Option pricing

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Wade Hansen

Taylor Bright

Options are priced according to the Black-Scholes formula. This formula takes into account price of the underlying, the strike price, the time to expiration, and the volatility of the underlying to calculate an appropriate price for the option premium. From the Black-Scholes formula, the option "greeks" for a particular option position are derived. The greeks are a measure of different aspects risk present in a particular option.

Here are the five key Greeks you need to pay attention to:

• Delta
• Gamma
• Theta
• Vega
• Rho

Delta—Describes how the value of an option changes as a result of small changes in the underlying asset, assuming that all the other factors influencing option pricing are constant. The delta of an option can also be viewed as the required hedge for the option against changes in the underlying stock, i.e. the position in the stock which ensures that the Profit/Loss on the option is offset by the Profit/Loss on the stock position.

Gamma—Describes how the delta of the option changes when the underlying asset changes. Hence, the gamma also describes how you should change your hedge to remain delta neutral when the spot moves. All purchased standard options, calls and puts, have positive gamma.

The gamma position also provides insight into the investor's view on the volatility of the underlying asset, as a long position shows expectations of a volatile market while a short position indicates that he/she expects a calm market.

Theta—Describes the change in the value of the option when time passes and everything else remains constant. This change stems from the fact that the time to an option's expiration is reduced with the passage of time. This change in value is also commonly referred to as how much the option 'bleeds' the speculator. The theta (sensitivity) is often noted in pips lost in value per day that passes.

Vega—Describes the change in the value of the option when the volatility changes. The volatility represents how large the swings are in the underlying asset and is the cornerstone in option pricing. Larger swings imply that the underlying asset is more likely to take on more extreme values. While the option holder's risk is limited to the premium, his/her upside is unlimited for vanilla options. Hence, an increase in the volatility of the underlying asset increases the value of the option.

Rho—Describes the sensitivity of the option price, based on the Black-Scholes model, with regards to changes in the interest rate.

Additional option pricing models include the Binomial Pricing model and the Bjerksund-Stensland American Option Approximation model.