This presentation is given by Greg Boland, who is Chief Executive Officer and an authorised representative of Tiger Fintech (NZ) Limited.
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July 2023
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Greg Boland - Copyright strictly reserved. No part of these course notes may be reproduced orcopied in any form or by any means without the written permission of Greg Boland.
The Greeks
The valuation models, such as Black Scholes, are not only used to price options but also used to predict the effect on option values of changes in any of the variable factors: share price, time and volatility.
Delta
The Delta of an option measures the change in option premium for a $1 increase in share price (or underlying asset price). The delta is quoted as a proportion of the share price change, for example if the share price rises by 10 cents and the option price rises by 2 cents the delta is 0.20 or 20%.
The delta will change during the life of an option. An option which is in-the-money will have a high delta whereas an option which is out-of-the-money will have a low delta.
In mathematical terms the delta of an option is calculated by taking the first derivative of the Black Scholes formula with respect to price N(d1).
The delta of an option is valuable to the option trader who does not intend to hold the option until it expires as the delta will indicate the rate at which the option premium will respond to changes in the share price.
For example:
Suppose an option trader believes that Apple’s share price is going to increase in the near future.
Apple currently trading at $185
Apple $180 calls : premium $11.85 and delta = 0.66
Apple $185 calls : premium $8.60 and delta = 0.56
Apple $190 calls : premium $5.90 and delta = 0.47
As the $185 options is at-the-money it has a greater chance of being exercised.
The trader purchases $185 calls with a premium of $8.60 cents each and a delta of 0.56.As the trader had believed, Apple rises rise by $2 the next day . The premium of the $185 calls increases by $1.15 to $9.75 per share. By buying the $185 calls the trader has made a profit of $1.15 per contract (13% in a day / 4880% pa).
The delta of a call option is positive, which is to be expected, since an increase in the stock price would make the call worth more.
• A deep In-The-Money call behaves as if one is long the underlying, and hence the corresponding delta is 1.
• A deep Out-of-The-Money call would have very little change in price as the underlying moves, hence the delta is 0.
• The range of delta for a call is [0,1].
Similarly, the delta of a put option is negative, since a decrease in the stock price would
make the put worth more.
• A deep ITM put’s delta behaves as if one is short the underlying, and hence the corresponding delta is -1.
• A deep OTM put would have very little change in price as the underlying moves, hence the delta is 0.
• The range of delta for a put is [−1,0][−1,0].
An ITM call will approach a delta of 1 as it gets closer to expiry, since the extrinsic value is minimal, and the intrinsic value has a delta of 1. Likewise, an OTM call will approach a delta of 0 as it gets close to expiry, since the intrinsic value has a delta of 0. Conversely, the further out to expiry, the closer the delta of a call will get to 0.5. This is because we are less certain if the call will be ITM or OTM.
As the strike price increases, as per the table above, the delta of a call decreases. One interpretation of this is that for the same move in the underlying, the price of the higher strike call is not going to be worth more. Another interpretation is that the higher strike call is less likely to end up in the money, hence it has a lower delta.
The easiest way to graph the delta of a call, would be to consider what happens to the options value as the share price increases. As the price rises the more an option moves to be in-the-money and therefore delta will approach +1. Conversely the more the share price drops the more the option will move towards being out-of-the money and hence delta approaches zero.
We can do the same to graph the delta of the put which is negative and the more the share price drops the more likely the put will end in the money and delta approaches-1 as it is similar to a sold position in the underlying. The higher the price the more the option will move out-of-the-money so delta approaches zero.Note that the delta of the at-the-money call is just slightly over 0.5 while an at-the money put will be slightly less than 0.5.
Note that the delta of the at-the-money call is just slightly over 0.5 while an at-the money put will be slightly less than 0.5.
Delta changes over time and volatility
The following graph is the effect of a decrease in time or volatility on Delta. The blue curve represents an option with higher time value caused by either more time to expiry or higher volatility (or both), and the red curve represents an option with the same strike with less time value (time to expiry or volatility). As time passes, the Delta curve starts to approach zero (out-of the-money) or plus one (in-the-money)
Usefulness of Delta
The delta of the call has several different uses / interpretations / approximations:
1. The rate of change in the price of the call with respect to the underlying
2. Hedge ratio required to remain delta neutral.
3. Probability that the Call will end up in-the-money at expiry.
4. The negative of the rate of change in the price of the call as the strike price increases.The Greeks
Gamma
Gamma measures the rate at which the option delta changes with movements in the share price. Specifically, the gamma of an option tells us by how much the delta of an option would increase by when the underlying moves by $1. Since delta is a first derivative,gamma is a second derivative of the price of the option. It is calculated by taking the second derivative of the Black Scholes price formula, with respect to the share price.It allows us to make predictions about how much the delta will change as the underlying changes. This in turn allows us to predict how much the option value would change as the underlying changes.
The trader, holding many option positions, will use the gamma to determine how quickly their position will change in value. Alternatively, the Gamma can be used to establish a delta neutral position where no change in net option premium occurs when share prices change.
Graph of Gamma
The best way to understand the graph of gamma, is to take the graph of delta and differentiate it point-wise. We take the delta graph (red), find the tangent at each point (blue line), whose slope gives us the value of gamma (blue circle), which we then connectup to get the gamma curve (yellow).
Gamma changes over time and volatility
Just as we understood the graph of Gamma from the graph of Delta, we should look at the effect of Delta changes over time and volatility.
As time decreases and volatility decreases, the Delta curve starts to look more like its graph at expiry, and therefore the Gamma curve starts to look like a normal distribution with lower variance.
The gamma of an option is always positive. We can show this by considering the case of a call option. As the underlying increases, we know that the delta increases, since it is more likely to be ITM. Hence, this tells us that gamma, which is the rate of change of delta,is positive.
Theta – time decay
Theta measures the rate of change of the option premium as time passes. This is also known as the time decay factor.
Theta is useful for the trader when planning how long to hold an option. By convention,theta is negative, which means that if you are long an option, it loses value over time.
It is better to say "paying theta" or "collecting theta" to be more explicit about the position.This is because while being long options means that you are paying theta, it is possible to be long an option strategy (like butterfly or calendar spreads) and be collecting theta.
Vega – change in volatility
Vega measures the rate of change of the price of the option with respect to volatility.Specifically, the vega of an option tells us by how much the price of an option would increase when volatility increases by 1%. It allows us to make predictions about how much the option value would change as volatility changes.
The vega of an option is always positive because when volatility goes up, options price increase and buyers will pay more for the protection/insurance which the options offer.
Graph of Vega against the underlying share price
When we consider the vega of an option, we look at the option value. Since the intrinsic value is constant as volatility changes, we should focus on the extrinsic value.
When the stock is far away from the strike, the extrinsic value is low and an increase in volatility would not affect the payoffs by much. Hence the extrinsic value will not increase significantly, so the vega is low.
When the stock is near the strike, an increase in volatility has a direct effect on the payoffs. Hence the extrinsic value will increase significantly, so the vega is higher.
Vega Changes over Time
Let's consider how vega changes over time:
The blue curve represents an option with more time to expiry, and the red curve represents an option with the same strike with less time to expiry.
As the time to expiry decreases, the at-the-money vega decreases. And at higher and lower share prices a similar affect happens in the wings, since the underlying has less time to move and thus is less likely to affect the extrinsic value.
Graph of Vega Changes over Volatility
Let's consider the graph of vega against volatility:
At-the-money vega is pretty constant - actually, it is slowly decreasing as volatility increases, but not noticeably so. This is represented by the red line above.
When an option is further away from being at-the-money and if the underlying stock is volatile enough to result in a payoff, the extrinsic value would start to increase and thus vega becomes larger. Of course, the further away an option is from ATM, the higher the volatility will have to be before this effect takes place. This explains the difference between the green and blue curves.
For a given volatility, the at-the-money option has the largest vega, and this sets a maximum limit on the vega of other options as per the red line in the above graph.
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