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European option contracts and payoffs

What you will be able to do

Section titled “What you will be able to do”

After this lesson, you should be able to:

  • identify the holder, writer, underlying, strike, and expiry of a European option;
  • calculate long and short call and put payoffs;
  • distinguish current value, premium, expiry payoff, and profit.

Rights and obligations

Section titled “Rights and obligations”

A European call gives its holder the right, but not the obligation, to buy one unit of the underlying for the option-strike-price K\explain{option-strike-price}{K} at the option-expiry-time Topt\explain{option-expiry-time}{T_{\mathrm{opt}}}. A European put gives the holder the corresponding right to sell. If the holder exercises, the writer must carry out the other side of the exchange. Hull introduces call and put rights and European exercise [1].

Expiry payoff

Section titled “Expiry payoff”

Let the derivative-underlying-value at expiry be STopt\explain{derivative-underlying-value}{S}_{\explain{option-expiry-time}{T_{\mathrm{opt}}}}. The local long-call-expiry-payoff and long-put-expiry-payoff, per unit, are

ΠToptcall=max⁡ ⁣(STopt−K,0)\explain{long-call-expiry-payoff}{\Pi_{\explain{option-expiry-time}{T}_{\mathrm{opt}}}^{\mathrm{call}}} =\max\!\left(\explain{derivative-underlying-value}{S}_{\explain{option-expiry-time}{T_{\mathrm{opt}}}}-\explain{option-strike-price}{K},0\right)
(4.2.1)
ΠToptput=max⁡ ⁣(K−STopt,0)\explain{long-put-expiry-payoff}{\Pi_{\explain{option-expiry-time}{T}_{\mathrm{opt}}}^{\mathrm{put}}} =\max\!\left(\explain{option-strike-price}{K}-\explain{derivative-underlying-value}{S}_{\explain{option-expiry-time}{T_{\mathrm{opt}}}},0\right)
(4.2.2)

The maximum with zero expresses the holder’s right not to exercise when exercise would give a negative payoff. A writer’s signed payoff is the negative of the matching holder payoff.

For STopt=115\explain{derivative-underlying-value}{S}_{\explain{option-expiry-time}{T_{\mathrm{opt}}}}=115 and K=100\explain{option-strike-price}{K}=100, one call pays 15 and one put pays zero. For STopt=70\explain{derivative-underlying-value}{S}_{\explain{option-expiry-time}{T_{\mathrm{opt}}}}=70 and K=85\explain{option-strike-price}{K}=85, one put pays 15. A writer of three such puts has a signed payoff of −45-45.

Value, premium, payoff, and profit are different

Section titled “Value, premium, payoff, and profit are different”

The European call value ct\explain{call-option-value}{c_t} or European put value pt\explain{put-option-value}{p_t} is a current value. At inception, the holder pays the local option-premium-paid π0\explain{option-premium-paid}{\pi_0}, under the stated settlement convention. The payoff is determined later, by the state at expiry.

If financing is ignored, the local option-holder-expiry-profit of a call is the payoff minus the premium:

GTopt=ΠToptcall−π0\explain{option-holder-expiry-profit}{G_{\explain{option-expiry-time}{T}_{\mathrm{opt}}}} =\explain{long-call-expiry-payoff}{\Pi_{\explain{option-expiry-time}{T}_{\mathrm{opt}}}^{\mathrm{call}}}-\explain{option-premium-paid}{\pi_0}
(4.2.3)

For example, a call bought for 6 that later pays 9 has a payoff of 9 and a profit of 3, if financing is ignored. A profit calculation that includes financing first moves the premium and all other cash flows to one common date. Hull’s option examples distinguish premium, exercise, payoff, and profit [2].

Knowledge check 4.2.1 European option contracts and payoffs

Link to Knowledge check 4.2.1: European option contracts and payoffs

Model boundary and review note

Section titled “Model boundary and review note”

This lesson covers European calls and puts only. It does not value them or cover American or Bermudan exercise, automatic exercise, assignment, settlement delay, corporate actions, collateral, counterparty default, or barrier and other path-dependent terms.

The payoff functions are implemented in a pure tested domain module. Sources, notation, examples, code, and answer keys remain draft pending independent human review.

References

Section titled “References”
  1. Hull, Options, Futures, and Other Derivatives (8th ed., 2012). Ch. 1 §1.5, printed pp. 7-9, call and put option rights and European exercise. draft ↩
  2. Hull, Options, Futures, and Other Derivatives (8th ed., 2012). Ch. 9 practice questions 9.1-9.2 and 9.9-9.10, printed pp. 211-212, payoff versus profit. draft ↩
Notation used on this page (9)
ΠToptcall\Pi_{T_{\mathrm{opt}}}^{\mathrm{call}}Long call expiry payoffdraft

Non-negative amount received by the holder of a European call at expiry. Per unit it is the positive part of underlying expiry value minus strike.

Units: stated currency at option expiry

ΠToptput\Pi_{T_{\mathrm{opt}}}^{\mathrm{put}}Long put expiry payoffdraft

Non-negative amount received by the holder of a European put at expiry. Per unit it is the positive part of strike minus underlying expiry value.

Units: stated currency at option expiry

GToptG_{T_{\mathrm{opt}}}Option holder expiry profitdraft

Expiry payoff less the premium carried to expiry under the explicitly stated financing convention. In the simplest examples financing is ignored, so the initially paid premium is subtracted directly.

Units: stated currency at option expiry

π0\pi_0Option premium paiddraft

Positive valuation-time amount paid by the option holder to acquire the contractual right. Its financing convention must be stated before converting an expiry payoff into profit.

Units: stated currency at premium payment time

ctc_tEuropean call valuedraft

Current non-negative value to the holder of a European call under the stated model, before any financing or transaction costs.

Units: stated currency at model time

StS_tDerivative underlying valuedraft

Value at model time t of one unit of the asset or claim named as the derivative's underlying; a positive quoted value, while a position in the underlying carries its own signed quantity.

Units: stated currency per unit of underlying at model time

ToptT_{\mathrm{opt}}Option expiry timedraft

Future model time when a European option's exercise decision and payoff are determined; a contract date shared by holder and writer, distinct from a bond maturity.

Units: model-years from the stated valuation time

KKOption strike pricedraft

Contractual price per unit of underlying used to determine the option's exercise payoff; a positive contractual amount, with payoff signs depending on call or put and holder or writer perspective.

Units: expiry-time currency per unit of underlying

ptp_tEuropean put valuedraft

Current non-negative value to the holder of a European put under the stated model, before any financing or transaction costs.

Units: stated currency at model time