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finance Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of production, distribution, and consumption of money, assets, goods and services (the discipline of f ...
, a price (premium) is paid or received for purchasing or selling
options Option or Options may refer to: Computing *Option key, a key on Apple computer keyboards *Option type, a polymorphic data type in programming languages *Command-line option, an optional parameter to a command *OPTIONS, an HTTP request method ...
. This article discusses the calculation of this premium in general. For further detail, see: for discussion of the mathematics; Financial engineering for the implementation; as well as generally.


Premium components

This price can be split into two components: intrinsic value, and time value.


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 In finance, a call option, often simply labeled a "call", is a contract between the buyer and the seller of the call option to exchange a security at a set price. The buyer of the call option has the right, but not the obligation, to buy ...
, 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 In finance, a put or put option is a derivative instrument in financial markets that gives the holder (i.e. the purchaser of the put option) the right to sell an asset (the ''underlying''), at a specified price (the ''strike''), by (or at) a ...
, 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 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)


Time value

The option premium is always greater than the intrinsic value. 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: 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 An interest rate is the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed (called the principal sum). The total interest on an amount lent or borrowed depends on the principal sum, the interest rate, t ...
) 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.


Pricing models

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 Rational pricing is the assumption in financial economics that asset prices - and hence asset pricing models - will reflect the arbitrage-free price of the asset as any deviation from this price will be "arbitraged away". This assumption is use ...
(i.e.
risk neutral In economics and finance, risk neutral preferences are preferences that are neither risk averse nor risk seeking. A risk neutral party's decisions are not affected by the degree of uncertainty in a set of outcomes, so a risk neutral party is indif ...
ity),
moneyness In finance, moneyness is the relative position of the current price (or future price) of an underlying asset (e.g., a stock) with respect to the strike price of a derivative, most commonly a call option or a put option. Moneyness is firstly a thr ...
, option time value and
put–call parity In financial mathematics, put–call parity defines a relationship between the price of a European call option and European put option, both with the identical strike price and expiry, namely that a portfolio of a long call option and a short pu ...
. 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 The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing derivative investment instruments. From the parabolic partial differential equation in the model, known as the Black� ...
for equities, to the Heath–Jarrow–Morton framework for interest rates, to the Heston model where volatility itself is considered
stochastic Stochastic (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselv ...
. See
Asset pricing In financial economics, asset pricing refers to a formal treatment and development of two main pricing principles, outlined below, together with the resultant models. There have been many models developed for different situations, but correspon ...
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 In finance, the binomial options pricing model (BOPM) provides a generalizable numerical method for the valuation of options. Essentially, the model uses a "discrete-time" ( lattice based) model of the varying price over time of the underlying f ...
; 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 method In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm. Mathem ...
s, usually requiring sophisticated derivatives-software, or a numeric package such as
MATLAB MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementat ...
. 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 The risk-free rate of return, usually shortened to the risk-free rate, is the rate of return of a hypothetical investment with scheduled payments over a fixed period of time that is assumed to meet all payment obligations. Since the risk-free ...
, and then averaged. For the analytic methods, these same are subsumed into a single probabilistic result; see .


Post crisis

After the
financial crisis of 2007–2008 Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of production, distribution, and consumption of money, assets, goods and services (the discipline of ...
,
counterparty credit risk A credit risk is risk of default on a debt that may arise from a borrower failing to make required payments. In the first resort, the risk is that of the lender and includes lost principal and interest, disruption to cash flows, and increased ...
considerations must enter into the valuation, previously performed in an entirely "
risk neutral In economics and finance, risk neutral preferences are preferences that are neither risk averse nor risk seeking. A risk neutral party's decisions are not affected by the degree of uncertainty in a set of outcomes, so a risk neutral party is indif ...
world". There are then Derivatives Pricing after the 2007-2008 Crisis: How the Crisis Changed the Pricing Approach
Didier Kouokap Youmbi,
Bank of England The Bank of England is the central bank of the United Kingdom and the model on which most modern central banks have been based. Established in 1694 to act as the English Government's banker, and still one of the bankers for the Government o ...
Prudential Regulation Authority
three major developments re option pricing: #For discounting, the
overnight indexed swap An overnight indexed swap (OIS) is an interest rate swap (''IRS'') over some given term, e.g. 10Y, where the periodic fixed payments are tied to a given fixed rate while the periodic floating payments are tied to a floating rate calculated from ...
(OIS) curve is now typically used for the "risk free rate", as opposed to
LIBOR The London Inter-Bank Offered Rate is an interest-rate average calculated from estimates submitted by the leading banks in London. Each bank estimates what it would be charged were it to borrow from other banks. The resulting average rate is u ...
as previously ( LIBOR is due to be phased out by the end of 2021, with replacements including
SOFR Secured Overnight Financing Rate (SOFR) is a secured interbank overnight interest rate. SOFR is a reference rate (that is, a rate used by parties in commercial contracts that is outside their direct control) established as an alternative to LIBOR. ...
and TONAR); see . Relatedly, the "
Multi-curve framework In finance, an interest rate swap (IRS) is an interest rate derivative (IRD). It involves exchange of interest rates between two parties. In particular it is a "linear" IRD and one of the most liquid, benchmark products. It has associations wi ...
" is now standard in the valuation of
interest rate derivatives In finance, an interest rate derivative (IRD) is a derivative whose payments are determined through calculation techniques where the underlying benchmark product is an interest rate, or set of different interest rates. There are a multitude of diff ...
and for
fixed income analysis Fixed income analysis is the process of determining the value of a debt security based on an assessment of its risk profile, which can include interest rate risk, risk of the issuer failing to repay the debt, market supply and demand for the sec ...
more generally. #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). #The risk neutral value, no matter how determined, is then adjusted for the impact of
counterparty credit risk A credit risk is risk of default on a debt that may arise from a borrower failing to make required payments. In the first resort, the risk is that of the lender and includes lost principal and interest, disruption to cash flows, and increased ...
via a
credit valuation adjustment Credit valuation adjustments (CVAs) are accounting adjustments made to reserve a portion of profits on uncollateralized financial derivatives. They are charged by a bank to a risky (capable of default) counterparty to compensate the bank for taking ...
, or CVA, as well as various of the other XVA which may also be appended.


References

{{Authority control Options (finance) Mathematical finance