Chapter 5 of 9
Put–call parity and synthetic exposures
Connect calls, puts, the underlying and discounted strike value, then distinguish a theoretical relationship from an executable arbitrage.
In this chapter
A payoff identity comes before a pricing claim #
At expiry, a long call and short put with the same strike and underlying have combined intrinsic payoff max(S − K, 0) − max(K − S, 0) = S − K. Below strike the put creates the negative portion; above strike the call creates the positive portion.
The resulting linear payoff resembles a forward obligation at strike, not a free long position with no funding requirement. Premiums, financing and settlement must still be accounted for. A synthetic exposure can carry risk similar to the exposure it replicates.
The standard European relationship #
Under matched European exercise assumptions, no arbitrage frictions and a continuous income yield q, the relationship is C − P = S × exp(−qT) − K × exp(−rT). C and P are present option values, S is spot, r is the relevant financing rate and T is years. Without income, q is zero.
The present value of the strike appears because paying K at expiry differs from paying K today. Comparing C − P directly with S − K before expiry ignores that time-value relationship. Discrete dividends, exercise style and financing constraints can require a different treatment.
Why an apparent mismatch may not be tradable #
A screen can combine stale last trades, different timestamps, wide bid–ask spreads and quantities unavailable for execution. An apparent parity gap calculated from midpoints is not automatically a profit that can be captured. Borrowing the underlying or funding the strike can involve additional constraints.
Physical settlement, charges, taxes and early-exercise features can matter. Before using a textbook relationship in a real valuation study, verify that every leg has the same underlying, strike, expiry, quantity and applicable convention.
Use the identity to understand a portfolio #
Synthetic relationships are useful for recognizing that different-looking collections of positions can share economic exposures. They are not evidence that adding more legs removes risk. A long call plus short put can retain a large adverse exposure and funding needs.
An educational comparison should show all legs, premium cash flows and the conditions under which the relationship is derived. Do not advertise the identity as guaranteed arbitrage income.
Decision lab: A parity equation requires compatible contracts #
Put–call parity relates matching underlying, strike, expiry and specified carrying assumptions. Apparent deviations may reflect dividends, funding, exercise style, short-sale constraints, stale quotes or transaction costs rather than an accessible arbitrage.
Use bid/ask prices that could coexist and add the costs of every required leg. Mark which inputs are observed and which are estimated. If replication requires financing or borrow you cannot obtain, the algebra alone does not provide an executable transaction.
Hypothetical worked example
A financing difference explains the premium difference
With fictional S = K = 100, r = 5%, T = 1 year and q = 0, discounted strike is 100 × exp(−0.05) ≈ 95.1229. The model relationship implies C − P ≈ 4.8771.
If a call value is 10.4506, a matching put consistent with this relationship is about 5.5735. Those are theoretical example values, not quotes. The difference is not zero merely because spot equals strike.
At expiry with S = 80, the call-minus-put intrinsic payoff is −20; at S = 120 it is +20. The identity explains the linear exposure without promising a favorable outcome.
Common mistakes
- Applying European parity to unmatched exercise or dividend conventions.
- Calling a midpoint-based discrepancy an executable arbitrage.
Why does discounted strike appear in pre-expiry put–call parity?
Make this chapter useful
Can you defend the reasoning?
Before moving on, answer the self-check in your own words and write why each alternative fails. Use the worked example to test that explanation. Then apply it in the module research task; keep the assumptions and the evidence that could change your answer.
Sources and further reading
- NSE individual-security derivatives specifications ↗
- NSE Clearing: equity derivatives margins ↗
- SEBI investor education ↗
Further-reading references from the curriculum source list. Contract sizes, expiry calendars, settlement references, margin parameters and tax treatment can change; verify the current operative documents for the actual product. Worked figures are illustrative, not current quotes.
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