamma
Consider a deep OTM call. Its delta is about 0. If the stock price were to jump by a small amount, the option price would hardly move and delta would still be 0.

Gamma of deep OTM options is 0

Gamma
The delta of a deep ITM call is about 1, and will not change if the underlying changes by a small amount.

The gamma of a deep ITM options is also 0.

Gamma
Gammas are maximised when the option is ATM. If the underlying changes by a small amount, delta can change by a large amount, depending on the size of gamma.

Consider the graph on the following page. It shows the gamma 0f a 60 option struck at $90:

Gamma and Time to Expiry
Gamma, like delta, also depends on the time to expiry: If it is a long way off, delta will not change that much and gamma is low. But an ATM option that has a large gamma and is near to expiry will be incredibly unstable. Consider the diagram overleaf:

Gamma and Time to Expiry

90 Days

60 Days

30 Days

1 Day

Short Gamma Trades
Short option positions are, by definition, short gamma or short convexity and are very difficult to hedge delta neutral

The higher the absolute gamma, the more risky the position. In fact it is impossible to hedge a short gamma position delta neutral without losing money

Consider the short gamma (short a call) delta neutral trade overleaf where delta is 0.40 and gamma is a relatively high 0.20:

Short Call

Long Stock (inverted)

Short Gamma Delta Neutral Position
Gain on stock < loss on option Loss on stock > gain on option

Short Gamma Trades
In the above diagram because of the convexity in the option price curve we will always lose regardless of the market direction.
We will cover short gamma trades (short volatility) in the volatility section of the notes.

However, some important points should be noted at this juncture: In the above trade to remain delta neutral we will be buying the underlying as the market rises and selling the underlying as the market falls. Consider the diagram overleaf:

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 Gamma Trades
In the previous diagram note that more convex the option the greater is delta, and hence the more stock will need to be bought and sold, therefore the greater the loss when you try to re-hedge delta neutral.
At this point, the essential point to remember is that you “lock in a loss” if you try to delta hedge a short gamma position.
The opposite is true if you are long gamma (long convexity). Consider the diagram overleaf of a long gamma (long call) delta neutral position:

Underlying
Premium
Long Call

Short Stock (inverted)
Loss on stock < gain on option Long Gamma Delta Neutral Position Gain on stock >loss on option

Long Gamma Trades
Because of the convexity in the option price curve, we make more on the long option position than we lose on the short stock position if the underlying trades up, and we gain more on the short stock position than we lose on the long option position when the underlying trades down.
This means that we should lock in a profit when trading a long gamma (convexity) delta neutral position. Consider diagram overleaf:

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 Gamma Trades
In the long volatility ( long gamma) delta neutral trade, the trader is locking in a profit by rehedging: selling stock on the way up and buying it back on the way down

The risk here is that, apart from volatility collapsing and large time decay effects, the trader has not realised that gamma is large.

Long Gamma Trades
If volatility does increase, and the trader rehedges using delta, then he certainly won’t lose money. The point is, if he ignores a high gamma then the trader will not make nearly as much money:

Hedging Short Gamma
The only way of hedging short gamma exposure is to “buy gamma”. This simply means buying options.

Example:

Suppose that a portfolio is delta neutral and has a gamma of -.03. The delta and gamma of a particular traded option are 0.62 and .015 respectively.

Hedging Short Gamma
To make the portfolio gamma neutral we need to include a position of :

.03/.015 = 2 long call options in the portfolio. Intuition:
We were short convexity so to counterbalance we need a long convexity position.

-.03 + (.015)(2) = 0.

Hedging Short Gamma
However, the original delta exposure of the portfolio will change from 0 to 0.62 x 2 = 1.24. We need to sell short 124 (assuming each option gives the right to 100 shares) of the underlying to go delta neutral again.

Theta
The theta, or time decay factor, is simply the rate at which an option loses value as time passes. It is usually expressed in points (pence, cents…) lost per day.
Long option positions have negative thetas, while short option positions have positive thetas.
Consider the following graph. It shows the theta of a 60 day option struck at $90.

Theta
It should be clear from the previous diagram that near the money options suffer the most time decay. As one moves further away from the exercise price, the effects of time decay get smaller and smaller.

Theta and Time to Expiry
As expiration approaches, the gamma of an ATM option becomes increasingly large. The same is true of theta, except it is opposite in sign. As we get closer to expiry the rate at which an option decays begins to accelerate. This can be seen from the following diagram:

Theta and Time to Expiry
0.0
-0.01
-0.02
-0.03
100
70
130

90 Days

60 Days

30 Days

1 Day

Theta
As a general rule, an option will have a gamma and theta of opposite signs. Moreover, the relative size of gamma and theta will correlate. A large positive gamma goes hand in hand with a large negative theta, while a large negative gamma goes hand in hand with a large positive theta.

Theta
Every option position is a trade-off between market movement and time decay. If price movement in the underlying will help a trader ( positive gamma), the passage of time will hurt ( negative theta). The trader can’t have it both ways.
Deep OTM and deep ITM options have virtually no time decay at all. It would appear tempting to use these kinds of options in the long gamma (volatility) trade.

Theta
However, the very options that have little time decay also have very low curvature, and we need curvature for the volatility trade. And options with the most curvature unfortunately suffer from the most time decay.
Time decay can and often does, ruin the long volatility trade. When we put on this trade we hope to experience price movements. Everyday that the underlying price stays still is a day of time value lost. The price curve, over time, collapses
into the intrinsic value line as time value is lost:

90 Days
Option Price Curve Collapses

60 Days

30 Days

10 Days

Into Intrinsic Value Line
Premium
Underlying

Theta
If the market one is involved in becomes completely stagnant then the losses due to time decay can build up. Long volatility traders in such situations talk of “bleeding to death” through theta.

What traders tend to do is establish what degree of volatility or price movement is necessary to cover the cost of time decay.
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Theta
Theta only works for short option trades. If the market is stagnant, option premiums will decay and the positions can be closed out at a profit.

Vega
While the terms delta, gamma and theta are found in most option texts, there is no generally accepted term for the sensitivity of an options theoretical value to a change in volatility. Since vega is not a Greek letter, sometimes referred to as Kappa.

The vega of an option is usually given in point change in theoretical value for each one percentage change in volatility.

Vega
Since all options gain value with rising volatility, the vega for both puts and calls is positive.

Long positions gain from increases in volatility and lose from decreases. The opposite is true of short positions. The following diagram illustrates the effect of volatility on the option price curve for a short put position:

Vega
Note in the following diagram of a 60 day call struck at $90, that the ATM options have the greatest vega. Essentially, this is again due to the probability of ending ITM at expiry.

Furthermore, longerdated options are more sensitive to vega than shorter dated ones. With 60 days to expiry (At the Money) vega = 0.14, with 1 day to expiry, vega = close to 0:

Vega and Time to Expiry
Because of the longer time to expiry there is, given a certain level of volatility, a greater probability that the option will expite ITM. The relationship between time to expiry and vega is perhaps better illustrated in the following diagram:

Vega and Time to Expiry

90 Days

60 Days

30 Days

1 Day

Vega
Short option positions lose from increases in volatility because premiums are increasing. The contrary is true for decreases in volatility. Consider the following diagram, which illustrates a short call position. The option price curve is pushed upwards as volatility increases:

Premium
Vol = 5%

Vol = 10%

Vol = 15%

Vol = 20%

Effect of Volatility on Option Premiums
Short Call

Vega
Note that volatility will be covered in depth in the volatility section of the lectures.

In the next section we will examine in more detail the delta and gamma hedges discussed previously:

Delta-Gamma hedging
In this section we will:
Explore an equity delta hedge in more detail over several days
Use gamma to better approximate the change in the option price (delta-gamma approximation)
Examine a simple algebraic example of the delta-gamma approximation

Delta-Gamma hedging
We will use the following inputs:
S = $40; X = $40, = 0.30, r=0.08, T-t = 91 days (91/365).
Given these inputs the price of the option (using the B-S model) is $2.7804- just accept that as given.
Suppose now that a market-maker sells one call (contract size = 100 shares), and hedges the position long delta shares.
We will consider the risk of a delta-hedged position by assuming that the market-maker delta-hedges and marks-to-market daily:

Delta-Gamma hedging
Day 0:
The market-maker is short so receives $278.04. Delta is 0.5824, therefore the position is hedged by buying 58.24 shares.
The net investment (cost) is: (58.24*$40)-$278.04 = ($2051.56)
At an 8% interest rate, the market-maker has an overnight financing charge of:

Delta-Gamma hedging
Consider the marked to market P&L of the market-maker (without considering re-balancing to maintain delta-neutrality yet) when the market goes up to $40.50 after a day (new call price = $3.0621):
Day 1
Gain on 58.24 shares 58.24*($40.50-$40) = $29.12
Gain on short call $278.04-$306.21 = -$28.17
Interest -$0.45
Overnight profit $0.50

Delta-Gamma hedging
Day 1
Rebalancing the portfolio- the new delta (at $40.50) is 0.6142, so we must buy an additional 61.42-58.24 = 3.18 shares to remain delta neutral. This does not affect our marked to market profit for day 1 ($0.50) because we will be buying the shares at $40.50. The cost ($40.50*3.18 = $128.79) will be reflected in the next day’s marked-to-market profit

Delta-Gamma hedging
Day 2:
Stock falls to $39.25
Option falls to $2.3282
Investment from day 1 to day 2(cost) = (61.42*$40.50)-$306.21 =( $2,181.3)
Overnight financing
So the marked-to-market P&L for day 1-day 2 is:

Delta-Gamma hedging
Day 2:
Gain on 61.42 shares 61.42*($39.25-$40.50) = -$76.78
Gain on short call $306.21-$232.82 = +$73.39
Interest -$0.48
Overnight profit -$3.87
We are now getting to the interesting bit! Why do we make $0.50 over the first day, and lose $3.87 over the second day, whilst keeping a delta neutral position??

Delta-Gamma hedging
Consider the following table which summarises the delta, net investment and daily profit over 5 days:
90

91
0 1 2 3 4 5
Stock $ 40 40.50 39.25 38.75 40.00 40.00
Call $ 278.04 306.21 232.82 205.46 271.04 269.27
Option Delta 0.5824 0.6142 0.5311 0.4956 0.5806 0.5801
Investment $ 2051.58 2181.30 1851.65 1715.12 2051.35 2051.29
Interest -0.45 -0.48 -0.41 -0.38 -0.45
Capital Gain $ 0.95 -3.39 0.81 -3.62 1.77
Daily Profit $ 0.50 -3.87 0.40 -4.00 1.32

Delta-Gamma hedging
Note on the previous slide that the capital gain from day 0 – day 1: $0.95 = $29.12-$28.17 (see slide 83). The same calculation (the difference between the marked-to-market Gain/loss on the long delta hedge and the marked to market gain/loss on the short option position) is done for the other 4 days.

Delta-Gamma hedging
So why the disparity in P&L over the 5 days? There are three effects, attributable to gamma, theta and the carrying cost of the position.
Gamma: For the largest moves in the stock, the market-maker loses money.We have already examined the short gamma effect, but there is no harm in a little repetition:
As the stock price rises (38.75 – $40.00), the delta of the call increases and it loses money faster than the stock makes money.
As the stock price falls ($40.50 – $39.25), the delta of the call decreases and it makes money more slowly than the fixed stock position loses money

Delta-Gamma hedging
In effect, the market-maker becomes unhedged net long as the stock price falls and unhedged net short as the stock price rises. The losses on day 2 and 4 are attributable to gamma.
Theta: Time decay is especially evident in the profit on day 5, but is also responsible for the profit on days 1 and 3.
Cost of carry: In order to hedge, the market-maker must purchase stock which, as we have seen, carries a day-to-day cost of financing

Delta-Gamma hedging
The table confirms something that we already know: that the market-maker who is short the call wants small stock price moves and can suffer a substantial loss with a big move.
So, clearly delta, gamma and theta play a role in determining the profit on a delta hedged position.
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Rho
Rho measures the rate of change of premium to changes in interest rates.
Call options on stock will have a positive Rho because an increase in interest rates will make calls a more desirable alternative to buying the stock because of the relative carrying cost. If interest rates increase call premiums will increase.
Put options on stock will have a negative Rho because puts will be a less desirable alternative to selling the stock. If interest rates increase put premiums will decrease.
Deep ITM options have the highest Rho because they require the greatest cash outlay.
The more time there is to expiry, the higher the Rho. The cost of carry on the trade is effectively higher.

Rho
Short puts have a positive Rho because if interest rates increase it is cheaper to sell the puts (positive Delta exposure) than to buy the shares
Short calls have a negative Rho because if interest rate increase more money can be made in absolute terms on selling the underlying then selling calls.
Rho is typically expressed as the amount of money, per share of the underlying, that the value of the option will gain or lose as the risk free interest rate rises or falls by 1.0% per annum (100 basis points)
Rho is seldom used because interest rates are usually fairly stable.

Rho
Summary:
Long calls, short puts= Long Rho
Long puts, short calls = Short Rho
98

Position Analysis
Position analysis involves analysing the overall exposure of the position to changing market conditions. Whether the position is a straddle or a condor, constant monitoring of the option price sensitivities is required.

Position Analysis: Call Ratio Spread
Analysing the position risk of a call ratio spread:

A call ratio spread involves selling more higher strike options relative to lower strike options bought. In other words we are net short vega and gamma, but long theta. Why are we long theta? The payoff profile of this position can be seen in the following diagram:

100

101

95
105
Call Ratio Spread
Long 10 June calls, strike = 95

Short 30 June calls, Strike = 105

Position Analysis: Call Ratio Spread
102

Position Analysis: Call Ratio Spread
Why is the position delta 0??

The position sensitivities are obtained by simply multiplying each sensitivity by the number of relevant options. Intuitively the position sensitivities make sense because we are essentially net short options.

How do we interpret the sensitivities:

103

Position Analysis: Call Ratio Spread
Position gamma is:[(10x+.043)+(30x-0.044)= -0.89

What does this mean:. The negative gamma is an indication that any large move in the underlying will hurt this position.

Under current market conditions, if the stock price begins to fall and we want to remain delta neutral, for each point decline (rise)in the stock price we need to sell (buy)approx. 890 shares (LIFFE options)

104

Position Analysis: Call Ratio Spread
In other words we are selling into a decline and buying into a rise. Note that we do not have to remain delta neutral. But if we do not the gamma risk is even greater.

If the market rises to 105 and stays there, as we move towards expiry the position will have much larger gamma risk because the 105 calls are now ATM where gamma is highest.

105

Position Analysis: Call Ratio Spread
When the 105 calls are ATM, for any given move in the stock price the rate of change of the option price is much greater, therefore riskier.

As the stock price increases beyond 105, the position delta becomes net short and by how much depends on time to expiry and gamma.

106

Position Analysis: Call Ratio Spread
With this kind of gamma risk the trader should consider a number of alternatives:

He could buy some positive gamma to try and offset the negative gamma risk, close out the position or hedge himself delta neutral.

The position, however, does have a positive theta. This is greatest when the underlying is trading at 105.

107

Position Analysis: Call Ratio Spread
If the market trades up to 105 and remains there, because we are net long theta, we will make more money from time decay on the short positions than we lose on the long ones.

Working against theta is the negative vega of the position. The higher the volatility the more the position suffers, especially when the short calls are ATM

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Position Analysis: Bullish Vertical Spread
In the previous example we were short more calls than we were long (at different strikes).

If we sell 1 higher strike call against 1 lower strike call we create what is known as a bullish vertical spread.

Consider the following example:

109

Position Analysis: Bullish Vertical Spread
Buy a 19 (strike)call and sell a 20 (strike) call when the underlying is trading at 20, 90 days to expiry and volatility is 22%
110
price
delta
gamma
theta
vega
Long119
call
-1.43
+.6951
+.0015
-4.09
+.034
Short 1 20
call
+.865
-.5177
-.00181
+4.09
-.039
+.565
+.178
-.00022
0
-.005
The position payoff profile looks like this:

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19
20
Bullish Vertical Spread

Long 1 call, strike = 19

Sell 1 call, strike = 20

Position Analysis: Bullish Vertical Spread
The net delta of the position (which costs £565) is long which means that to be delta neutral we need to short 178 shares (assume this is a LIFFE option)

Note the very low value for gamma, theta and vega which we shall return to below.

112

Position Analysis: Bullish Vertical Spread
If the underlying ends up at 19 both options expire worthless and I lose the £565 premium. If the underlying finishes at 20 or above I make £1000 {(S-X = 1) x 1000 shares} on the 19 call less the £565 premium = £435.This is the most I can make. Therefore the bias of this type of spread is bullish but has a bounded upper value.

113

Position Analysis: Bullish Vertical Spread
Why would anyone buy such a spread?: Time decay neutrality and volatility neutrality: the individual gamma, vega and theta’s should theoretically offset each other, as they do in the above example.

If you think that over the next 90 days the underlying will increase but you do not know when, volatility will go down but you do not want to get hurt and you don’t want any exposure to time decay, you could set up a bullish vertical spread.

114

Position Analysis: Bullish Vertical Spread
If the market does nothing for 30 days and then moves upwards you won’t have lost money through time decay and can profit from the long call. You are also essentially neutral to volatility as the two vega’s offset each other
Also notice that the position has very little curvature as shown by the very low gamma. Essentially this is a low risk position with limited upside and downside. This is sometimes called bounded convexity.

115

Summary of the Greeks
(These slides are based on Natenburg, Chapter 8 & 11).
The following tables show how various option positions’ exposure to time, curvature, time decay, volatility and the underlying can be easily summarised by the various Greeks. Understanding these sensitivities is crucial to understanding options. The following table is a summary of plain vanilla sensitivities:

116

Summary of the Greeks
117
Delta Exposure Gamma Exposure
(Curvature) Theta Exposure
(Time Decay) Vega Exposure
(Volatility)
Long Underlying Long 0 0 0
Short Underlying Short 0 0 0
Long Puts
Short Long Short Long
Short Puts Long Short Long Short
Long Calls Long Long Short Long
Short Calls Short Short Long Short

Summary of the Greeks
118
If Delta Exposure is:
Long
Short You want the underlying to:
Rise in price
Fall in price
If Gamma Exposure is:
Long
Short You want the underlying to:
Move very quickly, regardless of direction
Move slowly, or not at all
If your Theta Exposure is:
Long
Short The passage of time will:
Increase the value of your position
Decrease the value of your position
If your Vega Exposure is:
Long
Short You want volatility to:
Rise
Fall

Summary of the Greeks
A summary of the general trading strategies available to traders:
Scalping: buy at bid, sell at offer as often as possible without regard to theoretical value
Directional trading
Spreading: involves taking simultaneous and usually opposing positions in different instruments

119

Summary of the Greeks
There are a number of spreads that qualify as volatility spreads- straddle, strangle, call ratio (vertical) spread, put ratio (vertical) spread, call ratio back spread, long time spread-and we shall be looking at these in more detail in the volatility part of the course. Here, we will summarise the exposure of volatility spreads to time, volatility, curvature and the underlying.

120

Summary of the Greeks
It is important to note that a trader who sets up a volatility spread is concerned primarily with the magnitude of movement, and secondarily with the direction: volatility spreads will always be approximately delta neutral. If a trader has a large positive or negative delta, such that directional considerations become more important, then the position is no longer a volatility spread.
Consider the following table, where all positions are delta neutral

121

122
Spread Type Delta Exposure Gamma Exposure Theta Exposure Vega Exposure
Backspread 0 + – +
Long Straddle 0 + – +
Long Strangle 0 + – +
Short Butterfly 0 + – +
Ratio Vertical Spread 0 – + –
Short Straddle 0 – + –
Short Strangle 0 – + –
Long Butterfly 0 – + –
Long Time Spread 0 – + +
Short Time Spread 0 + – –

123
Spread Type Delta Exposure Gamma Exposure Theta Exposure Vega Exposure
Bullish Vertical Spread + 0 0 0
Bearish Vertical Spread – 0 0 0
Long Condor 0 – + –
Short Condor 0 + – +
Not Volatility Spreads
Collar + 0 0 0
Zero Cost Collar + 0 0 0

Summary of the Greeks
Note: all spreads which are helped by movements in the underlying market, have a positive gamma: backspreads, long straddles and strangles, short butterflies and short time spreads. Spreads which are hurt by movements in the underlying are short gamma: ratio vertical spreads, short straddles and strangles, long butterflies and long time spreads.

124

Summary of the Greeks
A volatility spread may start out delta neutral, but this will change as the underlying moves. Changes in volatility and time also affect delta. Continuous adjustments (as per the Black Scholes) are impossible. So when should a volatility trader adjust the delta?

125

Summary of the Greeks
Adjust at regular intervals: volatility is measured over regular time intervals (daily, weekly, monthly). Logical to adjust delta at similar intervals
Adjust when the position becomes a predetermined number of deltas long or short: a trader should know how much directional risk he/she is willing to take. How much directional exposure depends on the individual trader. If you are the head of a huge options book then being long 10% of the underlying probably won’t matter a great deal.

126

Summary of the Greeks
Adjust by feel: Some traders have a good sense of when the market is going to move. If this is the case then a trader can simply adjust the delta at the appropriate time.

127

Applied Exercise
Option Strategist Exercise 2

128

Henley_Business_School_Logo_HYBRID

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C
S

D
D
C
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d

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¶
P
S

D
D
p
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¶
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2
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70.00
75.26
80.53
85.79
91.05
96.32
101.58
106.84
112.11
117.37
1.00
7.21
13.42
19.63
25.84
32.05
38.26
44.47
50.68
56.89
0.00
0.05
0.10
0.15
0.20
0.25
Gamma
Underlying Asset Price
Tenor to Expiration (Days)
Rotate Right
Rotate Left
Goto Inputs
Goto Output
Data
Print

60.00
67.37
74.74
82.11
89.47
96.84
104.21
111.58
118.95
126.32
1.00
7.21
13.42
19.63
25.84
32.05
38.26
44.47
50.68
56.89
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0.00
Theta
Underlying Asset Price
Tenor to Expiration (Days)
Rotate Right
Rotate Left
Goto Inputs
Goto Output
Data
Print

60.00
71.05
82.11
93.16
104.21
115.26
126.32
1.00
4.11
7.21
10.32
13.42
16.53
19.63
22.74
25.84
28.95
32.05
35.16
38.26
41.37
44.47
47.58
50.68
53.79
56.89
60.00
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
Vega
Underlying Asset Price
Tenor to Expiration (Days)

Rotate Right

Rotate Left

Goto Inputs

Goto Output
Data

3-D Charting Key

Sheet: Start
Sheet: Inputs
Sheet: Chart
Sheet: Output Data
3-D Charting Input Sheet
Underlying Asset Price
Inputs
Variable
Table of Values
60.0
63.6842105263158
67.3684210526316
71.0526315789474
74.7368421052632
78.4210526315789
82.1052631578947
85.7894736842105
89.4736842105263
93.1578947368421
96.8421052631579
100.526315789474
104.210526315789
107.894736842105
111.578947368421
115.263157894737
118.947368421053
122.631578947368
126.315789473684
130.0
1.0
Model
5.0
5.675324109272264E-276
1.2617590757604336E-152
3.709858140397165E-69
1.3507753142723673E-20
0.010177247294191453
6.460308952685013E-12
4.3832756073617865E-45
1.528836494318413E-99
5.851404983558597E-173
2.6457690300946983E-263
1.0
1.0
Type
6.0
4.10526315789474
0.0
9.870533840647895E-233
8.639142143040876E-164
8.958345109466958E-110
1.2771836049612548E-68
1.2965684834933605E-38
2.5534835272826276E-18
1.5852617149250003E-6
0.03313093551596499
1.7648508104737992E-4
1.365712174892813E-12
6.909106323240241E-26
8.437405437681354E-44
7.773767420044966E-66
1.4663598130798738E-91
1.362401779195582E-120
1.3530895351850853E-152
2.852937842789205E-187
2.349165109774584E-224
1.3002637517288064E-263
90.0
Underlying Asset Price
7.21052631578947
2.6942571157077717E-182
3.815138443216537E-133
6.411171308965621E-94
3.431927489030166E-63
8.763143981567393E-40
1.0096643971013301E-22
3.4446966092950816E-11
1.7016547177346855E-4
0.04686494877720933
0.002278057085090913
5.263672221490462E-8
1.3610710267573678E-15
8.28490103649613E-26
2.2711145836957795E-38
4.9492092567371445E-53
1.413237650783469E-69
8.219201423689124E-88
1.4388224044234448E-107
1.0725874529186872E-128
4.638274743184445E-151
90.0
Exercise Price
10.3157894736842
8.0525080782633E-128
1.7460251233344637E-93
4.379683638091427E-66
1.2667931100907112E-44
2.800813730122362E-28
2.2761963778149683E-16
2.53231968636382E-8
0.0011693557210180597
0.05750399174493201
0.0067383200350955505
3.7602186862139023E-6
1.8178862237479005E-11
1.28029958357277E-18
2.0670035899524724E-27
1.1379840971322255E-37
3.0295389452321907E-49
5.308001928725651E-62
8.041665172200536E-76
1.3425433438295227E-90
3.06548571826831E-106
60.0
Days to Expiration
13.4210526315789
1.5430442648155197E-98
3.6736023948690275E-72
4.0821524569560114E-51
1.2394514706935806E-34
4.399689699188251E-22
6.104213459212664E-13
9.093070994530406E-7
0.0034110762870886414
0.06647449400390572
0.012497229828215689
3.85891342069624E-5
3.0998002490720693E-9
9.657665742122505E-15
1.6534635763652395E-21
2.1108303447438403E-29
2.6279352731792747E-38
4.043492360025638E-48
9.483778669266952E-59
4.0852309428159906E-70
3.815599377037035E-82
0.1
Volatility
16.5263157894737
2.9923161353370647E-80
7.663547860517981E-59
9.210613344650837E-42
2.181555746821809E-28
3.315038567210915E-18
8.611448305206742E-11
8.687224273743008E-6
0.006800257778260593
0.07436467768950673
0.018779223440663375
1.68696466261286E-4
7.831137258765826E-8
2.5982080522020624E-12
8.17554579203771E-18
3.125950625476563E-24
1.8060024452463666E-31
1.9109917850135384E-39
4.391118056931603E-48
2.5490425095901494E-57
4.277668265268424E-67
0.05
Interest Rate
19.6315789473684
9.653202752579755E-68
9.920664198236772E-50
2.3179339755313897E-35
4.120032856040272E-24
1.5046222158435684E-15
2.575931746656137E-9
4.12447020319526E-5
0.011060286437229904
0.08148274564611918
0.025176035720806924
4.6932652352171524E-4
7.228195010151062E-7
1.2083586793325719E-10
2.7821875448021367E-15
1.0869046228420366E-20
8.65515992789438E-27
1.651756182546441E-33
8.718950362448614E-41
1.4458931764539093E-48
8.437919225717588E-57
0.02
Yield Rate
22.7368421052632
1.205718017593038E-58
4.210407955596314E-43
1.0523791893219285E-30
5.341283437525732E-21
1.2979476673803507E-13
3.078175284112622E-8
1.2933106755485366E-4
0.015919742881778044
0.08801357809600001
0.03148845943089376
9.979168948785265E-4
3.6746813442193263E-6
1.990059405956984E-9
1.9467460726210534E-13
4.11864872618033E-18
2.2078787108313667E-23
3.448789857500885E-29
1.776548916189682E-35
3.368612116228844E-42
2.5931311638784894E-49
0.0
Market Option Price
25.8421052631579
9.889643246382413E-52
4.6002609331226586E-38
3.6604242712351814E-27
1.2468669004737385E-18
3.866870189808558E-12
2.0428560193751096E-7
3.1064054516093803E-4
0.021163919928231027
0.09407825919038687
0.037624184183446936
0.001784353575976033
1.2740793512267985E-5
1.6851615940032897E-8
4.947116845039026E-12
3.7768110579562667E-16
8.619058383692085E-21
6.648895934841722E-26
1.9331947891193346E-31
2.3336623200115213E-37
1.2748377597660773E-43
28.9473684210526
2.682277091572248E-46
4.197627262814272E-34
2.2276631886097447E-24
9.091906069754692E-17
5.5968217183175676E-11
9.090320047214109E-7
6.221691212259994E-4
0.026635871268200825
0.09976115082854757
0.04354481839876911
0.002834395903487896
3.404592549301699E-5
9.080473315127082E-8
6.319619208312013E-11
1.321963710824525E-14
9.412275494179824E-19
2.5455064178864315E-23
2.881775863731997E-28
1.4888142698010622E-33
3.7906471019841773E-39
Tenor to Expiration (Days)
32.0526315789474
6.475412887069417E-42
6.5780117079379065E-31
3.9344164566951156E-22
2.9023469022576348E-15
4.851227121731277E-10
3.044392832667634E-6
0.0010947471588541723
0.03222475449916904
0.10512379652193503
0.04923919386727525
0.004137148910730297
7.55834186155794E-5
3.548535052951004E-7
4.952997081494935E-10
2.335245803578075E-13
4.161054756936145E-17
3.093956652283879E-21
1.0480255180785735E-25
1.7483420468783905E-30
1.5396959443177897E-35
19.0
35.1578947368421
2.6402099491379706E-38
2.8222216507075305E-28
2.797618594379186E-20
5.046129015604082E-14
2.883307585281176E-9
8.26998114155039E-6
0.001750560473634604
0.03785363602230179
0.11021271957139771
0.05470977546215639
0.005672012699984315
1.4635659198717368E-4
1.0945442443330799E-6
2.709566109591448E-9
2.49433752197777E-12
9.459027537168522E-16
1.6173341683185567E-19
1.3506729514267949E-23
5.914971888805403E-28
1.447090718172832E-32
19.0
38.2631578947368
2.802314113443331E-35
4.542705784670277E-26
9.991292208572906E-19
5.538316560697093E-13
1.2876716645004059E-8
1.9167903232116082E-5
0.002602920957893634
0.04346972931144666
0.11506410040143948
0.05996562403913799
0.007413795384192517
2.5546236963380856E-4
2.821807780812193E-6
1.1288875094989505E-8
1.820249869617167E-11
1.2995752635148678E-14
4.463737861499449E-18
7.938408995269527E-22
7.802820655116515E-26
4.4926881552621256E-30
20.0
41.3684210526316
1.0481118949573093E-32
3.4200151731721744E-24
2.092239948794167E-17
4.254819372008373E-12
4.607111276785259E-8
3.92750680905444E-5
0.0036572913030029643
0.04903723061313341
0.11970673452331053
0.06501870421226445
0.009336044196799344
4.113417758948407E-4
6.329279840367355E-6
3.807494698058499E-8
9.885526841850379E-11
1.2083634268613204E-13
7.508594543927672E-17
2.538594708304539E-20
4.960382596873543E-24
5.911223408807335E-28
20.0
Axis
Variable
Min
Max
Data Points
Range
Change
44.4736842105263
1.7183216693066387E-30
1.4117737785795173E-22
2.8722945886402204E-16
2.4650494550758224E-11
1.3830528149047406E-7
7.298795749420684E-5
0.004912810464250306
0.054532235659706677
0.12416398510897014
0.06988193942313017
0.011413027725318398
6.212791380466598E-4
1.2714146372865692E-5
1.0864112014353001E-7
4.249571525668987E-10
8.250101740375449E-13
8.537499503568306E-16
5.016541394328163E-19
1.7703506152555952E-22
3.944701387346876E-26
X
70.0
3.68421052631579
47.578947368421
1.4509949876387256E-28
3.593764511598589E-21
2.807393636274088E-15
1.1379768199386638E-10
3.604816707110432E-7
1.25373651963769E-4
0.00636387163284723
0.059939177585053044
0.1284551184308255
0.07456819860578985
0.013620804126708491
8.911398803151971E-4
2.336840916496701E-5
2.7093330819045937E-7
1.5134582498690823E-9
4.393141583616687E-12
7.083419082808238E-15
6.730124751145585E-18
3.970996120917115E-21
1.5246480364693062E-24
5.0
Y
59.0
3.10526315789474
50.6842105263158
7.128377353113537E-27
6.164675135592252E-20
2.079180738576562E-14
4.363883948691534E-10
8.371055996403296E-7
2.019312518812232E-4
0.008001532853268765
0.06524833748652363
0.132596244884647
0.07908978717196431
0.015937710049233975
0.0012252961841854515
3.994039915186555E-5
6.052474123337999E-7
4.622055445544449E-9
1.90952349639472E-11
4.543489627133873E-14
6.581162294947153E-17
6.096046189015396E-20
3.772915477062274E-23
5.0
Z
53.7894736842105
2.2375314444333443E-25
7.629314224460868E-19
1.2240873068190224E-13
1.4353185529534617E-9
1.7667078167250515E-6
3.0834348362112E-4
0.009814679875068786
0.0704541011379772
0.136600999187226
0.0834582132537721
0.01834450134999947
0.001626681733609405
6.427652383776515E-5
1.2343368488453014E-6
1.2429432207814594E-8
7.016620156794138E-11
2.3554665855984726E-13
4.954255375008379E-16
6.838774610874215E-19
6.456902056506033E-22
56.8947368421053
4.828517372461751E-24
7.1853690484906E-18
5.947593348547584E-13
4.1517599246071675E-9
3.4418489036726706E-6
4.5024731968325315E-4
0.011790934887847859
0.07555373313595948
0.1404810429747199
0.08768410398781766
0.020824296015029366
0.0020969242231087463
9.835302584109041E-5
2.3323929236119624E-6
3.004852228129262E-8
2.2401969697901915E-10
1.021885371351259E-12
2.996468020271945E-15
5.9013541988886485E-18
8.116956055667916E-21
60.0
7.59199187450256E-23
5.3725626748424314E-17
2.456919843616693E-12
1.0773376604241685E-8
6.266463838389884E-6
6.330406093426216E-4
0.013917338512533323
0.08054651064165815
0.14424644336761006
0.0917772023579684
0.0233624134099263
0.002636516718019257
1.4420620565769315E-4
4.131862211852529E-6
6.638885982811534E-8
6.351002048625587E-10
3.8136538642146475E-12
1.5063220105530633E-14
4.079681182698178E-17
7.862315457431714E-20
60.0
60.0
60.0
60.0
60.0
60.0
60.0
60.0
60.0
60.0
60.0
60.0
60.0
60.0
60.0
60.0
60.0
60.0
60.0
60.0
63.6842105263158
63.6842105263158
63.6842105263158
63.6842105263158
63.6842105263158
63.6842105263158
63.6842105263158
63.6842105263158
63.6842105263158
63.6842105263158
63.6842105263158
63.6842105263158
63.6842105263158
63.6842105263158
63.6842105263158
63.6842105263158
63.6842105263158
63.6842105263158
63.6842105263158
63.6842105263158
67.3684210526316
67.3684210526316
67.3684210526316
67.3684210526316
67.3684210526316
67.3684210526316
67.3684210526316
67.3684210526316
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1.776548916189682E-35
1.9331947891193346E-31
2.881775863731997E-28
1.0480255180785735E-25
1.3506729514267949E-23
7.938408995269527E-22
2.538594708304539E-20
5.016541394328163E-19
6.730124751145585E-18
6.581162294947153E-17
4.954255375008379E-16
2.996468020271945E-15
1.5063220105530633E-14
0.0
2.349165109774584E-224
1.0725874529186872E-128
1.3425433438295227E-90
4.0852309428159906E-70
2.5490425095901494E-57
1.4458931764539093E-48
3.368612116228844E-42
2.3336623200115213E-37
1.4888142698010622E-33
1.7483420468783905E-30
5.914971888805403E-28
7.802820655116515E-26
4.960382596873543E-24
1.7703506152555952E-22
3.970996120917115E-21
6.096046189015396E-20
6.838774610874215E-19
5.9013541988886485E-18
4.079681182698178E-17
0.0
1.3002637517288064E-263
4.638274743184445E-151
3.06548571826831E-106
3.815599377037035E-82
4.277668265268424E-67
8.437919225717588E-57
2.5931311638784894E-49
1.2748377597660773E-43
3.7906471019841773E-39
1.5396959443177897E-35
1.447090718172832E-32
4.4926881552621256E-30
5.911223408807335E-28
3.944701387346876E-26
1.5246480364693062E-24
3.772915477062274E-23
6.456902056506033E-22
8.116956055667916E-21
7.862315457431714E-20
3D Chart Data
Underlying Asset Price
Table of Values
60.0
63.6842105263158
67.3684210526316
71.0526315789474
74.7368421052632
78.4210526315789
82.1052631578947
85.7894736842105
89.4736842105263
93.1578947368421
96.8421052631579
100.526315789474
104.210526315789
107.894736842105
111.578947368421
115.263157894737
118.947368421053
122.631578947368
126.315789473684
130.0
5.675324109272264E-276
1.2617590757604336E-152
3.709858140397165E-69
1.3507753142723673E-20
0.010177247294191453
6.460308952685013E-12
4.3832756073617865E-45
1.528836494318413E-99
5.851404983558597E-173
2.6457690300946983E-263
4.10526315789474
0.0
9.870533840647895E-233
8.639142143040876E-164
8.958345109466958E-110
1.2771836049612548E-68
1.2965684834933605E-38
2.5534835272826276E-18
1.5852617149250003E-6
0.03313093551596499
1.7648508104737992E-4
1.365712174892813E-12
6.909106323240241E-26
8.437405437681354E-44
7.773767420044966E-66
1.4663598130798738E-91
1.362401779195582E-120
1.3530895351850853E-152
2.852937842789205E-187
2.349165109774584E-224
1.3002637517288064E-263
7.21052631578947
2.6942571157077717E-182
3.815138443216537E-133
6.411171308965621E-94
3.431927489030166E-63
8.763143981567393E-40
1.0096643971013301E-22
3.4446966092950816E-11
1.7016547177346855E-4
0.04686494877720933
0.002278057085090913
5.263672221490462E-8
1.3610710267573678E-15
8.28490103649613E-26
2.2711145836957795E-38
4.9492092567371445E-53
1.413237650783469E-69
8.219201423689124E-88
1.4388224044234448E-107
1.0725874529186872E-128
4.638274743184445E-151
10.3157894736842
8.0525080782633E-128
1.7460251233344637E-93
4.379683638091427E-66
1.2667931100907112E-44
2.800813730122362E-28
2.2761963778149683E-16
2.53231968636382E-8
0.0011693557210180597
0.05750399174493201
0.0067383200350955505
3.7602186862139023E-6
1.8178862237479005E-11
1.28029958357277E-18
2.0670035899524724E-27
1.1379840971322255E-37
3.0295389452321907E-49
5.308001928725651E-62
8.041665172200536E-76
1.3425433438295227E-90
3.06548571826831E-106
13.4210526315789
1.5430442648155197E-98
3.6736023948690275E-72
4.0821524569560114E-51
1.2394514706935806E-34
4.399689699188251E-22
6.104213459212664E-13
9.093070994530406E-7
0.0034110762870886414
0.06647449400390572
0.012497229828215689
3.85891342069624E-5
3.0998002490720693E-9
9.657665742122505E-15
1.6534635763652395E-21
2.1108303447438403E-29
2.6279352731792747E-38
4.043492360025638E-48
9.483778669266952E-59
4.0852309428159906E-70
3.815599377037035E-82
16.5263157894737
2.9923161353370647E-80
7.663547860517981E-59
9.210613344650837E-42
2.181555746821809E-28
3.315038567210915E-18
8.611448305206742E-11
8.687224273743008E-6
0.006800257778260593
0.07436467768950673
0.018779223440663375
1.68696466261286E-4
7.831137258765826E-8
2.5982080522020624E-12
8.17554579203771E-18
3.125950625476563E-24
1.8060024452463666E-31
1.9109917850135384E-39
4.391118056931603E-48
2.5490425095901494E-57
4.277668265268424E-67
19.6315789473684
9.653202752579755E-68
9.920664198236772E-50
2.3179339755313897E-35
4.120032856040272E-24
1.5046222158435684E-15
2.575931746656137E-9
4.12447020319526E-5
0.011060286437229904
0.08148274564611918
0.025176035720806924
4.6932652352171524E-4
7.228195010151062E-7
1.2083586793325719E-10
2.7821875448021367E-15
1.0869046228420366E-20
8.65515992789438E-27
1.651756182546441E-33
8.718950362448614E-41
1.4458931764539093E-48
8.437919225717588E-57
22.7368421052632
1.205718017593038E-58
4.210407955596314E-43
1.0523791893219285E-30
5.341283437525732E-21
1.2979476673803507E-13
3.078175284112622E-8
1.2933106755485366E-4
0.015919742881778044
0.08801357809600001
0.03148845943089376
9.979168948785265E-4
3.6746813442193263E-6
1.990059405956984E-9
1.9467460726210534E-13
4.11864872618033E-18
2.2078787108313667E-23
3.448789857500885E-29
1.776548916189682E-35
3.368612116228844E-42
2.5931311638784894E-49
25.8421052631579
9.889643246382413E-52
4.6002609331226586E-38
3.6604242712351814E-27
1.2468669004737385E-18
3.866870189808558E-12
2.0428560193751096E-7
3.1064054516093803E-4
0.021163919928231027
0.09407825919038687
0.037624184183446936
0.001784353575976033
1.2740793512267985E-5
1.6851615940032897E-8
4.947116845039026E-12
3.7768110579562667E-16
8.619058383692085E-21
6.648895934841722E-26
1.9331947891193346E-31
2.3336623200115213E-37
1.2748377597660773E-43
28.9473684210526
2.682277091572248E-46
4.197627262814272E-34
2.2276631886097447E-24
9.091906069754692E-17
5.5968217183175676E-11
9.090320047214109E-7
6.221691212259994E-4
0.026635871268200825
0.09976115082854757
0.04354481839876911
0.002834395903487896
3.404592549301699E-5
9.080473315127082E-8
6.319619208312013E-11
1.321963710824525E-14
9.412275494179824E-19
2.5455064178864315E-23
2.881775863731997E-28
1.4888142698010622E-33
3.7906471019841773E-39
Tenor to Expiration (Days)
32.0526315789474
6.475412887069417E-42
6.5780117079379065E-31
3.9344164566951156E-22
2.9023469022576348E-15
4.851227121731277E-10
3.044392832667634E-6
0.0010947471588541723
0.03222475449916904
0.10512379652193503
0.04923919386727525
0.004137148910730297
7.55834186155794E-5
3.548535052951004E-7
4.952997081494935E-10
2.335245803578075E-13
4.161054756936145E-17
3.093956652283879E-21
1.0480255180785735E-25
1.7483420468783905E-30
1.5396959443177897E-35
35.1578947368421
2.6402099491379706E-38
2.8222216507075305E-28
2.797618594379186E-20
5.046129015604082E-14
2.883307585281176E-9
8.26998114155039E-6
0.001750560473634604
0.03785363602230179
0.11021271957139771
0.05470977546215639
0.005672012699984315
1.4635659198717368E-4
1.0945442443330799E-6
2.709566109591448E-9
2.49433752197777E-12
9.459027537168522E-16
1.6173341683185567E-19
1.3506729514267949E-23
5.914971888805403E-28
1.447090718172832E-32
38.2631578947368
2.802314113443331E-35
4.542705784670277E-26
9.991292208572906E-19
5.538316560697093E-13
1.2876716645004059E-8
1.9167903232116082E-5
0.002602920957893634
0.04346972931144666
0.11506410040143948
0.05996562403913799
0.007413795384192517
2.5546236963380856E-4
2.821807780812193E-6
1.1288875094989505E-8
1.820249869617167E-11
1.2995752635148678E-14
4.463737861499449E-18
7.938408995269527E-22
7.802820655116515E-26
4.4926881552621256E-30
41.3684210526316
1.0481118949573093E-32
3.4200151731721744E-24
2.092239948794167E-17
4.254819372008373E-12
4.607111276785259E-8
3.92750680905444E-5
0.0036572913030029643
0.04903723061313341
0.11970673452331053
0.06501870421226445
0.009336044196799344
4.113417758948407E-4
6.329279840367355E-6
3.807494698058499E-8
9.885526841850379E-11
1.2083634268613204E-13
7.508594543927672E-17
2.538594708304539E-20
4.960382596873543E-24
5.911223408807335E-28
44.4736842105263
1.7183216693066387E-30
1.4117737785795173E-22
2.8722945886402204E-16
2.4650494550758224E-11
1.3830528149047406E-7
7.298795749420684E-5
0.004912810464250306
0.054532235659706677
0.12416398510897014
0.06988193942313017
0.011413027725318398
6.212791380466598E-4
1.2714146372865692E-5
1.0864112014353001E-7
4.249571525668987E-10
8.250101740375449E-13
8.537499503568306E-16
5.016541394328163E-19
1.7703506152555952E-22
3.944701387346876E-26
47.578947368421
1.4509949876387256E-28
3.593764511598589E-21
2.807393636274088E-15
1.1379768199386638E-10
3.604816707110432E-7
1.25373651963769E-4
0.00636387163284723
0.059939177585053044
0.1284551184308255
0.07456819860578985
0.013620804126708491
8.911398803151971E-4
2.336840916496701E-5
2.7093330819045937E-7
1.5134582498690823E-9
4.393141583616687E-12
7.083419082808238E-15
6.730124751145585E-18
3.970996120917115E-21
1.5246480364693062E-24
50.6842105263158
7.128377353113537E-27
6.164675135592252E-20
2.079180738576562E-14
4.363883948691534E-10
8.371055996403296E-7
2.019312518812232E-4
0.008001532853268765
0.06524833748652363
0.132596244884647
0.07908978717196431
0.015937710049233975
0.0012252961841854515
3.994039915186555E-5
6.052474123337999E-7
4.622055445544449E-9
1.90952349639472E-11
4.543489627133873E-14
6.581162294947153E-17
6.096046189015396E-20
3.772915477062274E-23
53.7894736842105
2.2375314444333443E-25
7.629314224460868E-19
1.2240873068190224E-13
1.4353185529534617E-9
1.7667078167250515E-6
3.0834348362112E-4
0.009814679875068786
0.0704541011379772
0.136600999187226
0.0834582132537721
0.01834450134999947
0.001626681733609405
6.427652383776515E-5
1.2343368488453014E-6
1.2429432207814594E-8
7.016620156794138E-11
2.3554665855984726E-13
4.954255375008379E-16
6.838774610874215E-19
6.456902056506033E-22
56.8947368421053
4.828517372461751E-24
7.1853690484906E-18
5.947593348547584E-13
4.1517599246071675E-9
3.4418489036726706E-6
4.5024731968325315E-4
0.011790934887847859
0.07555373313595948
0.1404810429747199
0.08768410398781766
0.020824296015029366
0.0020969242231087463
9.835302584109041E-5
2.3323929236119624E-6
3.004852228129262E-8
2.2401969697901915E-10
1.021885371351259E-12
2.996468020271945E-15
5.9013541988886485E-18
8.116956055667916E-21
60.0
7.59199187450256E-23
5.3725626748424314E-17
2.456919843616693E-12
1.0773376604241685E-8
6.266463838389884E-6
6.330406093426216E-4
0.013917338512533323
0.08054651064165815
0.14424644336761006
0.0917772023579684
0.0233624134099263
0.002636516718019257
1.4420620565769315E-4
4.131862211852529E-6
6.638885982811534E-8
6.351002048625587E-10
3.8136538642146475E-12
1.5063220105530633E-14
4.079681182698178E-17
7.862315457431714E-20

45
.
0
$
)
1
(
*
56
.
2051
$
365
/
08
.
0
=
–
e

48
.
0
$
)
1
(
*
3
.
2181
$
365
/
08
.
0
=
–
e

Position
Delta
Gamma
Theta
Long 10 June 95
calls
.76
+.043
-.0344
Short 30 June 105
calls
.26
-.044
+.0278
Position
0
-0.89
+.4900

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