Optionsintermediate

Synthetic Call

A synthetic long call is built by owning the stock and buying a put at the desired strike. The put floors the loss at $K - S_0 - P$ and the stock provides unlimited upside, exactly the payoff profile of a long call. Put-call parity confirms the equivalence, $S_0 + P = C + K e^{-rT}$ , so the synthetic call differs from a direct call only by a zero-coupon bond of face value $K$ . This construction is also known as a protective put and is widely used by investors who already hold a stock and want downside insurance.

Why it matters

Owning a stock exposes you to all upside and all downside. Adding a put with strike $K$ truncates the downside at $K$ , so your loss can never exceed the premium $P$ plus the gap between the entry price and $K$ . The resulting payoff slopes flat below $K$ and slopes upward one-for-one above $K$ , the hockey-stick of a long call. The stock supplies the upside, the put supplies the floor, and parity ties the package back to a vanilla call.

Formulas

Synthetic call via parity

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

Rearranged put-call parity. Owning stock + long put is equivalent in payoff to owning a call + bond paying $K$ at expiry.

Synthetic call payoff at $T$

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

Floor at

K

, full upside above

K

. Subtracting the strike gives the equivalent call payoff

\max(S_T - K, 0) + K

Worked examples

Scenario

An investor owns 100 CSL shares at A$280. She is worried about a 3-month downside but does not want to sell. She buys 1 CSL 3-month European put struck at A$275 for A$8.50.

Solution

Initial cost on top of the stock equals A$8.50 per share, total premium A$850. At expiry, if CSL closes at A$310, the put expires worthless and her shares are worth A$310, net A$310 - 8.50 = A$301.50 per share equivalent. If CSL closes at A$240, the put pays $275 - 240 = $35 per share and her shares are worth A$240, total $240 + 35 - 8.50 = A$266.50 per share, floor enforced at A$275 minus the A$8.50 premium. The payoff profile is identical to a long 275-strike call plus a riskless investment of $275 e^{-0.04 \times 0.25} = $272.27$, consistent with parity.

Scenario

Comparing the cost of a real call versus the synthetic call. CSL 3-month call struck at A$275 trades at A$13.27. Spot A$280, put A$8.50, $r = 4\%$ .

Solution

Parity check, $C + K e^{-rT} = 13.27 + 275 e^{-0.01} = 13.27 + 272.27 = 285.54$ . The synthetic equivalent, $S_0 + P = 280 + 8.50 = 288.50$ . The synthetic costs A$2.96 more, which is essentially the cost of carrying the stock instead of holding the bond. In a frictionless market the two are equivalent. The A$2.96 gap reflects dividends, financing, and bid-ask spreads, not a true arbitrage.

Common mistakes

✗A synthetic call is cheaper than buying a real call. By parity the two cost the same in a frictionless market. Any apparent price gap reflects the carry on the stock leg (dividends, financing) and transaction costs.
✗The protective put eliminates all downside risk. Loss is capped, but the put premium is paid upfront and is lost if the stock rises. The position underperforms a naked stock holding in flat or rising markets by exactly the premium.
✗Synthetic call and protective put are different strategies. They are the same package, viewed differently. Protective put framing emphasises hedging an existing stock position. Synthetic call framing emphasises constructing call-like exposure from stock and option components.

Revision bullets

•Long stock + long put at strike $K$ = synthetic long call
•Payoff at $T$ is $\max(S_T, K)$ , hockey-stick shape
•Same as a protective put, different framing
•By parity, equivalent to call + zero-coupon bond of face $K$
•Max loss equals $S_0 - K + P$ (or just $P$ if entered at $K$ )
•Premium $P$ underperforms naked stock in flat markets

Quick check

A synthetic long call is constructed by:

By put-call parity, a synthetic long call ( $S_0 + P$ ) differs from a direct long call ( $C$ ) by:

Connected topics

Long/Short Long 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 the equivalence between protective put and fiduciary call.
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.
Original parity paper that established the equivalence of the protective put and fiduciary call 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.
Establishes the no-arbitrage foundation for replicating any option payoff using stock, bond, and other options.
OIC, Protective Put strategy
Options Industry Council. Protective Put. OIC Strategy Education.
Industry educational reference for the protective put strategy, the most common synthetic call construction.

How to cite this page

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

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