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Conversion and Reversal Arbitrage: Executable Put-Call Parity

Build conversion and reversal ledgers from dividend-aware European put-call parity, executable bid and ask prices, financing, stock borrow, assignment, and settlement terms.

Updated

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

Direct answer

A conversion is +stock+put−call; a reversal, also called a reverse conversion, is −stock+call−put. The call and put must refer to the same underlying claim, strike K, expiration T, multiplier, deliverable, exercise style, and settlement method, and their quantities must match the stock-equivalent amount.

For matched European options and known discrete cash dividends D_i paid at dates t_i, deterministic continuously compounded rate r, and current stock S₀, put-call parity is C−P=S₀−ΣD_i e^(−rt_i)−Ke^(−rT). Equivalently, C−P=PrepaidForward₀,T−PV(K)=DF(0,T)×(F₀,T−K). With continuous dividend yield q, it becomes C−P=S₀e^(−qT)−Ke^(−rT).

The conversion’s option-and-stock value at expiration is S_T+max(K−S_T,0)−max(S_T−K,0)=K, while the reversal has value −K. The conversion also receives intervening stock dividends; the reversal owes equivalent payments to the stock lender. These identities define theoretical cash flows, not an automatically executable profit.

How to test a conversion or reversal

  1. Lock the exact claims and timeline: stock or security, option root, call and put, K, T, quantity, multiplier, deliverable, adjustment, American or European style, physical or cash settlement, currency, timestamp, last trading time, exercise cutoff, and settlement dates.
  2. Write the signed inventory and verify the expiration identity state by state. Do not substitute a synthetic position with a different deliverable, settlement reference, expiration, or corporate-action adjustment.
  3. Choose the correct carry equation. For known discrete dividends, calculate each dated PV(D_i) and PV(K); for continuous yield use the stated yield model; for forwards use internally consistent discount and forward curves. Uncertain dividends, rates, and borrow are estimates, not locked cash flows.
  4. Build both executable ledgers from one synchronized market snapshot and sufficient size. Before fees, conversion cost is X_conv=S_ask+P_ask−C_bid; reversal proceeds are X_rev=S_bid+P_bid−C_ask. Last prices, midpoints, asynchronous quotes, and a theoretical surface do not establish a tradable package.
  5. Add dated carrying cash. A conversion candidate starts with E_conv=PV(K)+ΣPV(D_i)−X_conv; a reversal starts with E_rev=X_rev−PV(K)−ΣPV(D_i). Then apply the account’s actual borrowing or lending rate, short-sale-proceeds restrictions, locate, borrow fee or rebate, collateral, payment in lieu of dividends, commissions, exchange charges, margin, tax, and model buffer.
  6. Stress the lifecycle before trading. Model partial and rejected legs, stock gaps, American call or put assignment, dividend dates, borrow recall and forced buy-in, corporate actions, physical shares and strike cash, cash settlement, broker liquidation, and mismatched security and option settlement.
  7. Use an eligible linked or complex limit order where available, but verify the venue and broker behavior. Reconcile fills, remaining legs, stock, options, borrow, dividends, interest, collateral, exercise or assignment, cash settlement, fees, taxes, and the final statement before calling the result realized arbitrage.

For European claims, the fair time-zero conversion value including known dividends is V_conv=PV(K)+ΣPV(D_i); the reversal’s signed position value is its negative, while its fair initial cash proceeds have the same positive magnitude. A positive screen residual is only a candidate. It becomes a locked net result only if every future obligation can actually be funded, borrowed, executed, held, and settled under the assumed terms.

American exercise preserves the matched expiration payoff but can break the planned holding path. Assignment on one short option does not automatically exercise, sell, or close the other option or the stock. A cash-settled index call-put pair also does not create ownership of the index; an ETF or future hedge introduces basis, carry, quantity, and settlement differences.

Worked examples

  • European conversion with a theoretical positive residual. Let T=1, r=5% continuously compounded, K=100, no dividends, S_ask=100, P_ask=2, and C_bid=8. Then PV(K)=100e^(−0.05)=95.12294245, executable conversion cost is 100+2−8=94, and the present-value candidate edge is 95.12294245−94=1.12294245 per share. Borrowing 94 until expiration requires 94e^0.05=98.81948306; the package has expiration value 100, leaving gross profit 1.18051694 per share, or $118.051694 for a 100-share package, before every fee, spread, margin, tax, and operational cost.
  • Known dividend and stock borrow in a reversal. Let T=.5, r=4%, K=100, and a known D=$1.20 dividend at t=.25. With S_bid=102.30, P_bid=2.20, and C_ask=4.10, reversal proceeds are 102.30+2.20−4.10=100.40. The fair threshold is 100e^(−.02)+1.20e^(−.01)=99.20792713, so the initial candidate excess is 1.19207287 per share. Investing 100.40 to expiration produces 100.40e^.02=102.42821454; the dividend-equivalent payment has expiration value 1.20e^.01=1.21206020. After paying K=100 and an explicit $0.80 per share borrow fee, profit is 102.42821454−1.21206020−100−.80=$0.41615434 per share, or $41.615434 for 100 shares. A higher or changing borrow charge, restricted short proceeds, recall, or tax can erase it.
  • Fees and one partial leg reverse a small edge. With no dividends, K=50, T=.25, and r=2%, fair conversion value is 50e^(−.005)=49.75062396. For 10 packages, synchronized prices S_ask=50.05, P_ask=1.30, and C_bid=1.62 give X_conv=49.73 per share; gross present-value edge is (49.75062396−49.73)×1,000=$20.623960. Fees on 20 option contracts at $0.65 are $13, leaving only $7.623960. If all options fill but only 700 shares fill at $50.05, and the remaining 300 shares fill after a rise to $50.85, extra stock cost is $240 and the result becomes $7.623960−$240=−$232.376040 before other costs.
  • Early assignment dismantles an American conversion. Hold 2 packages of a physical equity conversion when the short calls with K=$50 are assigned before the ex-dividend date. The account delivers 200 shares, receives 2×$50×100=$10,000, and the short calls and stock disappear, but the 2 long puts remain. Selling the puts at an executable bid of $0.08 receives $16; after 2×$0.65=$1.30 option fees and a $10 assignment fee, disposal cash is $10,000+$16−$1.30−$10=$10,004.70. Losing eligibility for a $1 per share dividend also forgoes $200. This ledger excludes the original package cost and financing, so $10,004.70 is not profit.

Contract, execution, and carry risks

  • Match the exact option root, underlying claim, strike, expiration, and legal series.
  • Match multiplier, deliverable, corporate-action adjustment, currency, and quantity ratio.
  • Distinguish European parity from American early-exercise rights.
  • Model every known dividend amount and ex-date with the correct dated present value.
  • Treat uncertain, special, reduced, or omitted dividends as risk rather than fixed carry.
  • Use the correct prepaid-forward, discount-factor, forward, and strike-present-value conventions.
  • Buy stock and options at asks and sell them at bids for the relevant package direction.
  • Reject stale, asynchronous, midpoint, last-sale, and insufficient-size parity screens.
  • Control rejected legs, partial fills, legging, gaps, halts, corrections, and order cancellation.
  • Use the account’s actual borrowing and lending rates rather than one risk-free rate.
  • Confirm that short-sale proceeds can be used or invested as the model assumes.
  • Obtain a valid locate and confirm continuing stock-borrow availability before a reversal.
  • Stress borrow fees, rebates, collateral haircuts, recalls, buy-ins, and hard-to-borrow changes.
  • Include payments in lieu of dividends and their potentially different tax treatment.
  • Model early assignment on short calls near dividends and deep-in-the-money short puts.
  • Do not assume a long option will automatically offset or exercise after assignment.
  • Separate physical stock and strike cash from index or other cash-settlement obligations.
  • Allow for stock, option, exercise, assignment, and cash-posting timing mismatches.
  • Include margin, buying power, concentration, broker liquidation, fees, slippage, and taxes.
  • Reconcile corporate actions, adjusted contracts, final positions, cash, borrow, and records.

Common misconceptions

  • Expiration direction neutrality means the package is riskless throughout its lifecycle.
  • Any positive put-call-parity residual is free and executable profit.
  • A reversal needs no locate or stock borrow because its options offset the short stock.
  • Implied volatility determines the matched expiration identity or guarantees the edge.
  • Theoretical residual equals realized account profit after financing, assignment, fees, and tax.

Authoritative sources

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