Optionsintermediate

Synthetic Stock Position

A synthetic long stock is built by buying a call and selling a put at the same strike and expiry. By put-call parity the combined payoff equals owning the stock minus the present value of the strike, $C - P = S_0 - K e^{-rT}$ . The position behaves like a leveraged stock holding without actually buying shares. Traders use synthetic stock when the underlying is hard to borrow or short, when capital efficiency matters, or as one leg of a conversion arbitrage. The mirror trade, short call + long put, creates a synthetic short stock.

Why it matters

The long call gives the right to buy at $K$ , so all upside above $K$ is captured. The short put obligates buying at $K$ , so all downside below $K$ is absorbed. Together you participate fully in price moves above and below $K$ , exactly like holding the stock. The only difference is timing of cash flows, because the synthetic delays paying $K$ until expiry while the actual stock holder pays $S_0$ today. That delay is worth $K e^{-rT}$ in present value, and parity exactly accounts for it.

Formulas

Synthetic long stock from options

C - P \equiv S_0 - K e^{-rT}

Rearranging put-call parity. The combined option position replicates the stock minus a zero-coupon bond paying

K

at expiry.

Synthetic long stock payoff at $T$

(\max(S_T - K, 0)) - (\max(K - S_T, 0)) = S_T - K

The combined payoff is linear in

S_T

with slope 1, the same as owning the stock. The constant

-K

reflects the deferred payment.

Worked examples

Scenario

BHP shares trade at A$45. A trader buys a 6-month European call struck at A$45 for A$2.80 and sells a 6-month European put struck at A$45 for A$1.95. Risk-free rate $r = 4\%$ per annum.

Solution

Net cost today equals $C - P = 2.80 - 1.95 = 0.85$ . Parity predicts $S_0 - K e^{-rT} = 45 - 45 e^{-0.04 \times 0.5} = 45 - 44.109 = 0.891$ . The market price of the synthetic is within 4 cents of the parity benchmark, consistent with a small bid-ask spread. At expiry, if BHP closes at A$52, the call delivers a $7 gain and the put expires worthless, net payoff $7 - 0.85 = 6.15$. If BHP closes at A$40, the put is assigned with a $5 loss and the call expires worthless, net payoff $-5 - 0.85 = -5.85$ . The profit pattern matches owning the stock funded by a 6-month loan of $K e^{-rT}$ .

Scenario

Capital efficiency. An institutional desk wants long exposure to S&P/ASX 200 but the futures market is closed for an overnight session, and stock borrow is required for shorting.

Solution

The desk constructs a synthetic long using SPI 200 ETOs. Buying the at-the-money call and selling the at-the-money put gives equivalent index exposure. Initial capital is only the net premium plus margin, far less than the cash required to buy the basket of 200 stocks. The trade is unwound the next morning by closing both option legs, leaving a clean directional P&L tied to the overnight move.

Common mistakes

✗A synthetic stock is identical to holding the stock. The synthetic does not receive dividends, carries no voting rights, and may have different margin treatment. After-tax outcomes can differ because dividends and option premiums are taxed differently.
✗Synthetic stock is automatically cheaper than buying the actual share. By parity the cost is equivalent in a frictionless market. Any apparent saving is offset by the deferred payment, opportunity cost of margin, and risk-free interest.
✗Synthetic short stock is just an unhedged short option position. A synthetic short is short call + long put, a combined position with linear $-S_T + K$ payoff. It is fundamentally different from a naked short call, which has limited gain and unlimited loss.

Revision bullets

•Long call + short put at same $K$ , $T$ replicates stock
•Combined payoff equals $S_T - K$ , linear in spot
•Cost equals $S_0 - K e^{-rT}$ by parity
•Useful for capital efficiency and stock-borrow workarounds
•No dividends or voting rights in synthetic
•Mirror trade gives synthetic short stock

Quick check

A synthetic long stock position is constructed by:

An investor builds a synthetic long position in ANZ shares by buying a 50-strike call and selling a 50-strike put. Compared with simply buying ANZ shares, the investor does NOT receive:

Connected topics

Long Call Short Put PCP

In learning paths

Options Path

Sources

Hull (2022), §11.4
Hull, John C. Options, Futures, and Other Derivatives. 11th ed. Pearson, 2022. ISBN 978-0-13-693997-9.
Derives put-call parity and shows how synthetic stock and bond positions follow directly from the relationship.
Stoll (1969), Put-Call Parity
Stoll, Hans R. The Relationship Between Put and Call Option Prices. Journal of Finance, Vol. 24, No. 5, December 1969, pp. 801-824.
Seminal paper that established put-call parity as the equality between call-bond and put-stock portfolios.
Merton (1973), Theory of Rational Option Pricing
Merton, Robert C. Theory of Rational Option Pricing. Bell Journal of Economics and Management Science, Vol. 4, No. 1, Spring 1973, pp. 141-183.
Provides the rigorous no-arbitrage foundation for replicating any payoff using options and the underlying.
OIC, Synthetic positions
Options Industry Council. Synthetic Long Stock and Synthetic Short Stock. OIC Strategy Education.
Industry reference for constructing and managing synthetic stock positions in listed equity options markets.

How to cite this page

Dr. Phil's Quant Lab. (2026). Synthetic Stock Position. Derivatives Atlas. https://phucnguyenvan.com/concept/synthetic-stock

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