How do I construct a synthetic call from a put?

Question

AcadiFi · Accepted Answer

From parity C = P + S - PV(K), a synthetic long call = long put + long stock + short zero-coupon bond with face K. You're borrowing PV(K) today and promising to repay K at expiry.

Example: Zenith Utilities stock at $42, European put K=$40, T=6 months, r=3.5%, put price $1.25. Synthetic call = $1.25 + $42 - $39.305 (borrow PV of $40) = $3.945. If market call trades at $4.20, sell market call, buy synthetic, lock in $0.255.

Payoff check at expiry: if S_T=$50, put expires worthless (0), stock worth $50, owe $40 on bond, net = 0 + 50 - 40 = $10, matching call payoff max(50-40,0)=$10. If S_T=$35, put pays $5, stock worth $35, owe $40, net = 5 + 35 - 40 = 0, matching max(35-40,0)=0.

Practical applications: investors who already own stock and want upside while protecting downside sometimes buy a put to create a synthetic call position. Or when calls trade at a premium to theoretical value, arbitrageurs short the call and buy synthetic to capture the spread. Transaction costs, borrowing costs for shorting stock, and dividend assumptions can create apparent parity violations that aren't truly arbitrageable.

How do I construct a synthetic call from a put?

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