Implied Forward Price from Calls and Puts

For educational purposes only; not investment advice.

Direct answer

An implied forward price is the delivery price inferred from a call and put with the same underlying, strike, and expiry. Under European exercise and continuously compounded rate r:

C - P = e^(-rT)(F - K)

so F = K + e^(rT)(C-P). It is useful for locating forward at-the-money, checking option-chain consistency, and comparing volatility across strikes or expiries. It is a cost-of-carry price shaped by financing, dividends, stock borrow, and contract terms—not the market’s average forecast of the future spot price.

Synthetic-forward mechanism

Long one call and short one put at the same K and T produces S_T - K at expiry: the payoff of a long forward. The option pair’s current net value must therefore equal the discounted value of F-K. With continuous dividend yield q, cash-and-carry gives F = S e^[(r-q)T]; with known discrete dividends, a common form is F = [S-PV(dividends)]e^(rT).

Parity requires identical contract specifications and synchronized, tradable inputs. U.S. listed equity options are generally American-style, so early-exercise value can disturb the exact European equality. Hard-to-borrow stock, funding spreads, taxes, stale quotes, settlement differences, and adjusted deliverables can also move the option-implied result away from a simple spot-carry calculation.

Six-month example and quote range

Suppose K = $100, T = 0.5, C = $6.50, P = $4.00, and r = 4%:

F = 100 + e^(0.04×0.5)(6.50-4.00) = 100 + 1.0202×2.50 ≈ $102.55.

Ignoring discounting gives $102.50, close here but increasingly inaccurate for longer maturities or higher rates. Nearby strikes should imply similar forwards. For example, midpoint pairs (K,C,P) of ($95,$9.20,$1.70), ($100,$6.50,$4.00), and ($105,$4.10,$6.60) all give the same un-discounted estimate of $102.50.

Execution creates a band. If the $100 call is $6.40/$6.60 and the put is $3.90/$4.10, buying the call and selling the put costs $2.70, while selling the call and buying the put receives $2.30. Applying the interest factor gives an implied-forward interval of about $102.35–$102.75, not one arbitrage-ready point.

Practical checklist and risks

Match underlying, expiry, strike, exercise style, settlement, multiplier, and deliverable exactly.
Snapshot the call and put simultaneously; use executable Bid/Ask combinations for trading conclusions.
Apply the rate matching the expiry and stated compounding convention rather than silently using F ≈ K+C-P.
Estimate from several liquid strikes near the forward; investigate dispersion before averaging or discarding it.
Separate ordinary dividends, special dividends, funding, and stock-borrow costs instead of assigning every gap to one cause.
Check OCC adjustment notices when mergers, splits, distributions, or special dividends alter the deliverable.
Account for American early exercise, especially deep-in-the-money puts and calls immediately before an ex-dividend date.
Verify last-trading day, settlement price, and AM versus PM settlement for index and cash-settled contracts.
Treat a theoretical difference smaller than spreads, fees, financing, borrow, and leg risk as non-executable.

Common misconceptions

“The forward is the expected future stock price.” It is an arbitrage-linked carrying price, not a mean forecast.
“Spot plus Call minus Put is always enough.” Strike and discounting are part of the formula.
“Every strike must return exactly the same number.” Frictions, American exercise, stale quotes, and rounding create dispersion.
“A midpoint discrepancy is free arbitrage.” A trade must cross Bid/Ask and survive all carrying costs and leg risk.
“Dividends can be ignored because they are not in the parity formula.” Their effect is embedded in spot carry and relative option prices.
“A low implied forward is bearish.” Dividends or expensive stock borrow can lower it without expressing direction.