Advanced Financial Management · Application of option pricing theory in investment decisions

Put-Call Parity and Factors Affecting Option Value

Updated 11 October 2026 · Fact-checked

Put-call parity links a European call and put with the same exercise price and expiry on a share paying no dividends: c + Pe e^(−rt) = p + Pa. To find a put, use p = c − Pa + Pe e^(−rt). Option value also depends on share price, exercise price, volatility, time and interest rate.

Understand Put-Call Parity and Option Value Determinants

An option gives you the right, not the obligation, to trade. A call is the right to buy at the exercise price (Pe). A put is the right to sell at Pe. A European option can only be exercised at expiry. An American option can be exercised at any time up to expiry.

Option value has two parts: intrinsic value and time value. Intrinsic value is what you would gain by exercising now. For a call it is the larger of (Pa − Pe) and zero, where Pa is the current share price. For a put it is the larger of (Pe − Pa) and zero. Time value is the extra the market pays for the chance that the option becomes more valuable before expiry.

Five factors drive value. A higher share price raises call value and lowers put value. A higher exercise price lowers call value and raises put value. Higher volatility raises both, because the holder gains from big favourable moves but is protected from big adverse ones. More time to expiry usually raises both, because there is more time for the price to move. A higher risk-free rate raises call value and lowers put value, because the present value of the exercise price falls. Expected dividends lower call value and raise put value, because they reduce the share price.

Put-call parity comes from a simple idea. Buy a share and buy a put. Alternatively, buy a call and put the present value of the exercise price in a risk-free investment. At expiry both give the same payoff: the share price if it is above Pe, otherwise Pe. Same payoff means same cost today. That gives c + Pe e^(−rt) = p + Pa. It holds for European options on non-dividend shares. If market prices break it, an arbitrage profit exists.

The Greeks measure sensitivity. Delta is the change in option value for a small change in share price. A call delta is N(d1), between 0 and 1. A put delta is N(d1) − 1, between −1 and 0. Gamma is the change in delta for a change in share price, so it tells you how often a delta hedge needs rebalancing. Theta is time decay, vega is sensitivity to volatility and rho is sensitivity to interest rates.

Key rules to remember

Put-call parity (European, no dividends): c + Pe e^(−rt) = p + Pa; c = call value, p = put value, Pa = current share price, Pe = exercise price, r = risk-free rate, t = years to expiry. Both options must share the same Pe and expiry.
Put value from call value: p = c − Pa + Pe e^(−rt); This is the form given in the ACCA formula sheet. Use it when you have a call value from Black-Scholes.
Call value from put value: c = p + Pa − Pe e^(−rt); Rearrangement of parity. Useful if the question gives the put value.
Black-Scholes call value: c = Pa N(d1) − Pe N(d2) e^(−rt); Where d1 = [ln(Pa ÷ Pe) + (r + 0.5s²)t] ÷ (s√t) and d2 = d1 − s√t. s = annual volatility (standard deviation).
Delta of a call and a put: Call delta = N(d1); Put delta = N(d1) − 1; Put delta is negative. A rise in share price lowers a put's value.
Delta hedge: Options needed = number of shares to hedge ÷ |delta of one option|; Shares held are hedged by writing calls (or buying puts). The hedge must be rebalanced as delta changes; gamma measures how fast.
Intrinsic value: Call = max(Pa − Pe, 0); Put = max(Pe − Pa, 0); Option value = intrinsic value + time value, so value is never below intrinsic value for an American option.

How to solve Put-Call Parity and Option Value Determinants questions

Use this order for any question on option value factors or put-call parity.

1Identify what is asked: a put value, a call value, a direction of change, a hedge ratio, or an arbitrage check.
2Check the conditions for parity: European options, same exercise price and expiry, and no dividends before expiry. State these as assumptions.
3List the inputs: Pa, Pe, r, t and the given call or put value. Make sure r and t use the same time unit, usually years.
4Calculate the present value of the exercise price: Pe × e^(−rt). Work with at least four decimal places for the exponent.
5Apply the rearranged parity formula and compute the missing value. Do a sense check: a put cannot be negative, and it should be at least Pe e^(−rt) − Pa.
6For factor questions, take each factor in turn, say whether call and put value rise or fall, and give a one-line reason.
7For delta questions, find the delta, divide shares held by delta and state how many options to buy or write. Mention that gamma means rebalancing is needed.
8Finish with a short comment for the client or board: what the value means and what limits the model.

Quickest way: Three-line parity calculation

When to use it: When the call value is given or already calculated and you need the put, with marks mainly for the number and a short comment.

Write p = c − Pa + Pe e^(−rt) from the formula sheet.
Compute Pe e^(−rt) first and note it down as the present value of the exercise price.
Substitute: p = c − Pa + that present value. Check the answer is positive, then state the assumptions in one line.

Common mistakes in Put-Call Parity and Option Value Determinants

Using Pe instead of its present value, Pe e^(−rt), in the parity formula.
Students remember 'call + exercise price = put + share price' without the discounting.
Fix: Always discount Pe at the risk-free rate over the option life. Write the PV as a separate line before substituting.
Getting the signs wrong, giving p = c + Pa − Pe e^(−rt).
Mixing up which side the share price sits on.
Fix: Remember the pairs: call plus PV of exercise price equals put plus share. Then rearrange for the put and sense check that the put is positive.
Saying that higher interest rates raise the value of both options.
Students link 'higher rates' with 'higher value' without thinking.
Fix: A higher rate lowers the PV of the exercise price. That helps a call holder, who pays Pe later, and hurts a put holder, who receives Pe later. Calls rise, puts fall.
Treating a put delta as positive, or forgetting to subtract 1 from N(d1).
Students remember delta = N(d1) and apply it to both option types.
Fix: Put delta = N(d1) − 1, which is always between −1 and 0. Use its absolute size when finding the number of options to hedge.
Applying put-call parity to options with dividends or to American options without comment.
The formula looks general, so conditions are forgotten.
Fix: State that parity holds exactly for European options on non-dividend shares. Say that dividends and early exercise make it only approximate.
Listing factors without explaining the direction of effect for calls and puts separately.
Students recall the factor names but not the logic.
Fix: Use a two-part sentence for each factor: effect on calls, effect on puts, plus the reason. Volatility and time are the ones where both move the same way.

Worked examples

Example 1

A European call option on a non-dividend-paying share has an exercise price of $11.00 and six months to expiry. The share price is $10.00. The risk-free rate is 4% a year and the call is valued at $0.90. Use put-call parity to find the value of a European put with the same terms.

Show the solution

Use p = c − Pa + Pe e^(−rt). Here c = 0.90, Pa = 10.00, Pe = 11.00, r = 0.04, t = 0.5.
Calculate rt = 0.04 × 0.5 = 0.02. Then e^(−0.02) = 0.980199.
PV of exercise price = 11.00 × 0.980199 = $10.7822.
p = 0.90 − 10.00 + 10.7822 = $1.6822.
Sense check: the put is positive and the share is below the exercise price, so it is in the money. A value of about $1.68 is sensible.

Answer: The put value is about $1.68, assuming European options, the same exercise price and expiry, and no dividends.

Example 2

Your client holds 62,000 shares in a listed company and wants protection against a fall in the share price. A put option on the share has a Black-Scholes N(d1) of 0.62. Calculate the put delta and the number of put options needed for a delta hedge. Explain what happens to the hedge if the share price rises sharply.

Show the solution

Put delta = N(d1) − 1 = 0.62 − 1 = −0.38.
Each put changes in value by about $0.38 for every $1 change in share price, in the opposite direction to the share.
Number of puts = shares held ÷ |put delta| = 62,000 ÷ 0.38 = 163,158 puts (rounded up).
Check: 163,158 × 0.38 ≈ 62,000, so the put position offsets the share position for small price moves.
If the share price rises sharply, the put moves further out of the money and its delta moves towards zero. The hedge then offsets less of the share movement. This sensitivity of delta to the share price is gamma.
The client must therefore rebalance, buying more puts as delta changes. This adds transaction cost, and the hedge is accurate only for small price changes.

Answer: Put delta is −0.38. About 163,158 puts are needed. A sharp rise in the share price reduces the put's delta in size, so the position needs rebalancing because of gamma.

Exam tips

Show the formula, the present value of the exercise price and the substitution as separate lines. Method marks are earned even if the arithmetic slips.
State the assumptions of parity in a sentence: European, same exercise price and expiry, no dividends. Examiners reward this commercial awareness.
For 'explain the factors' requirements, cover calls and puts separately and give a reason for each direction. Link to the scenario, for example a volatile sector.
Use the professional skills marks by commenting on what the value means for the client and on model limits, such as constant volatility and the European-option assumption.
If delta is asked, work out the hedge ratio and then comment on rebalancing and gamma. A calculation alone gives only part of the marks.

Practice questions from Application of option pricing theory in investment decisions

Put-Call Parity and Option Value Determinants in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Put-Call Parity and Option Value Determinants: frequently asked questions

What is put-call parity in ACCA AFM?

It is the relationship c + Pe e^(−rt) = p + Pa between a European call and put with the same exercise price and expiry. It holds for shares paying no dividends. You use it to get a put value when you already have the call value from Black-Scholes.

How do I calculate a put option value from a call?

Use p = c − Pa + Pe e^(−rt). Discount the exercise price at the risk-free rate over the option life, subtract the current share price from the call value, and add the discounted exercise price. Check that the answer is positive.

Which factors increase the value of a call option?

A higher share price, a lower exercise price, higher volatility, more time to expiry and a higher risk-free rate all raise call value. Expected dividends lower it. For a put, the effects of share price, exercise price, interest rate and dividends are reversed. Volatility raises both.

Do I need to know the Greeks for AFM?

You should understand delta and gamma and be able to use delta to set a hedge. Delta is N(d1) for a call and N(d1) − 1 for a put. Gamma shows how much delta changes as the share price moves, so it tells you the hedge needs rebalancing. Theta, vega and rho are worth knowing in words.