Put–call parity and covered interest parity: compare matching cash flows

Put–call parity matches two terminal payoffs

For European options with the same strike and maturity on a non-dividend-paying stock, one share plus a put pays the greater of the terminal stock price and the strike. A call plus enough cash to deliver the strike at maturity pays the same amount. Under the idealised assumptions, equal future cash flows imply equal present costs: S + P = C + discounted K.

Use the put–call-parity experiment to move the terminal stock price across the strike. Stoll’s primary paper on put and call prices provides the historical research context. The board focuses on the payoff identity and matched financing. It omits dividends, early exercise, trading fees, bid–ask spreads and restrictions on short sales, so its displayed gap is a model consistency check.

Worked example: replicate a one-hundred-unit floor

Take the board’s invented inputs: stock price 100, strike 100, one year to maturity, annual effective rate 5% and put price 8. These are teaching values rather than market quotations.

1. Buy the stock and put for a combined cost of 108. If the terminal stock price is 70, the stock pays 70 and the put pays 30, producing 100. If the terminal price is 130, the put expires and the package pays 130.
2. Discount the strike: 100 / 1.05 is about 95.24. A call priced about 12.76 plus that deposit also costs 108. At maturity the deposit becomes 100 and the call adds any stock value above 100.
3. Use an overpriced-option preset and compare both initial costs while varying the terminal price. The model’s payoff equality remains. Before translating the gap into a strategy, add borrowing terms, spreads, shorting availability and every cash flow omitted by the board.

Covered interest parity compares currency funding routes

A domestic deposit and a foreign deposit with a forward exchange contract can be arranged to produce the same future domestic-currency amount. The forward contract fixes the conversion rate in advance, so the comparison does not require forecasting the future spot rate. A consistent exchange-rate quote convention is essential: reversing the quote reverses which factors multiply and divide.

In the covered-interest-parity simulation, the spot quote is domestic units per foreign unit. Starting with 100 domestic units, the domestic route yields 105. At spot 2, the foreign route buys 50 foreign units, grows to 51 and converts at the forward rate. Equality requires a forward rate of 105/51. Change the terminal spot rate to see why only the unhedged comparison changes with it.

Observed deviations require an institutional explanation

The BIS analysis of covered interest parity explains why balance-sheet capacity, hedging demand and intermediation costs matter when comparing real markets. A price relationship can remain a useful reference even when trading it is costly or constrained. Calling every deviation irrational skips the question of which participant could actually execute and finance the offsetting positions.

Do not infer from covered parity that the uncovered foreign deposit has the same expected or realised return as a domestic deposit. Its currency conversion is uncertain. Similarly, a payoff floor in the option example is part of a paid-for package under contractual assumptions, not a general promise that stock ownership has become riskless or that an advertised product carries no counterparty risk.

A cash-flow replication checklist

These questions make the comparison inspectable even if you never plan to trade the instruments. They help distinguish arithmetic consistency from economic feasibility.

• List each instrument’s initial payment, intermediate obligations and terminal payoff. Check the result below, at and above any option strike.
• Use one currency quote convention throughout. Label whether the quoted number means domestic per foreign unit or its reciprocal.
• State whether rates are simple, periodically compounded or continuous, and use matching maturities when discounting.
• Add spreads, funding terms, collateral, taxes, execution timing and constraints before interpreting a theoretical cost difference as an implementable opportunity.

Equal payoffs and desirable risks are separate questions

Parity reasoning compares contractual packages under a model. Choosing how much risk to retain is another problem. The insurance-deductible activity compares outcome distributions with a stated utility function, while drawdown recovery traces how losses change the base from which later returns compound. Neither replaces the cash-flow identity.

A useful reading habit is to name the type of claim: an algebraic identity, an equilibrium relationship, a historical observation or a preference-based decision. The strength of one type does not automatically transfer to another. Matching terminal values can be exact within a small model while the choice of a real position remains dependent on circumstances the model does not contain.