Lesson 6.1: Option payoffs and put-call parity
Learn what calls and puts pay at expiry, how to combine options into straddles, spreads and covered calls, and how put-call parity ties option prices together.
By the end of this lesson you will be able to read and draw the profit of a call and a put at expiry, combine options into the common strategies, find the break-even points and the worst loss of a position, state put-call parity and use it to find a fair put price from a call price, spot an arbitrage when parity fails, and infer a forward price from option quotes. You will show that a straddle bought for 11 breaks even at 89 and 111, that a call at 8.00 on a share at 100 with a strike of 100, 6 per cent interest and six months to run implies a put at 5.04, and that a put quoted at 4.00 in that market lets you lock in a riskless profit of 1.04. You should have read Module 1 on markets and instruments. The mathematics is the maximum of two numbers, discounting with an exponential and simple algebra. This course is education about how markets and research work, not investment advice.
The first five modules treated prices as things you trade directly. Module 6 turns to derivatives: contracts whose value comes from the price of something else. Among them, options have a special place in quantitative finance. Their payoff is not a straight line, so their value depends on how much the underlying price moves and not only on where it goes. That makes them the natural instrument for trading volatility, and the pricing of options is the oldest and most developed part of the mathematics of finance. We begin with the simplest thing about them: what they pay.
What an option is
An option gives its holder a right, not an obligation.
A call option gives the right to buy the underlying asset at a fixed price, the strike K, on or before a fixed date, the expiry or maturity T.
A put option gives the right to sell the underlying at the strike K.
The buyer pays a price up front, the premium. The seller, or writer, receives it and takes on the obligation to deliver if the buyer chooses to exercise. A European option can be exercised only at expiry. An American option can be exercised at any time up to expiry. Index options in India, such as those on the Nifty, are European and cash-settled: nobody delivers the index, and at expiry the holder simply receives the difference in cash. Options trade in standard lots, so a position is a number of lots, and the contract specifications, lot sizes and expiry days are set by the exchange and change from time to time, so look them up when you trade.
The word option is exact. The holder of a call will exercise it only if the share is worth more than the strike. If it is worth less, the call is worthless and expires, and the holder has lost the premium, but nothing more. That one-sided bargain, a limited loss with an open-ended gain, is why options cost money.
Payoffs and profits at expiry
At expiry, with S the price of the underlying then, the payoff of each option is:
Position · Payoff at expiryLong call (you bought it) · max(S − K, 0)
Short call (you sold it) · −max(S − K, 0)
Long put · max(K − S, 0)
Short put · −max(K − S, 0)
Subtract the premium paid, or add the premium received, to get the profit. Take a call and a put on a share, each with a strike of 100 and a premium of 5.
Price at expiry S · 80 · 90 · 100 · 105 · 110 · 120Long call profit · −5 · −5 · −5 · 0 · +5 · +15
Long put profit · +15 · +5 · −5 · −5 · −5 · −5
The call loses its premium of 5 for any price up to the strike, breaks even at 105 (the strike plus the premium) and gains one for one beyond that. The put loses 5 for any price above the strike, breaks even at 95 (the strike minus the premium) and gains one for one below that. The seller of each has the mirror-image profit: a small fixed gain and a loss that can grow large. The long call's gain has no ceiling, and the long put's gain is limited only by the fact that the price cannot fall below zero.
Moneyness, intrinsic value and time value
An option is in the money if exercising it now would pay something: a call with S above K, or a put with S below K. It is at the money when S is close to K, and out of the money otherwise. The intrinsic value is what it would pay if exercised now, max(S − K, 0) for a call. The rest of the premium is the time value: the extra that buyers pay because the price might still move in their favour before expiry. At expiry the time value is zero. An at-the-money option is all time value, and Lesson 6.2 shows how to price it.
Combining options into strategies
Because options can be added together, a trader can build almost any shape of payoff. The common ones:
Strategy · Built from · Shape · Used whenProtective put · Own the share, buy a put · Floor under the loss · You want insurance on a holding
Covered call · Own the share, sell a call · Gains capped, small cushion · You will sell at a higher price anyway and want the premium
Straddle · Buy a call and a put at the same strike · Gains if the price moves a lot in either direction · You expect a large move but not its direction
Strangle · Buy an out-of-the-money call and put · Like a straddle, cheaper, needs a larger move · The same, with less premium
Bull call spread · Buy a call, sell a higher-strike call · Limited gain, limited loss · You expect a moderate rise
Butterfly · Buy low and high strikes, sell two middle ones · Gains if the price ends near the middle · You expect little movement
Collar · Own the share, buy a put, sell a call · Both a floor and a cap · You want low-cost protection
Worked example 1: the straddle. A share trades at 100. You buy a call and a put, both with a strike of 100, paying 6 for the call and 5 for the put, 11 in all. At expiry the call pays max(S − 100, 0) and the put pays max(100 − S, 0), so exactly one of them pays, and the total payoff is the distance of S from 100. The profit is that distance minus 11. At S = 100 you lose the whole 11. At S = 89 or S = 111, the distance is 11 and you break even. At S = 120 or S = 80 the profit is 9, and at S = 130 or S = 70 it is 19. The most you can lose is the premium of 11, and there is no limit to the gain on the upside. A straddle is a bet that the price will move by more than 11 in either direction: it is a bet on volatility.
Worked example 2: protective put and covered call. You own a share at 100. Add a put with a strike of 95 costing 3 for protection, or sell a call with a strike of 105 for 3. The profits at expiry are:
Price at expiry · 70 · 80 · 90 · 100 · 110 · 120 · 130Share alone · −30 · −20 · −10 · 0 · +10 · +20 · +30
With protective put (strike 95, cost 3) · −8 · −8 · −8 · −3 · +7 · +17 · +27
With covered call (strike 105, premium 3) · −27 · −17 · −7 · +3 · +8 · +8 · +8
The protective put caps the loss at 8 (the 5 between 100 and 95, plus the 3 premium), at the price of 3 off every outcome. The covered call adds 3 in every outcome, but gives up all gains above 105, and offers little protection against a fall.
Try a straddle of your own below. Change the strike and the two premiums, and find the break-even points.
Synthetic positions and put-call parity
A long call and a short put with the same strike and expiry have a combined payoff of max(S − K, 0) − max(K − S, 0). If S is above K the first term is S − K and the second is zero. If S is below K the first is zero and the second is K − S, subtracted, giving S − K again. In both cases the payoff is S − K, exactly that of a forward contract to buy at K. So the two options together make a synthetic forward. By combining forms in this way we can make a synthetic stock (long call, short put and cash), a synthetic call (stock, put and borrowing) and so on.
This leads to the most important relationship in options, put-call parity. Compare two portfolios, each held to expiry T:
Portfolio A: one call plus cash of K e^(−rT), which grows at the risk-free rate r to K at expiry.
Portfolio B: one put plus one share.
At expiry, if S is above K, A holds a call worth S − K plus K in cash, S in total, and B holds a worthless put and a share worth S. If S is below K, A holds a worthless call and K in cash, and B holds a put worth K − S and a share worth S, K in total. In both cases the two portfolios are worth max(S, K). Two portfolios that are worth the same at expiry in every circumstance must cost the same today, or someone could buy the cheaper and sell the dearer for a riskless profit. So
C + K e^(−rT) = P + S, or equivalently C − P = S − K e^(−rT)
where C and P are the European call and put prices, and S is today's share price. This holds for a share that pays no dividends, whatever the volatility, the direction or the pricing model.
Worked example 3. A share trades at 100. A European call with a strike of 100 and six months to expiry trades at 8.00, and the risk-free rate is 6 per cent a year with continuous compounding. The present value of the strike is 100 × e^(−0.06 × 0.5) = 100 × 0.9704 = 97.04. Parity says C − P = 100 − 97.04 = 2.96, so the put should be worth 8.00 − 2.96 = 5.04.
An arbitrage. Suppose the put is quoted at 4.00 instead. Then C − P = 4.00, above the 2.96 that parity allows: the call is dear relative to the put. Sell the call for 8.00, buy the put for 4.00 and buy the share for 100. The net outlay is 100 + 4.00 − 8.00 = 96. At expiry, whatever the price, the position is worth exactly 100: if S is above 100 the call is exercised against you and you deliver the share for 100, and if S is below 100 you exercise your put and sell it for 100. A riskless 100 in six months costs 96 now, whereas borrowing it today would cost 97.04. The difference, 1.04, is a riskless profit. In practice such gaps vanish in milliseconds, because the traders who watch them trade until the prices realign.
Dividends, forwards and index options
If the share pays dividends, the holder of the share receives them but the holder of the option does not, so parity is adjusted: C − P = S − PV(dividends) − K e^(−rT). For an index, with a continuous dividend yield q, the share price in the formula is replaced by S e^(−qT). The neatest way to write it uses the forward price F, the price today for delivery at T, which equals S e^((r − q)T):
C − P = e^(−rT) × (F − K)
Traders use this the other way round: the market's call and put prices at one strike imply a forward price. Take an index at 24,000 (an illustrative level), r = 6 per cent, q = 1.2 per cent and one month, T = 1 ÷ 12. The forward is F = 24,000 × e^(0.048 ÷ 12) = 24,096.2. At a strike of 24,000 parity says C − P = e^(−0.005) × 96.2 = 95.7. If a trader sees the call at 410 and the put at 330, the gap is 80, below 95.7, which means the market's implied forward is lower than the 24,096 implied by the interest rate and the dividend yield, perhaps because the market expects a higher dividend or because borrowing the index is costly. Reading the implied forward from option prices is the first step in building a volatility surface, which Lesson 6.3 takes up.
Bounds and early exercise
Parity also gives simple no-arbitrage bounds. A European call can never be worth less than max(S − K e^(−rT), 0): the call is at least as good as a forward bought at K with the cost deferred. For S = 100, K = 90, r = 6 per cent and one year, the lower bound is 100 − 90 × e^(−0.06) = 100 − 84.76 = 15.24, greater than the intrinsic value of 10. A call can also never be worth more than the share. Another consequence is that an American call on a share with no dividends is never exercised early: exercising throws away the time value and the interest earned on the strike, so selling the option is always better. An American put, however, may be worth exercising early when it is deep in the money, since the interest on the cash received exceeds the remaining time value, a point that Lesson 7.4 shows with numbers.
Why options matter to a quant
They are non-linear. A stock's profit is a straight line. An option's profit bends at the strike, and the amount of bend, called convexity, is what Lesson 6.2 measures as gamma.
They are the way to trade volatility. The price of an option rises with the expected size of the price moves, so a position in options is a position on how much the market will move.
They are leveraged. A small premium controls a large notional amount, which magnifies both gains and losses.
They have a set of exact relationships. Parity is one. Others, such as the bounds above and the no-arbitrage conditions across strikes, give a quant checks that must hold in any model and tests for data errors.
They carry information. The market's option prices contain a view about the distribution of future prices: the implied volatility of Lesson 6.3 is among the most watched numbers in finance.
The practicalities deserve respect. Option quotes carry wider spreads than the underlying, liquidity is concentrated in a few strikes close to the current price, and sellers of options face margin requirements and potentially unlimited loss. Even in a model, an option position's risk is not just its premium.
The code below computes the payoffs, the profits of the strategies, parity, the arbitrage and the implied forward.
python
import numpy as np
from math import exp
payoffs and profits at expiry
S = np.array([70, 80, 90, 100, 105, 110, 120, 130.0])
K, premium = 100.0, 5.0
longcall = np.maximum(S - K, 0) - premium
longput = np.maximum(K - S, 0) - premium
print("price ", S)
print("long call ", longcall)
print("long put ", longput)
straddle: strike 100, premium 6 + 5
cost = 11.0
straddle = np.maximum(S - K, 0) + np.maximum(K - S, 0) - cost
print("straddle ", straddle, "| break-evens", K - cost, K + cost)
share alone, protective put (K=95, cost 3) and covered call (K=105, premium 3), bought at 100
share = S - 100
protectiveput = share + np.maximum(95 - S, 0) - 3
coveredcall = share - np.maximum(S - 105, 0) + 3
print("share alone ", share)
print("protective put", protectiveput)
print("covered call ", coveredcall)
put-call parity
s, k, r, t = 100.0, 100.0, 0.06, 0.5
pvk = k exp(-r t)
call = 8.00
print("PV of strike", round(pvk, 3), "| call minus put must be", round(s - pvk, 3), "| fair put", round(call - (s - pvk), 3))
putquoted = 4.00
print("arbitrage: outlay", s + putquoted - call, "vs PV of the 100 it returns", round(pvk, 3),
"| riskless profit", round(pvk - (s + putquoted - call), 3))
lower bound of a call and the forward implied by an index option
print("call lower bound (S=100, K=90, r=6%, T=1):", round(100 - 90 exp(-0.06), 2))
index, rr, q, tt, strike = 24000.0, 0.06, 0.012, 1 / 12, 24000.0
forward = index exp((rr - q) tt)
print("forward", round(forward, 1), "| call minus put at the strike", round(exp(-rr tt) (forward - strike), 1))
print("implied forward from a call at 410 and a put at 330:", round(strike + (410 - 330) exp(rr tt), 1))
Common mistakes
Confusing payoff with profit. The payoff ignores the premium. The profit includes it, so the break-even is the strike plus (or minus) the premium.
Thinking a cheap out-of-the-money option is a bargain. It is cheap because it rarely pays, and most such options expire worthless.
Selling options for the premium without counting the tail. The seller of a call has an unlimited loss, and the seller of a put loses down to zero.
Ignoring that parity needs the right inputs. The interest rate, the dividends and the borrowing cost of the share all enter. A parity "violation" is often just a wrong dividend.
Treating American and European options alike. Parity in this form holds for European options. American options obey inequalities.
Forgetting lot sizes and margins. A position is a number of lots, and the margin can be large for a seller.
Reading a straddle as a bet on direction. It is a bet on size, whichever way it goes.
Trading the payoff diagram and forgetting time. The diagram is for expiry. Before expiry, the option's value is higher than its payoff and moves with time, volatility and rates, which the next lesson covers.
Key takeaways
A call pays max(S − K, 0) and a put pays max(K − S, 0) at expiry. The profit subtracts the premium, so a long call breaks even at K plus the premium and a long put at K minus the premium.
The buyer's loss is limited to the premium, and the seller's is not. That asymmetry is why options cost money.
Options combine into strategies. A straddle bought for 11 at a strike of 100 breaks even at 89 and 111 and loses 11 at 100. A protective put at 95 for 3 caps the loss at 8, and a covered call gives 3 but gives up gains above 105.
A long call and short put make a synthetic forward. Put-call parity, C − P = S − K e^(−rT), follows from two portfolios with equal payoffs, and it does not depend on any pricing model.
With S = 100, K = 100, r = 6 per cent, T = 0.5 and a call at 8.00, the put should be 5.04. A put at 4.00 gives a riskless profit of 1.04 per share.
With dividends, parity becomes C − P = e^(−rT)(F − K), and traders read the implied forward from option prices.
A European call is worth at least S − K e^(−rT). A call on a share with no dividends is never exercised early, and an American put may be.
Check yourself
1. You buy a put with a strike of 250 for a premium of 12. What is the break-even price at expiry, and the maximum profit?
Break-even = 250 − 12 = 238. The maximum profit is when the price falls to zero: 250 − 12 = 238.
2. You sell a call with a strike of 500 for a premium of 20. What is your profit if the price at expiry is 530?
You receive 20 and pay 530 − 500 = 30, so the profit is −10.
3. A strangle buys a call at 110 for 2 and a put at 90 for 2. What are its break-even points and its maximum loss?
Total premium = 4. Break-evens are 110 + 4 = 114 and 90 − 4 = 86. The maximum loss is 4, if the price ends between 90 and 110.
4. S = 50, K = 50, r = 4 per cent, T = 1, no dividends, and the call is 4.20. What is the fair price of the put?
PV of the strike = 50 × e^(−0.04) = 48.04. P = C − S + PV(K) = 4.20 − 50 + 48.04 = 2.24.
5. Why does a straddle seller dislike a sudden big move in either direction?
The seller keeps the premium only if the price ends near the strike. A large move in either direction makes one of the two options pay out more than the premium received.
Try this
Set aside forty-five minutes. You need a spreadsheet or Python and, if you can, a quote of a few option prices from an exchange website.
1. Draw the profit at expiry of a long call, a long put, a short call and a short put on one share. Mark the break-even points.
2. Build a bull call spread by buying a call at 100 and selling a call at 110, with premiums of 6 and 2.5. Find the maximum profit, the maximum loss and the break-even.
3. Use the straddle simulator to find how far the price must move to break even when the strike is 100 and the premiums are 8 and 7.
4. Take one real option pair with the same strike and expiry on an index or share. Use the parity simulator with the market rate and a dividend estimate to find the implied put from the call and compare with the quoted put.
5. Compute the implied forward price from the call and put at the strike nearest the money, and compare it with the spot and the futures price.
6. Write down the payoff table, at five prices, of a collar: a share, a put at 95 and a sold call at 110.
7. Write a short note: one strategy that fits a view on volatility, its break-even points, its worst case and what could go wrong.
A good result is a set of payoff diagrams that you can draw from memory, a worked parity check with real prices and a clear statement of the worst case of each position. The next lesson gives the price itself: how the Black–Scholes model values an option, and what the Greeks tell you about its risks.
Read more
John Hull's Options, Futures and Other Derivatives is the standard introductory text and covers payoffs, parity and bounds with many examples. Sheldon Natenberg's Option Volatility and Pricing is the practitioner's classic on strategies. Hans Stoll's 1969 paper "The Relationship Between Put and Call Option Prices" in the Journal of Finance is the original statement of put-call parity, and Robert Merton's 1973 paper "Theory of Rational Option Pricing" in the Bell Journal of Economics and Management Science derives the bounds and the result on early exercise. For the specifications of Indian index options, read the current contract note on the exchange's website.
Tags: Quant Trader Options Payoff diagrams Put-call parity No-arbitrage