Synthetic Positions¶

Overview¶

Synthetic positions replicate the payoff of holding the underlying asset or a simple option position using a combination of options. They are based on put-call parity and are used for hedging, arbitrage, and capital efficiency.

Difficulty advanced Market Outlook: Varies by synthetic type Risk Level: Varies (can replicate any risk profile)

Put-Call Parity¶

The foundation of all synthetic positions:

C + K × e^{-rT} = P + S

Where:
C = Call price
P = Put price
K = Strike price
r = Risk-free rate
T = Time to expiration
S = Stock price

Rearranged:
C - P = S - K × e^{-rT}

does: the no-arbitrage identity tying European call, put, stock, and discounted strike — every synthetic position is just an algebraic rearrangement of this. Violations of parity (after carry, dividends, borrow costs) are riskless arbitrage; in practice it tells you any of the four legs can be replicated by the other three.

Synthetic Long Stock¶

Construction¶

Buy 1 ATM Call
Sell 1 ATM Put
Same strike, same expiration

Payoff: Identical to owning 100 shares

where: long call + short put at the same ATM strike and expiry. does: put-call parity equivalent of long stock (rearrange C − P = S − K·e^{−rT}). Bullish — used for capital efficiency (no full stock outlay), to bypass borrow/locate restrictions, and when options are cheaper than the implied parity cost of carrying stock.

Payoff¶

Profit/Loss
  |            /
  |           /
  |          /
  |         /
  |        /
  |       /
  |      /
  |     /
  |    /
  |   /
  |  /
  | /
  |/
--+-------------- Stock Price
  |\
  | \
  |  \
  |   \
  |    \
  |     \
  v

Cost and Risk¶

Net Cost = Call Premium - Put Premium ≈ 0 (ATM)
Effective Entry Price = Strike + Net Debit (or - Net Credit)
Delta ≈ 1.0 (same as stock)
Max Profit = Unlimited
Max Loss = Effective Entry Price (stock to zero)

where: Net Cost net premium paid (or received) for the call-put combo · Effective Entry Price strike adjusted by net debit/credit — your true cost basis · delta near 1 mirrors share exposure. does: quantifies the synthetic long as a stock-equivalent — same delta, unlimited upside, capped downside at the effective entry. Put-call parity says cost = S − K·e^{−rT} in equilibrium; deviations are the arbitrage signal.

Use Cases¶

Capital efficiency: control 100 shares for less capital
When stock is hard to borrow (for shorting later)
When options are mispriced relative to parity
Quick position entry without stock trading restrictions

Synthetic Short Stock¶

Construction¶

Sell 1 ATM Call
Buy 1 ATM Put
Same strike, same expiration

Payoff: Identical to shorting 100 shares

where: short call + long put at the same ATM strike and expiry. does: put-call parity equivalent of short stock (P − C = K·e^{−rT} − S). Bearish — used when borrow is hard-to-locate or expensive, to sidestep uptick rules, or for capital-efficient downside exposure.

Cost and Risk¶

Net Cost = Put Premium - Call Premium ≈ 0 (ATM)
Effective Entry Price = Strike - Net Credit (or + Net Debit)
Delta ≈ -1.0 (same as short stock)
Max Profit = Stock to zero
Max Loss = Unlimited

where: Net Cost net premium across both legs · Effective Entry Price strike adjusted by credit/debit — your synthetic short price · delta ≈ −1 mirrors a short share position. does: quantifies synthetic short as short-stock equivalent — same delta, capped upside, unlimited tail. By parity its cost is K·e^{−rT} − S; deviations net of dividend/borrow are the arb signal.

Use Cases¶

Shorting without borrowing stock
When borrow costs are high
Avoids uptick rule restrictions
Capital efficient short exposure

Synthetic Long Call¶

Construction¶

Buy 100 shares
Buy 1 ATM Put

Payoff: Identical to long call (protective put = synthetic long call)

where: long shares + long ATM put — the classic protective-put structure. does: put-call parity equivalent of long call (S + P = C + K·e^{−rT}). Bullish with floor — keeps upside while capping drawdown at the put strike; used for hedged accumulation and to manufacture call exposure when listed calls are illiquid or rich.

Cost and Risk¶

Net Cost = Stock Price + Put Premium
Delta ≈ Call Delta
Max Profit = Unlimited
Max Loss = Stock Price + Put Premium - Strike (limited by put)

where: Stock Price + Put Premium total outlay · max loss bounded by the gap between entry and strike plus premium paid. does: quantifies the synthetic call — unlimited upside, defined downside. Should equal listed call premium plus PV of strike under parity; gaps net of dividends are the trade signal.

Synthetic Long Put¶

Construction¶

Short 100 shares
Buy 1 ATM Call

Payoff: Identical to long put

where: short shares + long ATM call — capped-upside short. does: put-call parity equivalent of long put (−S + C = P − K·e^{−rT}). Bearish with cap — short stock hedged by long call gives the same convex downside payoff as a put; used when shorting is constrained or to replicate puts in size.

Cost and Risk¶

Net Cost = Call Premium - Short Stock Proceeds
Delta ≈ Put Delta
Max Profit = Strike - Net Cost
Max Loss = Unlimited (from short stock, limited by call)

where: Call Premium − Short Stock Proceeds net cash flow at entry · max loss bounded above by the long call. does: quantifies synthetic put — defined max loss via the call cap, max profit if stock drops to zero. Parity check: should mirror listed put premium net of carry and dividends.

Synthetic Short Call¶

Construction¶

Short 100 shares
Sell 1 ATM Put

Payoff: Identical to short call (covered put)

where: short shares + short ATM put — the "covered put" structure. does: put-call parity equivalent of short call. Bearish income — collects put premium against a short stock position; capped profit, unlimited upside risk if stock rallies. Used when you want to short and harvest theta on otherwise-rich puts.

Synthetic Short Put¶

Construction¶

Buy 100 shares
Sell 1 ATM Call

Payoff: Identical to short put (covered call = synthetic short put)

where: long shares + short ATM call — i.e. a covered call. does: put-call parity equivalent of short put. Mildly bullish to neutral income — caps upside at strike, collects call premium; identical payoff to a cash-secured short put written at the same strike.

Conversion and Reversal Arbitrage¶

Conversion (Arbitrage when Put-Call Parity is violated)¶

When: C + PV(K) > P + S

Execute:
- Sell Call (overpriced)
- Buy Put
- Buy Stock

Profit = (C + PV(K)) - (P + S) > 0 (riskless)

where: PV(K) = K·e^{−rT} present value of strike · trigger condition is a parity violation in the "calls rich" direction. does: locks in put-call parity arbitrage when the call side is overpriced — sell synthetic stock (call − put), buy real stock, pocket the spread riskless to expiry. Net of commissions, borrow, and dividends; opportunity disappears as soon as the legs realign.

Reversal¶

When: P + S > C + PV(K)

Execute:
- Buy Call
- Sell Put
- Short Stock

Profit = (P + S) - (C + PV(K)) > 0 (riskless)

where: PV(K) = K·e^{−rT} present value of strike · trigger condition is a parity violation in the "puts rich" direction. does: mirror of the conversion — when the put side is overpriced, buy synthetic stock (call − put) and short the real stock to lock in the spread. Both conversion and reversal are the workhorse arbitrage trades that keep put-call parity tight in liquid options markets.

Greeks of Synthetic Positions¶

Synthetic Position	Delta	Gamma	Theta	Vega
Long Stock (synthetic)	+1.0	0	0	0
Short Stock (synthetic)	-1.0	0	0	0
Long Call (synthetic)	≈ 0.50	≈ 0	≈ 0	≈ 0
Long Put (synthetic)	≈ -0.50	≈ 0	≈ 0	≈ 0

Note: Synthetic positions using ATM options replicate the underlying exactly in terms of Greeks (gamma, theta, vega cancel out).

Risk Considerations¶

1. Early Assignment Risk¶

Short options in synthetic positions can be assigned early, disrupting the position.

2. Dividend Risk¶

Synthetic short stock does not receive dividends; synthetic long stock does not pay them.

3. Transaction Costs¶

Three-legged positions (conversion/reversal) have significant commissions.

4. Margin Requirements¶

Synthetic positions may have different margin requirements than actual positions.

5. Execution Risk¶

Arbitrage opportunities disappear quickly; simultaneous execution is critical.

Checklist¶

[ ] Put-call parity verified before entering synthetic
[ ] Dividend dates checked (affects parity)
[ ] Borrow costs compared (synthetic vs actual short)
[ ] Transaction costs included in arbitrage calculation
[ ] Margin requirements understood
[ ] Early assignment risk assessed
[ ] Exit strategy defined

Assumptions & Limitations¶

Put-call parity assumes European options; American options can deviate
Dividend adjustments needed for parity with dividend-paying stocks
Transaction costs often eliminate small arbitrage opportunities
Execution must be simultaneous for true arbitrage
Early assignment of short options can create unexpected exposure
Margin treatment varies by broker

References¶

Hull, J.C. (2022). Options, Futures, and Other Derivatives (11th ed.). Pearson.
Stoll, H.R. (1969). "The Relationship Between Put and Call Option Prices." Journal of Finance, 24(5), 801-824.
Klemkosky, R.C. & Resnick, B.G. (1977). "Put-Call Parity and Market Efficiency." Journal of Finance, 32(5), 1549-1556.