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Implied Forward Price: Discounted Parity and Executable Bounds

Infer a delivery forward from matched European options, distinguish it from prepaid value and forecasts, and control discounting, dividends, borrow, exercise style, and execution.

Updated

For educational purposes only; not investment advice. Investing may result in loss.

Direct answer

For matched European options, C-P=D(0,T)*(F(0,T)-K), so F(0,T)=K+(C-P)/D(0,T). The prepaid-forward value is FP=D(0,T)*F(0,T)=C-P+K*D(0,T). F is the currency-per-share delivery price at maturity; FP and C-P are present values. They are not interchangeable.

With flat continuous carry, D(0,T)=e^(-rT) and F=S_0*e^((r-q)T). With deterministic cash dividends, F=(S_0-PV_div)/D(0,T). The inferred forward is an arbitrage-linked carry residual under assumptions, not the physical expected future spot and not automatically an executable futures or forward quote.

A controlled workflow

  1. Lock underlying, option class, strike, expiry or fixing, exercise style, settlement, multiplier, deliverable, currency and the current OCC memo.
  2. Build valuation, premium, stock, strike, fixing and final-settlement clocks, including exact year fractions, day count, holidays and AM or PM treatment.
  3. Capture synchronized executable stock, call and put bid/ask, displayed size and package quote. Record borrow, rebate, taxes and payment-in-lieu assumptions separately.
  4. Specify D(0,T), zero curve, compounding, funding or collateral convention, dividends and the output: current synthetic value C-P, prepaid forward FP, or delivery price F.
  5. For European-equivalent inputs calculate FP=C-P+K*D(0,T) and F=FP/D(0,T) at full precision. Build the lower side from C_bid-P_ask and upper side from C_ask-P_bid; leg sums do not guarantee package fills.
  6. For American or adjusted options, model early exercise and borrow or use defensible bounds and European instruments. Apply the live deliverable and official fixing instead of forcing European equality.
  7. Validate the cross-strike slope -D(0,T) and multiple expiries, then reconcile actual fills, stock, strike cash, dividends, borrow, exercise, assignment, fees, taxes and final settlement.

Worked examples

  • Discounted parity: Let K=100, T=0.5, continuously compounded r=4%, C=6.50 and P=4.00. Then D(0,T)=0.980198673307, FP=100.519867330676 and F=102.550503350067. The shortcut K+C-P=102.50 understates the forward by 0.050503350067 per share, or $5.0503350067 for multiplier 100.
  • Cross-strike linearity: With the same D(0,T) and F, theoretical C-P=D*(F-K) equals 7.400993366534 at K=95, 2.500000000000 at K=100, and -2.400993366534 at K=105. Each five-point strike step changes the difference by -4.900993366534=-5*D. The old no-discount shortcut would incorrectly make inconsistent pairs look identical.
  • Discrete dividends: Let S_0=100 and add deterministic $0.50 dividends at t=0.20 and t=0.45. Their present values are 0.496015957419 and 0.491080516179, so PV_div=0.987096473598, FP=99.012903526402, F=101.013096857576, and parity at K=100 requires C-P=0.993036195727. The no-dividend forward 102.020134002676 is higher by 1.007037145100 because timing and carry matter.
  • Executable interval: At K=100, call bid/ask is 6.40/6.60 and put bid/ask is 3.90/4.10. The short-synthetic reference 2.30 implies F_bid=102.346463082062; the long-synthetic reference 2.70 implies F_ask=102.754543618072. Width is 0.408080536011 per share before fees, size and leg risk; midpoint is not execution.

Risks and validation

  • Series risk: Underlying, strike, expiry or class mismatch breaks parity.
  • Style risk: American early-exercise value contaminates the European result.
  • Settlement risk: Cash, physical, AM, PM and official fixing define different claims.
  • Deliverable risk: Adjustments can change shares, multiplier and cash-in-lieu.
  • Currency risk: Per-share quotes and contract cash must use one currency and scale.
  • Timestamp risk: Asynchronous stock and option markets create false basis.
  • Quote risk: Stale, crossed or midpoint quotes are not executable.
  • Size risk: Displayed liquidity may not support the intended package.
  • Leg risk: Partial fills and latency can destroy the synthetic relationship.
  • Curve risk: A public yield is not automatically the correct zero discount factor.
  • Convention risk: Compounding and day-count mismatches move D(0,T).
  • Clock risk: Premium, stock, strike and final settlement can occur on different dates.
  • Funding risk: Collateral and desk financing differ from a textbook risk-free rate.
  • Dividend risk: Amounts, payment dates and undeclared distributions remain uncertain.
  • Adjustment risk: Ordinary and special distributions can receive different OCC treatment.
  • Borrow risk: Locate, fees, rebate, recall and payment in lieu alter carry.
  • Tax risk: Withholding and investor-specific treatment change executable economics.
  • Strike-dispersion risk: Stale quotes, skew and American value can break linearity.
  • Numerical risk: Nonpositive D, unit errors and rounding distort F and FP.
  • Interpretation risk: A low forward is not inherently bearish or free arbitrage.

Common misconceptions

  • “The forward is the expected future spot.” It is a carry-linked delivery price under stated assumptions.
  • “K+C-P always gives the forward.” The option difference must be capitalized by the correct discount factor.
  • “Every American call-put pair must return exactly one F.” Early exercise, borrow and quotes create model-dependent dispersion.
  • “A midpoint discrepancy is executable arbitrage.” Bid/ask, size, fees, funding and leg risk remain.
  • “A low forward is bearish or caused only by dividends.” Rates, borrow, tax, style and adjustments can enter the residual.

Authoritative sources

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