Valuation of options

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

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 the underlying: Any fluctuation in the price of the underlying (stock/index/commodity) obviously has the largest effect on premium of an option contract. An increase in the underlying price increases the premium of call option and decreases the premium of put option. Reverse is true when underlying price decreases.
• Strike price: How far is the strike price from spot also affects option premium. Say, if NIFTY goes from 5000 to 5100 the premium of 5000 strike and of 5100 strike will change a lot compared to a contract with strike of 5500 or 4700.
• Volatility of underlying: Underlying security is a constantly changing entity. The degree by which its price fluctuates can be termed as volatility. So a share which fluctuates 5% on either side on daily basis is said to have more volatility than e.g. 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 greater risk it brings to the seller.
• Payment of Dividend: Payment of Dividend does not have direct impact on value of derivatives but it does have indirect impact through stock price. We know that if dividend is paid, stock goes ex-dividend therefore price of stock will go down which will result into increase in Put premium and decrease in Call premium.

Apart from above, other factors like bond yield (or interest 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 option; he has to earn more than this because of higher risk he is taking.

Because the values of option 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 of rational pricing (i.e. risk neutrality), moneyness, option time value and put–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 the Heath–Jarrow–Morton framework for interest rates, to the Heston model where volatility itself is considered stochastic. See Asset 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 the Black–Scholes formula and the Black model.
• Lattice models (Trees): Binomial options pricing model; Trinomial tree
• Monte Carlo methods for option pricing
• Finite difference methods for option pricing
• More recently, the volatility surface-aware models in the local volatility and stochastic volatility families.

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

The final four are numerical methods, usually requiring sophisticated derivatives-software, or a numeric package such as MATLAB. 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 (for non-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 the risk-free rate, and then averaged. For the analytic methods, these same are subsumed into a single probabilistic result; see Black–Scholes model § Interpretation.

After the financial crisis of 2007–2008, counterparty credit risk considerations were brought into the valuation, previously using the risk-free rate to discount the payoff. Here, there are [2] three major developments re option pricing:

1. For discounting, the overnight indexed swap (OIS) curve is typically used for the "risk free rate", as opposed to LIBOR as previously (LIBOR is due to be phased out by the end of 2021, with replacements including SOFR and TONAR); see Interest rate swap § Valuation and pricing. Relatedly, the "Multi-curve framework" is now standard in the valuation of interest rate derivatives and for fixed income analysis more generally.
2. As mentioned, option pricing models must consider the volatility surface, and the numerics will then require a zeroth calibration step, such that observed prices are returned before new prices and / or "greeks" can be calculated. To do so, banks will apply local- or stochastic volatility models, such as Heston mentioned above (or less common, implied trees).
3. The risk neutral value, no matter how determined, is adjusted for the impact of counterparty credit risk via a credit valuation adjustment, or CVA, as well as various of the other XVA which may also be appended.