In
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 fina ...
, a price (premium) is paid or received for purchasing or selling
options. This article discusses the calculation of this premium in general. For further detail, see: for discussion of the mathematics;
Financial engineering
Financial engineering is a multidisciplinary field involving financial theory, methods of engineering, tools of mathematics and the practice of programming. It has also been defined as the application of technical methods, especially from mathema ...
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 an ...
, 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 s ...
, 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
SZ DJI Technology Co., Ltd. or Shenzhen DJI Sciences and Technologies Ltd. ( zh, c=深圳大疆创新科技有限公司, p=Shēnzhèn Dà Jiāng Chuàngxīn Kējì Yǒuxiàn Gōngsī) in full, more popularly known as its trade name DJI, which s ...
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
In finance, a bond is a type of security under which the issuer (debtor) owes the holder (creditor) a debt, and is obliged – depending on the terms – to repay the principal (i.e. amount borrowed) of the bond at the maturity date as well as ...
(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, th ...
) 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 indiff ...
ity),
moneyness
In 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 disc ...
,
option time value In finance, the time value (TV) (''extrinsic'' or ''instrumental'' value) of an option is the premium a rational investor would pay over its ''current'' exercise value ( intrinsic value), based on the probability it will increase in value before ex ...
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 put ...
.
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 Blac ...
for equities, to the
Heath–Jarrow–Morton framework The Heath–Jarrow–Morton (HJM) framework is a general framework to model the evolution of interest rate curves – instantaneous forward rate curves in particular (as opposed to simple forward rates). When the volatility and drift of the in ...
for interest rates, to the
Heston model
In finance, the Heston model, named after Steven L. Heston, is a mathematical model that describes the evolution of the volatility of an underlying asset. It is a stochastic volatility model: such a model assumes that the volatility of the asset ...
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 Price, pricing principles, outlined below, together with the resultant models.
There have been many models developed for different situations, but cor ...
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
The Black model (sometimes known as the Black-76 model) is a variant of the Black–Scholes option pricing model. Its primary applications are for pricing options on future contracts, bond options, interest rate cap and floors, and swaptions. It w ...
.
*
Lattice models
In mathematical physics, a lattice model is a mathematical model of a physical system that is defined on a lattice, as opposed to a continuum, such as the continuum of space or spacetime. Lattice models originally occurred in the context of co ...
(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
The trinomial tree is a lattice-based computational model used in financial mathematics to price options. It was developed by Phelim Boyle in 1986. It is an extension of the binomial options pricing model, and is conceptually similar. It can also ...
*
Monte Carlo methods for option pricing In mathematical finance, a Monte Carlo option model uses Monte Carlo methodsAlthough the term 'Monte Carlo method' was coined by Stanislaw Ulam in the 1940s, some trace such methods to the 18th century French naturalist Buffon, and a question he as ...
*
Finite difference methods for option pricing Finite difference methods for option pricing are numerical methods used in mathematical finance for the valuation of options. Finite difference methods were first applied to option pricing by Eduardo Schwartz in 1977.
In general, finite differ ...
*
More recently, the
volatility surface
Volatility smiles are implied volatility patterns that arise in pricing financial options. It is a parameter (implied volatility) that is needed to be modified for the Black–Scholes formula to fit market prices. In particular for a given exp ...
-aware models in the
local volatility A local volatility model, in mathematical finance and financial engineering, is an option pricing model that treats volatility as a function of both the current asset level S_t and of time t . As such, it is a generalisation of the Black–Sch ...
and
stochastic volatility
In statistics, stochastic volatility models are those in which the variance of a stochastic process is itself randomly distributed. They are used in the field of mathematical finance to evaluate derivative securities, such as options. The name ...
families.
The Black model extends Black-Scholes from equity to
options on futures
In finance, a futures contract (sometimes called a futures) is a standardized legal contract to buy or sell something at a predetermined price for delivery at a specified time in the future, between parties not yet known to each other. The asset ...
,
bond option
In finance, a bond option is an option to buy or sell a bond at a certain price on or before the option expiry date. These instruments are typically traded OTC.
*A European bond option is an option to buy or sell a bond at a certain date in futur ...
s,
swaption
A swaption is an option granting its owner the right but not the obligation to enter into an underlying swap. Although options can be traded on a variety of swaps, the term "swaption" typically refers to options on interest rate swaps.
Types of ...
s, (i.e. options on
swaps), and
interest rate cap and floor An interest rate cap is a type of interest rate derivative in which the buyer receives payments at the end of each period in which the interest rate exceeds the agreed strike price. An example of a cap would be an agreement to receive a payment for ...
s (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, implementation ...
. 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 ra ...
, 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 fi ...
,
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 indiff ...
world". There are then
[ Derivatives Pricing after the 2007-2008 Crisis: How the Crisis Changed the Pricing Approach](_blank)
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 of ...
– 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 a ...
(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 Tokyo Overnight Average Rate (TONA rate or TONAR) or Japanese Yen Uncollateralized Overnight Call Rate ( ja, 無担保コールO/N物レート) is an unsecured interbank overnight interest rate and reference rate for Japanese yen. Mutan rate and TO ...
); 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 with ...
" is now standard in the valuation of
interest rate derivatives 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 secu ...
more generally.
#As mentioned, option pricing models must consider the
volatility surface
Volatility smiles are implied volatility patterns that arise in pricing financial options. It is a parameter (implied volatility) that is needed to be modified for the Black–Scholes formula to fit market prices. In particular for a given exp ...
, 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
In statistics, stochastic volatility models are those in which the variance of a stochastic process is itself randomly distributed. They are used in the field of mathematical finance to evaluate derivative securities, such as options. The name ...
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
An X-Value Adjustment (XVA, xVA) is an umbrella term referring to a number of different “valuation adjustments” that banks must make when assessing the value of derivative contracts that they have entered into. The purpose of these is twofold: ...
which may also be appended.
References
{{Authority control
Options (finance)
Mathematical finance