Overview of the Greeks
When we purchase an option, we can trade the cash instrument (called "trading spot" or "trading the cash"), hoping to realize more profit from trading the cash than we pay initially in premium for the option. When we sell an option, we hope that the premium that we are paid upfront dwarfs the losses we will sustain from trading the cash.
When we buy options, we are said to be buying volatility. We make money if the spot rate is volatile enough for us to pay for the option. When we sell options, we are selling volatility. We make money if spot is calm enough that we don't have to hedge the exposure frequently.
However, delta hedging is not the only way for us to make money with options. The genius of derivatives is that it allows us to take positions in (or to hedge against fluctuations in) other aspects of the cash instrument's price evolution. Derivatives are dangerous if we do not understand or address each potential dimension of their risk.
Here are several examples. With a simple plain vanilla option, we can make money if implied volatility moves in our favor. With currency futures, currency forwards and currency options, we can speculate on the spread between interest rates in two different countries for a maturity date. With some exotic options, we can buy an option that appreciates in value with the passage of time (all other things being constant) and that also appreciates in value with movement lower in implied volatility.
Options dealers and savvy options traders use time-proven techniques to break down the risks in an options position or in a portfolio of options, futures, forwards and cash positions into information that is more readily comprehensible and therefore more easily positioned or hedged. This method of analysis employs tools called the "greeks", as well as using simulation, scenario analysis and value-at-risk analysis.
The greeks get their name from the fact that the sensitivities of an option to various market parametres are labelled with letters from the greek alphabet.
DELTA
The delta of an option is the sensitivity of the option's price to very small changes in the price of the underlying instrument. When we talked about trading spot around the options position in order to realize profit that would pay for the option's premium, we were talking about trading the delta.
By taking an opposite position equal in size to the option's delta, we immunize the option against profit and loss variability due to small changes in the spot rate.
For example, consider our equity call option with a strike price of $50 when the underlying price is $50. Because it is an at-the-money option, we know that the delta is 50. The delta is expressed in terms of a percentage of the notional amount. An option that is hopelessly out-of-the-money very near to expiration has a delta of 0. Also, near expiration, an option that is completely in-the-money with no danger of being thrown out-of-the-money has a delta of 100. Everything else is in between. At-the-money options have a delta of 50.
Our equity call has a positive delta because it is a long position in a call. If we exercise the call, we will end up being long the stock.An equity put struck at-the-money would have a negative delta of 50. If we exercise the put, we will end up being short the stock.Similarly, shorting a call implies a negative delta and shorting a put implies a positive delta.
To delta hedge our long at-the-money equity call struck at $50, we need to know the notional amount. Let it be $100 for the sake of argument. Therefore, the delta position implied by our option is $50 (i.e. 50/100 x $100).
If we take a short position in the cash market (assuming that shorting the stock is feasible and liquid enough) at the spot price of $50, we have immunized the option's sensitivity to small changes in the spot price.If spot goes to $48, the $2 we make on the short stock position will offset the $2 we will lose on the change in price of the option. Similarly, if spot goes to $52, the $2 we make on the option premium will be offset by the $2 we will lose on the short stock position.
Assuming that we own the option, if we plot the curve of the option premium (on the y-axis) against the price of the underlying instrument (on the x-axis), everything else remaining constant, we obtain a convex curve. The slope of this convex curve is the option's delta.
GAMMA
Things begin to get interesting for larger moves in the stock price.
If spot goes to $70, we might expect to make $1150 on the option price while only losing $1000 on the short stock position.
How does this work? Because of the convexity of the option's curve, the delta will change if spot moves enough.If spot goes to $52, the delta might change to 52. If spot goes to $55, the delta might change to 57. If spot goes to $60, the delta might change to 64. If spot goes to $70, the delta might be 80. The option position behaves as if it is a miraculous trade that seemingly gets longer as spot goes higher in a non-linear fashion.
Since we have only hedged our exposure to a position that is long $50 at $50, the hedged option position will continue to make money on the incremental position, i.e. the part that appeared to get longer from $50 to $70 at an average rate of say $65.
The greater the convexity of the option curve, the more bang for our long option buck and the more pain we will endure if we are short the option, in a volatile environment.
Convexity is described by the greek letter called "gamma". Mathematically, gamma is the second derivative of the option's price with respect to the underlying cash price. Intuitively, it is the sensitivity of the delta (or rate of change of the delta) with respect to the cash price.
VEGA
We know that options will be expensive when volatility is actually high or when volatility is thought to be heading higher. We also know that options are cheap when volatility is low or when volatility is believed to be heading lower.
There are two kinds of volatility between which we must distinguish: actual volatility and implied volatility.
Actual volatility is a measure of how much the spot price moves around, in fact, for a given time period. Implied volatility is the volatility used in the calculation of the option's price. Without going into the mathematics of it at this point, suffice it to say that we can back out (or "imply") the volatility used to calculate an option's price, if we know with certainty the value of each of the other variables used in the option valuation formula. For the Black-Scholes-Merton model, the list of these remaining variables typically includes the underlying cash price, the maturity date, the delivery date, the strike price and the risk-free rate of interest.
Some of the more developed derivatives markets, such as the foreign exchange options market, actually trade in terms of implied volatility or "vol" instead of specifying a price at which may buy or sell the option in question.
The sensitivity of an option's price to changes in its implied volatility, all other things being constant, is called the "vega". Let us consider the case where we have just bought and delta hedged the long $50 equity call in the stock of company ABC Inc. Spot does not move for a couple of hours until a headline tells the market that DEF Inc. has made a hostile bid for ABC Inc.
Even though spot does not move immediately because traders are confused about the implications of the DEF bid, implied volatilities jump much higher because of the additional uncertainty for ABC's future prospects posed by the DEF initiative.
We will make money, not from delta hedging, but from the jump up in the value of the option. We now own something that has become more valuable in the blink of an eye because the market believes that the volatility of the ABC stock price will be greater than previously thought. Pity the poor short option holder.
This article was courtesy of Chand Sooran.
When we buy options, we are said to be buying volatility. We make money if the spot rate is volatile enough for us to pay for the option. When we sell options, we are selling volatility. We make money if spot is calm enough that we don't have to hedge the exposure frequently.
However, delta hedging is not the only way for us to make money with options. The genius of derivatives is that it allows us to take positions in (or to hedge against fluctuations in) other aspects of the cash instrument's price evolution. Derivatives are dangerous if we do not understand or address each potential dimension of their risk.
Here are several examples. With a simple plain vanilla option, we can make money if implied volatility moves in our favor. With currency futures, currency forwards and currency options, we can speculate on the spread between interest rates in two different countries for a maturity date. With some exotic options, we can buy an option that appreciates in value with the passage of time (all other things being constant) and that also appreciates in value with movement lower in implied volatility.
Options dealers and savvy options traders use time-proven techniques to break down the risks in an options position or in a portfolio of options, futures, forwards and cash positions into information that is more readily comprehensible and therefore more easily positioned or hedged. This method of analysis employs tools called the "greeks", as well as using simulation, scenario analysis and value-at-risk analysis.
The greeks get their name from the fact that the sensitivities of an option to various market parametres are labelled with letters from the greek alphabet.
DELTA
The delta of an option is the sensitivity of the option's price to very small changes in the price of the underlying instrument. When we talked about trading spot around the options position in order to realize profit that would pay for the option's premium, we were talking about trading the delta.
By taking an opposite position equal in size to the option's delta, we immunize the option against profit and loss variability due to small changes in the spot rate.
For example, consider our equity call option with a strike price of $50 when the underlying price is $50. Because it is an at-the-money option, we know that the delta is 50. The delta is expressed in terms of a percentage of the notional amount. An option that is hopelessly out-of-the-money very near to expiration has a delta of 0. Also, near expiration, an option that is completely in-the-money with no danger of being thrown out-of-the-money has a delta of 100. Everything else is in between. At-the-money options have a delta of 50.
Our equity call has a positive delta because it is a long position in a call. If we exercise the call, we will end up being long the stock.An equity put struck at-the-money would have a negative delta of 50. If we exercise the put, we will end up being short the stock.Similarly, shorting a call implies a negative delta and shorting a put implies a positive delta.
To delta hedge our long at-the-money equity call struck at $50, we need to know the notional amount. Let it be $100 for the sake of argument. Therefore, the delta position implied by our option is $50 (i.e. 50/100 x $100).
If we take a short position in the cash market (assuming that shorting the stock is feasible and liquid enough) at the spot price of $50, we have immunized the option's sensitivity to small changes in the spot price.If spot goes to $48, the $2 we make on the short stock position will offset the $2 we will lose on the change in price of the option. Similarly, if spot goes to $52, the $2 we make on the option premium will be offset by the $2 we will lose on the short stock position.
Assuming that we own the option, if we plot the curve of the option premium (on the y-axis) against the price of the underlying instrument (on the x-axis), everything else remaining constant, we obtain a convex curve. The slope of this convex curve is the option's delta.
GAMMA
Things begin to get interesting for larger moves in the stock price.
If spot goes to $70, we might expect to make $1150 on the option price while only losing $1000 on the short stock position.
How does this work? Because of the convexity of the option's curve, the delta will change if spot moves enough.If spot goes to $52, the delta might change to 52. If spot goes to $55, the delta might change to 57. If spot goes to $60, the delta might change to 64. If spot goes to $70, the delta might be 80. The option position behaves as if it is a miraculous trade that seemingly gets longer as spot goes higher in a non-linear fashion.
Since we have only hedged our exposure to a position that is long $50 at $50, the hedged option position will continue to make money on the incremental position, i.e. the part that appeared to get longer from $50 to $70 at an average rate of say $65.
The greater the convexity of the option curve, the more bang for our long option buck and the more pain we will endure if we are short the option, in a volatile environment.
Convexity is described by the greek letter called "gamma". Mathematically, gamma is the second derivative of the option's price with respect to the underlying cash price. Intuitively, it is the sensitivity of the delta (or rate of change of the delta) with respect to the cash price.
VEGA
We know that options will be expensive when volatility is actually high or when volatility is thought to be heading higher. We also know that options are cheap when volatility is low or when volatility is believed to be heading lower.
There are two kinds of volatility between which we must distinguish: actual volatility and implied volatility.
Actual volatility is a measure of how much the spot price moves around, in fact, for a given time period. Implied volatility is the volatility used in the calculation of the option's price. Without going into the mathematics of it at this point, suffice it to say that we can back out (or "imply") the volatility used to calculate an option's price, if we know with certainty the value of each of the other variables used in the option valuation formula. For the Black-Scholes-Merton model, the list of these remaining variables typically includes the underlying cash price, the maturity date, the delivery date, the strike price and the risk-free rate of interest.
Some of the more developed derivatives markets, such as the foreign exchange options market, actually trade in terms of implied volatility or "vol" instead of specifying a price at which may buy or sell the option in question.
The sensitivity of an option's price to changes in its implied volatility, all other things being constant, is called the "vega". Let us consider the case where we have just bought and delta hedged the long $50 equity call in the stock of company ABC Inc. Spot does not move for a couple of hours until a headline tells the market that DEF Inc. has made a hostile bid for ABC Inc.
Even though spot does not move immediately because traders are confused about the implications of the DEF bid, implied volatilities jump much higher because of the additional uncertainty for ABC's future prospects posed by the DEF initiative.
We will make money, not from delta hedging, but from the jump up in the value of the option. We now own something that has become more valuable in the blink of an eye because the market believes that the volatility of the ABC stock price will be greater than previously thought. Pity the poor short option holder.
This article was courtesy of Chand Sooran.
posted by John at 11:27 AM
5 Comments:
I am posting this message on all the forums so just bear with me here. In order to make sure we have some type of continuity on the flow of posts. The best way to post is simply to post under the latest article. I think this will make things easier since we don't have a regular message board. If people post under different articles, I'm worried that responses might be ignored. If you have to, you can make reference to a previous post and under which article it was if it's a follow up. I hope this helps things run smoother.
Well regarding the APPX trade, there was news after the close on Bloomberg that they did get approval for their breast cancer drug. I don't know how much that 4 pt spike before the stock got halted will play into this.
My guess is the stock opens in the high 40's and trades into the 50's come Monday. I had two long time spreads on that cost next to nothing. I had the Jan/Feb 50 and 55 spreads on. I paid about .30 total. If any of you were watching closely enough that Jan/Feb 55 time spread was trading for a long time at a dime on Thursday.
Congrats to any of you that got those prices. That truly could be the trade of a lifetime. Can't ask for a better payoff then that. Probably 50 to 1. You can't even get that in Vegas.
John
John, have you thought about just starting a group on yahoo? you could moderate it to screen members. and folks could elect to receive (or not receive) email or digests or whatever. it might be an easier way to keep threads alive and focused so people don't have to sort through what mey be irrelevant posts. plus its free (except everyone would have to open a yahoo account).
congrats on all the APPX winners here as well.
michael
Mike,
I will look into that that. I just don't know if I want to have people's attention diverted to two different sites. It looks like it only takes seconds to set one up so I might try it for a few days and see what happens. It's been my experience with message boards that they usually start active and then they dry up pretty fast for whatever reason. But we can experiment with it.
Members of this group, feel free to respond to what you think of having a yahoo user group dedicated for option trades.
John
Options? Bonds? Metal? I'd rather keep it all in forex futures trading.
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