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Essential Elements of Option Pricing - Getting to Know the Greeks  

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In this article we explain some of the key elements of option pricing and illustrate how an option pricing model can be used to design and monitor an options-based trading or hedging strategy.  We also introduce the concept of an “options book”, which is a way of netting the various risks associated with a multi-option position.  Our focus is on the concepts rather than the mathematics of option pricing. 

The value of a typical commodity option is driven by four key factors:  the strike price, the current market price of the underlying commodity, volatility and the time to expiration.  An option whose strike price is equal to the current market price of the underlying commodity is said to be at-the-money (ATM).  An option that is sitting in a payoff position (i.e., a call option whose strike price is below the current market price or a put option whose strike price is above the current market price) is referred to as being in-the-money (ITM).  An option that is not sitting in a payoff position (i.e., a call option whose strike price is above the current market price or a put option whose strike price is below the current market price) is said to be out-of-the-money (OTM).  The further ITM (or the less OTM), the more the option will be worth, as the probability of a payoff is greater.  The more volatile its underlying market, the more an option will be worth, again because there is a greater probability that it will move into a payoff position.  Finally, the more time to expiration, the more an option will be worth as the more time it has to move into a payoff position.

The price or value of an option consists of both intrinsic and extrinsic value.  Intrinsic value is the amount the option is ITM (if any).  An option will normally be worth at least as much as its intrinsic value (otherwise an arbitrage opportunity presents itself).  A June crude oil call with a strike price of $100/bbl and an underlying June futures price of $105/bbl has intrinsic value of $5/bbl.  If this call option happens to be trading at $9/bbl, then it has an intrinsic value of $5/bbl and an extrinsic value of $4/bbl.  Calculating the intrinsic value of an option is straightforward.  Thus the option pricing problem is really all about placing a value on the extrinsic value or “time premium” as it is sometimes called.

Using an Options Pricing Model
For exchange-traded options, pricing models such as our proprietary PositionBook™ software are used for two main purposes:  (1) by option market-makers to determine the prices they are willing to buy and sell options; and (2) by option traders to measure the risks involved in various types of option-based strategies.  In this article we will concern ourselves only with the latter. 

For an exchange-traded option, we don’t need to use a pricing model to determine the value of the option because the market tells us its value.  Rather, we use an option pricing model to better understand the risks and potential payoffs of a position in relation to changes in the underlying price, volatility and time.  For example, how much will it cost us in time value to hold a long options position as part of a hedging strategy?  How much of a payoff can we expect from a long put position if prices drop by 20% over the next six months?  How much value will be left in those options six months from now if prices stay flat?  How much risk are we taking on a short option position if volatility increases significantly prior to expiration?  These are the types of questions we can answer with a good pricing model. 

The “Greeks”
The risk measures associated with option pricing are affectionately known as the Greeks and the main ones are delta, gamma, theta and vega.  Delta and gamma relate to how the value of the option changes as it moves in and out of the money, as determined by changes in the price of the underlying commodity.  Theta concerns itself with measuring the impact of time on the value of the option.  Vega is our measure of the impact of changes in volatility.  Each of these measures calculates the impact on option value holding everything else constant (i.e., ceteris paribus).

Vega
Vega measures the impact of changes in volatility on the value of an option.  Vega tells us how much the value of the option will change given a one percentage point change in implied volatility (i.e., going from 24% to 25%).  Rather than thinking of this in terms of a single unit (e.g., bushel, tonne, barrel, etc.), we find it much more intuitive to think of it in terms of a position.  We like to call this the “position Greek” as opposed to the “raw Greek”.  For example, if we have a long option position with a vega of $15,650, then a one percentage point increase in implied volatility will increase the value of this position by about $15,650.  Conversely, a one percentage point decrease in implied volatility will decrease the value of this position by about $15,650.  For a short option position our exposure to volatility would be the opposite.  Vega helps us understand how sensitive a position is to changes in implied volatility.  

Theta
Theta is perhaps the easiest Greek to understand, as it simply measures the rate at which time decay is occurring in an option.  All else being equal, options decrease in value as time passes.  If we are long the option, then time decay is working against us.  If we are short the option, then time decay is working in our favor.  If we are short some options with a position theta of $1,200, we are earning about $1,200 per day in time decay.  Theta is valid for a short period of time since it changes also with the passage of time.  If we were long these same options, we would be incurring time decay of about $1,200 per day, or in other words we would be losing about $1,200 per day due to time decay. 

Delta
Delta is the king of the Greeks and the one you really need to master if you are going to trade options.  Delta helps us understand how the value of the option changes given a change in the price of the underlying commodity.  Depending on how far in or out of the money our option is, the delta can vary from zero for a deep OTM option, to 1.0 for a deep ITM option.  An option with a delta of 1.0 is acting just like the underlying futures, since its value is changing one-for-one along with the underlying futures price. 

An example of delta is a long position of 100 ATM crude oil call options, with a raw delta of 0.5.  The notional size of this position is 100,000 barrels, but the delta-equivalent is only 50,000 barrels, because for every $1/bbl change in the futures price, the value of the option only changes by about $0.50/bbl, or at half the rate of the futures.  So in essence, the delta-equivalent of the position is the same as the futures-equivalent of the options.

Gamma
Gamma is the other Greek that helps us understand how the value of our option will change given a change in the value of the underlying commodity.  Gamma tells us how fast our delta changes as the underlying futures price changes.  Our long crude oil call position from above has a delta of 50,000 barrels, but might have a gamma of say 3,100 barrels.  This means that our position delta increases by 3,100 barrels given a one dollar increase in the underlying futures, making us 3,100 barrels longer, on a delta-equivalent or futures-equivalent basis (i.e. delta would increase from 50,000 to 53,100).  For basic option trades and hedges, gamma doesn’t really help us that much and we can likely learn more about the dynamics of the option position and its delta through sensitivity analysis.  However, for certain types of option strategies, such as delta-neutral trading, understanding your gamma risk is critical.  Gamma is critical to appreciate in situations where prices are jumping or “gapping”, as a delta-neutral strategy fails to properly replicate a long option position. 

The Options Book Concept
While you can get a pretty good intuitive feel of the pricing dynamics of a simple option position, this quickly evaporates when you start getting a few different options on the books.  This is why, after more than twenty five years of dealing with options, I believe that nothing brings the Greeks to life better than the concept of an options book.  An options book involves combining all of your positions and expressing the Greeks in terms of their position-equivalents.  The nice thing about this is that the position-equivalent Greeks are additive, thus allowing you to calculate your “net” combined position.
The following hypothetical crude oil example is based on calculations performed by our proprietary PositionBook™ software.  In this position we are short calls and long puts, the type of collar that energy producers commonly use.  The notional size of this position is 740,000 bbl (740 contracts), but our combined delta-equivalent is only short 59,693 bbl, or about 60 contracts.  Note that both the short calls and long puts have a negative delta, as they are both essentially “short” positions (i.e., they benefit if the price goes down and lose if the price goes up).  The important thing to understand is that the delta-equivalent is a better indicator of the size of your position than the notional. 

Note that our short calls have a combined positive theta of $575/day.  Since we are short these calls, we are earning time value of $575/day.  Our long puts have a combined negative theta of $3,210/day, meaning that we are incurring time decay of this amount on these puts.   Since we are both long puts and short calls in our overall position, the positive theta on the short calls offsets the negative theta on the long puts, so our combined theta is -$2,635/day.  So overall, we are incurring time decay of about $2,635/day for holding this position.  

Our short calls have a combined negative vega of -$13,803.  So for a one percentage point increase in implied volatility, we would lose about $13,803, since they would go up in value by this amount (and vice versa if implied volatility drops by one percentage point).  Our long puts have a combined positive vega of $59,560, which means they will increase in value by this amount given a one percentage point increase in implied volatility.  On our overall position, we have a combined positive vega of $45,757, which for most traders would be considered a large volatility position.  

Illustration of an Options Book for Crude Oil (CL)

The above example illustrates how we can combine various options together into one book.  We could also add any futures positions to this book, which would allow us to net our entire futures and options position together.   Short futures positions have a delta of -1.0 and long futures positions have a delta of +1.0.   Gamma, theta and vega don’t apply to futures, since the delta remains fixed at -1.0 or +1.0, and there is no time decay or implied volatility associated with a futures position.  PositionBook™ also prices more exotic options such as average-rate and barrier options.

One very important point to remember is that the option Greeks are dynamic in nature, meaning that their values change as time passes, volatility changes and the underlying futures price moves up and down.  So while the Greeks are great analytical tools, they are no replacement for sensitivity analysis and stress testing of your position.

Ron R. Gibson

Note:  A trial version of PositionBook™ can be downloaded from:
www.positionbook.com

 

Gibson Capital Inc.
Calgary, Alberta, Canada

Email: info@gibsoncapital.ca


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