Essential Elements of Option Pricing  Getting to Know the GreeksIn 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 optionsbased 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 multioption 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 atthemoney (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 inthemoney (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 outofthemoney (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 an exchangetraded 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” Vega Theta Delta 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 deltaequivalent 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 deltaequivalent of the position is the same as the futuresequivalent of the options. Gamma The Options Book Concept 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 averagerate 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: 

