Option premium and valuation

In simple terms — The premium is the price at which someone may quote or complete a trade. Theoretical value is an estimate produced by a model. They may be close, but they are not the same thing, and neither one guarantees the price of the next execution.

The premium is the unit price of an option. It is formed in the market through the interaction of orders, quotes, intermediary risk and conditions in the underlying. The option chain normally shows it as bid and ask; an actual price exists only when an order is executed.

Theoretical value is the output of a model given a set of inputs and assumptions. It helps structure analysis, compare scenarios and calculate sensitivities; it does not proclaim the “true” price. A model does not necessarily see order-book depth, order urgency, costs, a market maker's inventory risk or a sudden market jump.

Before expiry, the observed premium and theoretical value are different measures: both change with price, time, volatility and market conditions.

From a quote to a cash amount

The displayed premium is often expressed per unit of underlying. The cash amount requires at least a quantity and a multiplier:

premium amount = unit price × multiplier × number of contracts

If a call is offered at 4.20, the multiplier is 100 and one contract is bought, the gross debit is 420 currency units, not 4.20. Commissions, taxes, foreign-exchange effects, exercise charges and other items may be added. A 100 multiplier is common for many U.S. equity options, but it is not universal: specifications, corporate actions and contracts on indices, futures or currencies may produce different units and deliverables.

A sound reading separates three levels:

Level	What it represents	What it does not establish
unit premium	price for the unit defined by the contract	total cash outlay
premium amount	premium × multiplier × quantity	total cost after every expense
position P&L	economic change in the whole position	only the underlying's move

Bid, mid and ask

The bid is the best displayed proposal to buy; the ask is the best displayed proposal to sell. The mid is an arithmetic average:

mid = (bid + ask) / 2

With a 3.80 bid and a 4.20 ask, the mid is 4.00. For a multiplier of 100, the distance between bid and ask is equivalent to 40 currency units per contract. The mid is useful as a reference, but it is not an executable quote: there may be no order available at that level. Bid and ask also have sizes, timestamps and conditions of validity; they may change before an order reaches the market, be insufficient for the entire quantity or be subject to the venue's rules.

The “last” price records the latest reported transaction. It may be stale, refer to a minimal quantity or belong to market conditions that no longer exist. Marking a position at the last, at the mid or through a model produces different figures: the marking method must be stated.

Intrinsic value and value beyond intrinsic

For a vanilla call and put, current intrinsic value is:

call intrinsic value = max(S − K, 0)
put intrinsic value  = max(K − S, 0)

S is the underlying price and K is the strike. The portion of premium above intrinsic value is often called time value or extrinsic value. The name does not mean it depends on the calendar alone: it also incorporates uncertainty, implied volatility, rates, dividends or carry, liquidity and contract features.

An out-of-the-money option can therefore have a positive premium even though its intrinsic value is zero. Conversely, an in-the-money option is not automatically a profitable trade: the buyer may have paid a premium greater than its current intrinsic value.

Valuation inputs

A general representation is:

theoretical value = f(S, K, time, volatility, rates, cash flows, style, settlement, model)

underlying and strike determine current moneyness;
time remaining determines how much evolution can still occur;
volatility represents the range of possible moves within the model;
rates, dividends and carry affect the relative value of money and the underlying through time;
exercise style and settlement change the available rights and cash flows;
model and conventions determine how inputs become an output.

Black–Scholes–Merton, binomial trees and numerical methods address different assumptions and products. No formula removes model risk. For American, barrier or digital options, contracts with discrete dividends or non-standard deliverables, a formula intended for a European vanilla option may be unsuitable.

Delta, gamma, vega and theta describe local changes in theoretical value. They are not promises of P&L and can change together with the inputs.

Premium, payoff and break-even

The terminal payoff of a long call is max(S_T − K, 0). Its terminal result, before costs and multiplier, subtracts the premium paid:

long-call P&L at expiry = max(S_T − K, 0) − premium

This gives the instructional break-even K + premium for a long call and K − premium for a long put. These points apply only at expiry, under the stated conventions and before costs. They are not targets, probabilities or required thresholds for closing early at a profit.

Before expiry, time value, implied volatility and other variables remain. A call with strike 100 bought for 4 may be worth 5 while the underlying is at 102, or it may be worth less than 4 even though the underlying has risen: the outcome also depends on elapsed time, volatility, spreads and the marking method. Terminal payoff and pre-expiry mark-to-market are not the same.

A reproducible comparison

To compare market and model without mixing measures:

identify the exact underlying, type, strike, expiry, style and deliverable;
record timestamp, session, bid, ask and their sizes;
state whether the measure uses bid, mid, ask, last or an actual trade;
record multiplier, quantity, currency and excluded costs;
record model, version, inputs and conventions;
compare theoretical value with the bid–ask interval, not just the mid;
repeat scenarios by changing one input and then several inputs together;
always separate current value, expiry payoff and actually realised P&L.

A gap between model and quote does not prove an arbitrage by itself. It may reflect different inputs, a non-flat volatility surface, hedging costs, illiquidity, jump risk or unsynchronised data.

Common mistakes

reading 2.50 as the total cost without checking the multiplier;
treating the mid as a guaranteed execution price;
considering the last more informative than bid, ask and timestamp;
calling a premium “expensive” or “cheap” without naming the model and comparison;
confusing implied volatility with certain future volatility;
using expiry break-even to describe today's P&L;
adding Greeks reported with different units or conventions;
ignoring spreads, commissions, slippage, margin and operational risk.

High-impact mistake — A theoretical value of 4.00 does not mean an option can be bought or sold at 4.00. The only certain price of a trade is its actual execution price, for the quantity actually executed.

Checklist

Is the premium a unit price or has it already been multiplied?
What are the bid, ask, size, timestamp and session status?
Is the data real-time, delayed, indicative or calculated?
Which price is used for mark-to-market?
Are model, inputs, style and conventions stated?
Are intrinsic and extrinsic value separated?
Is break-even presented only as a terminal measure?
Do scenarios include costs, liquidity and possible assignment?

Sources

The Options Clearing Corporation, Characteristics and Risks of Standardized Options — contract features, premium, exercise, settlement and risks of standardised listed options.
Options Industry Council, Options Pricing — intrinsic value, time value and the principal drivers of premium.
Options Industry Council, Understanding Options Greeks — pricing inputs and the theoretical nature of sensitivities.
U.S. Securities and Exchange Commission, Investor.gov, An Introduction to Options — call and put structure, premium and buyer/seller risks.
Cboe Options Institute, Options Calculator — comparison of inputs, theoretical price and Greeks.
Fischer Black and Myron Scholes, The Pricing of Options and Corporate Liabilities, 1973 — primary formulation of the valuation model and replication principle.
Options Industry Council, Options Quotes & Calculators — quotes, delayed data, volatility, Greeks and theoretical scenarios.