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Infinance, a price (premium) is paid or received for purchasing or sellingoptions. Thecalculation of this premium will require sophisticated mathematics.

This price can be split into two components:intrinsic value, andtime value (also called "extrinsic value").^[1]

Theintrinsic value is the difference between the underlying spot price and the strike price, to the extent that this is in favor of the option holder. For acall option, the option is in-the-money if the underlying spot price is higher than the strike price; then the intrinsic value is the underlying price minus the strike price. For aput option, the option is in-the-money if thestrike price is higher than the underlying spot price; then the intrinsic value is the strike price minus the underlying spot price. Otherwise the intrinsic value is zero.

For example, when aDJI call (bullish/long) option is 18,000 and the underlying DJI Index is priced at $18,050 then there is a $50 advantage even if the option were to expire today. This $50 is the intrinsic value of the option.

In summary, intrinsic value:

= current stock price − strike price (call option)

= strike price − current stock price (put option)

The option premium is always greater than the intrinsic value up to the expiration event. This extra money is for the risk which the option writer/seller is undertaking. This is called the time value.

Time value is the amount the option trader is paying for a contract above its intrinsic value, with the belief that prior to expiration the contract value will increase because of a favourable change in the price of the underlying asset. The longer the length of time until the expiry of the contract, the greater the time value. So,

Time value = option premium − intrinsic value

There are many factors which affect option premium. These factors affect the premium of the option with varying intensity. Some of these factors are listed here:

• Price of theunderlying: Any fluctuation in the price of the underlying stock/index/commodity obviously has the largest effect on the premium of an option contract. An increase in the underlying price increases the premium of call options and decreases the premium of put options. The reverse is true when the underlying price decreases.
• Strike price: The distance of the strike price from spot also affects option premium. IfNIFTY goes from 5000 to 5100, the premium of 5000 strike and of 5100 strike will change more than a contract with strike of 5500 or 4700.
• Volatility of underlying: The underlying security is a constantly changing entity. The volatility is the degree of its price fluctuations. A share which fluctuates 5% on either side on daily basis has more volatility than stable blue chip shares whose fluctuation is more benign at 2–3%. Volatility affects calls and puts alike. Higher volatility increases the option premium because of the greater risk it brings to the seller.
• Payment ofDividend: Payment of Dividend does not directly impact the value of derivatives but indirectly impacts it through the stock price. Whenever a dividend is paid, the stock goes ex-dividend, therefore the price will go down which will results in an increase in put premiums and decrease in call premiums.

Apart from above, other factors likebond yield (orinterest rate) also affect the premium. This is because the money invested by the seller can earn this risk free income in any case and hence while selling options. The seller has to earn more than this because of the higher risk it is taking.

Because the values ofoption contracts depend on a number of different variables in addition to the value of the underlying asset, they are complex to value. There are many pricing models in use, although all essentially incorporate the concepts ofrational pricing (i.e.risk neutrality),moneyness,option time value andput–call parity.

The valuation itself combines (1) a model of the behavior ("process") of the underlying price with (2) a mathematical method which returns the premium as a function of the assumed behavior.

The models in (1) range from the (prototypical)Black–Scholes model for equities, to theHeath–Jarrow–Morton framework for interest rates, to theHeston model where volatility itself is consideredstochastic. SeeAsset pricing for a listing of the various models here.

As regards (2), the implementation, the most common approaches are:

• Closed form, analytic models: the most basic of these are theBlack–Scholes formula and theBlack model.
• Lattice models (Trees):Binomial options pricing model;Trinomial tree
• Monte Carlo methods for option pricing
• Finite difference methods for option pricing
• More recently, thevolatility surface-aware models in thelocal volatility andstochastic volatility families.

The Black model extends Black-Scholes from equity tooptions on futures,bond options,swaptions, (i.e. options onswaps), andinterest rate cap and floors (effectively options on the interest rate).

The final four arenumerical methods, usually requiring sophisticated derivatives-software, or anumeric package such asMATLAB. For these, the result is calculated as follows, even if the numerics differ: (i) a risk-neutral distribution is built for the underlying price over time (fornon-European options, at least at each exercise date) via the selected model, as calibrated to the market; (ii) the option's payoff-value is determined at each of these times, for each of these prices; (iii) the payoffs are discounted at therisk-free rate, and then averaged. For the analytic methods, these same are subsumed into a single probabilistic result; seeBlack–Scholes model § Interpretation.

After the2008 financial crisis,counterparty credit risk considerations were brought into the valuation, previously using the risk-free rate to discount the payoff. There are three major developments here regarding option pricing:^[2]

1. For discounting, theovernight indexed swap (OIS) curve is typically used for the "risk free rate", as opposed toLIBOR as previously; seeInterest rate swap § Valuation and pricing. Relatedly, the "Multi-curve framework" is now standard in the valuation ofinterest rate derivatives and forfixed income analysis more generally.
2. To ensure that option prices are consistent with thevolatility surface, the numerics will incorporate a zerothcalibration step, such that observed prices are returned before new prices and / or"greeks" can be calculated. To do so, banks will applylocal orstochastic volatility models, such as the Heston model mentioned above (or less common,implied trees).
3. The risk neutral value, no matter how determined, is adjusted for the impact ofcounterparty credit risk via acredit valuation adjustment, or CVA, as well as various of the otherXVA which may also be appended.

• Option (finance) § Valuation
• Financial engineering
• Mathematical finance § Derivatives pricing: the Q world
• Financial modeling § Quantitative finance

• Options (finance)
• Mathematical finance

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