# Valuation of options

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Concept in finance

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In [finance](/source/Finance), a price (premium) is paid or received for purchasing or selling [options](/source/Option_(finance)). The [calculation of this premium](/source/Option_(finance)#Valuation) will require sophisticated mathematics.

## Premium components

This price can be split into two components: [intrinsic value](/source/Intrinsic_value_(finance)#Options), and [time value](/source/Option_time_value) (also called "extrinsic value").[1]

### Intrinsic value

The *intrinsic 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 a [call option](/source/Call_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 a [put option](/source/Put_option), the option is in-the-money if the *strike* 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 a [DJI](/source/Dow_Jones_Industrial_Average) 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)

### Extrinsic (Time) value

Main article: [Option time value](/source/Option_time_value)

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

### Other factors affecting premium

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](/source/Underlying): 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](/source/Strike_price): The distance of the strike price from spot also affects option premium. If [NIFTY](/source/CNX_Nifty) 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](/source/Volatility_(finance)) 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 of [Dividend](/source/Dividend): 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 like [bond yield](/source/Yield_to_maturity) (or [interest rate](/source/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 options. The seller has to earn more than this because of the higher risk it is taking.

## Pricing models

See also: [Mathematical finance § Derivatives pricing: the Q world](/source/Mathematical_finance#Derivatives_pricing:_the_Q_world), and [Financial modeling § Quantitative finance](/source/Financial_modeling#Quantitative_finance)

Because the values of [option](/source/Option_(finance)) 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](/source/Rational_pricing#Options) (i.e. [risk neutrality](/source/Risk_neutral)), [moneyness](/source/Moneyness), [option time value](/source/Option_time_value) and [put–call parity](/source/Put%E2%80%93call_parity).

The valuation itself combines (1) a model of the behavior (["process"](/source/Stochastic_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](/source/Black%E2%80%93Scholes_model) for equities, to the [Heath–Jarrow–Morton framework](/source/Heath%E2%80%93Jarrow%E2%80%93Morton_framework) for interest rates, to the [Heston model](/source/Heston_model) where volatility itself is considered [stochastic](/source/Stochastic). See [Asset pricing](/source/Asset_pricing) for a listing of the various models here.

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

- [Closed form](/source/Closed-form_expression), analytic models: the most basic of these are the [Black–Scholes formula](/source/Black%E2%80%93Scholes_model#Black–Scholes_formula) and the [Black model](/source/Black_model).

- [Lattice models](/source/Lattice_model_(finance)) (Trees): [Binomial options pricing model](/source/Binomial_options_pricing_model); [Trinomial tree](/source/Trinomial_tree)

- [Monte Carlo methods for option pricing](/source/Monte_Carlo_methods_for_option_pricing)

- [Finite difference methods for option pricing](/source/Finite_difference_methods_for_option_pricing)

- [More recently](/source/Financial_economics#Derivative_pricing), the [volatility surface](/source/Volatility_surface)-aware models in the [local volatility](/source/Local_volatility) and [stochastic volatility](/source/Stochastic_volatility) families.

The Black model extends Black-Scholes from equity to [options on futures](/source/Options_on_futures), [bond options](/source/Bond_option), [swaptions](/source/Swaption), (i.e. options on [swaps](/source/Swap_(finance))), and [interest rate cap and floors](/source/Interest_rate_cap_and_floor) (effectively options on the interest rate).

The final four are [numerical methods](/source/Numerical_method), usually requiring sophisticated derivatives-software, or a [numeric package](/source/List_of_numerical-analysis_software) such as [MATLAB](/source/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](/source/Option_style), 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](/source/Risk-free_rate), and then averaged. For the analytic methods, these same are subsumed into a single probabilistic result; see [Black–Scholes model § Interpretation](/source/Black%E2%80%93Scholes_model#Interpretation).

## Post crisis

Further information: [Financial economics § Derivative pricing](/source/Financial_economics#Derivative_pricing), and [Financial economics § Departures from normality](/source/Financial_economics#Departures_from_normality)

After the [2008 financial crisis](/source/2008_financial_crisis), [counterparty credit risk](/source/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, the [overnight indexed swap](/source/Overnight_indexed_swap) (OIS) curve is typically used for the "risk free rate", as opposed to [LIBOR](/source/LIBOR) as previously; see [Interest rate swap § Valuation and pricing](/source/Interest_rate_swap#Valuation_and_pricing). Relatedly, the "[Multi-curve framework](/source/Multi-curve_framework)" is now standard in the valuation of [interest rate derivatives](/source/Interest_rate_derivatives) and for [fixed income analysis](/source/Fixed_income_analysis) more generally.

1. To ensure that option prices are consistent with the [volatility surface](/source/Volatility_surface), the numerics will incorporate a zeroth [calibration step](/source/Stochastic_volatility#Calibration_and_estimation), such that observed prices are returned before new prices and / or ["greeks"](/source/Greeks_(finance)) can be calculated. To do so, banks will apply [local](/source/Local_volatility) or [stochastic volatility](/source/Stochastic_volatility) models, such as the Heston model mentioned above (or less common, [implied trees](/source/Implied_binomial_tree)).

1. The risk neutral value, no matter how determined, is adjusted for the impact of [counterparty credit risk](/source/Counterparty_credit_risk) via a [credit valuation adjustment](/source/Credit_valuation_adjustment), or CVA, as well as various of the other [XVA](/source/XVA) which may also be appended.

## See also

- [Option (finance) § Valuation](/source/Option_(finance)#Valuation)

- [Financial engineering](/source/Financial_engineering)

- [Mathematical finance § Derivatives pricing: the Q world](/source/Mathematical_finance#Derivatives_pricing:_the_Q_world)

- [Financial modeling § Quantitative finance](/source/Financial_modeling#Quantitative_finance)

## References

1. **[^](#cite_ref-1)** ["Extrinsic Value Definition | Britannica Money"](https://www.britannica.com/money/extrinsic-value). *www.britannica.com*. Retrieved 2023-05-09.

1. **[^](#cite_ref-Youmbi_2-0)** [Derivatives Pricing after the 2007-2008 Crisis: How the Crisis Changed the Pricing Approach](https://ssrn.com/abstract=2511585), Didier Kouokap Youmbi, [Bank of England](/source/Bank_of_England) – [Prudential Regulation Authority](/source/Prudential_Regulation_Authority_(United_Kingdom))

v t e Derivatives market Derivative (finance) * List of futures exchanges Options Terms Delta neutral Exercise Expiration Moneyness Open interest Pin risk Risk-free interest rate Strike price Synthetic position the Greeks Volatility Vanillas American Bond option Call Employee stock option European Fixed income FX Option styles Put Warrants Exotics Asian Barrier Basket Binary Callable bull/bear contract Chooser Cliquet Compound Forward start Interest rate Lookback Mountain range Rainbow Spread Swaption Strategies Backspread Box spread Butterfly Calendar spread Collar Condor Covered option Credit spread Debit spread Diagonal spread Fence Intermarket spread Iron butterfly Iron condor Jelly roll Ladder Naked option Straddle Strangle Protective option Ratio spread Risk reversal Vertical spread (Bear, Bull) Valuation Valuation methods Continuous-time stochastic processes: • Arithmetic diffusion: Bachelier • Geometric diffusion: Black, Black–Scholes, Garman–Kohlhagen, Margrabe • Stochastic volatility: Heston • Jump processes: Jump diffusion Discrete-time processes: • Binomial, Trinomial, Lattices Numerical methods: • Finite difference, MC Simulation, Real options Model-free:• Put–call parity, Vanna–Volga Swaps Amortising Asset Basis Commodity Conditional variance Constant maturity Correlation Credit default Currency Dividend Equity Forex Forward Rate Agreement Inflation Interest rate Overnight indexed Total return Variance Volatility Year-on-year inflation-indexed Zero Coupon Zero-coupon inflation-indexed Forwards Futures Contango Spot contract Normal backwardation Commodities future Currency future Dividend future Forward market Forward price Forwards pricing Forward rate Futures pricing Interest rate future Margin Perpetual futures Single-stock futures Slippage Stock market index future Exotic derivatives Energy derivative Freight derivative Inflation derivative Property derivative Weather derivative Other derivatives Collateralized debt obligation (CDO) Constant proportion portfolio insurance Contract for difference Credit-linked note (CLN) Credit default option Credit derivative Equity-linked note (ELN) Equity derivative Foreign exchange derivative Fund derivative Fund of funds Interest rate derivative Mortgage-backed security Power reverse dual-currency note (PRDC) Market issues Consumer debt Corporate debt Government debt Great Recession Municipal debt Tax policy Business portal

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Adapted from the Wikipedia article [Valuation of options](https://en.wikipedia.org/wiki/Valuation_of_options) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Valuation_of_options?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.