Replicating a Call Option with Stock and Forward Contracts

Summary

This question presents a one-period market with a stock and forward contracts, then asks how to replicate a call option using those traded instruments. The stock starts at 10, the forward price is 12, and the stock can finish at either 11 or 14; the call has a strike of 13. The author is unsure whether the usual stock-and-bond replication can be adapted by substituting a forward for the bond.

The example raises a central idea in option pricing: replication depends on matching the option’s payoffs across possible terminal states, while its arbitrage-free price follows from the cost of the replicating portfolio. No solution or replication calculation is provided, so the document does not establish whether the available contracts span the call payoff or determine its price. It also highlights that probabilities are not necessarily needed for replication-based pricing, though the market’s contract specifications and no-arbitrage assumptions would be needed to reach a definite conclusion.

Key ideas

• Option replication requires matching the option payoff in each possible terminal stock state.
• The question asks whether stock and forward contracts can span a call payoff in a one-period market.
• A replicating portfolio’s initial cost can determine an arbitrage-free option price when replication is possible.
• The document raises the role of probabilities but does not provide a solution or pricing calculation.

Tags

# Replicating call option in market which only trades stock and forward contracts


# Replicating call option in market which only trades stock and forward contracts












I am having a bit of trouble with a problem I've been given.

Consider a market which only trades a stock and forward contracts. There's only time 0 and 1.

Initial stock price S_0 is 10, the forward price F = 12, and at time 1, stock takes the price of either 11 or 14.

So I am asked to replicate a call option on the stock with strike price 13 with maturity at 1 and find the arbitrage free price of the call.

Some searching for similar problems told me to buy stock and sell short on a bond to replicate the call, but since a forward contract that can either benefit or incur a loss is in place of the bond, I am not so sure what to do. So I am replacing shorting a bond to shorting a forward contract? The problem doesn't give any probabilities on whether a stock would 11 or 14, so I am getting more and more confused.

I am still learning basics of finance, so any insights to understand the problem would be real helpful.

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