Using a Replicating Portfolio to Identify Call-Option Arbitrage
Summary
The example shows how to test whether a call option is mispriced using a two-state stock model. First, calculate the call’s payoff in each state and divide the payoff difference by the stock-price difference to find the option’s delta. That ratio gives the stock position needed to hedge one call, or equivalently the call position offset by one stock share.
After constructing the hedge, account for the initial cash flow and the value of the stock investment or borrowing at the risk-free rate. The portfolio’s terminal payoff should be the same in either state. Discount that payoff and compare its present value with the portfolio’s initial cost to determine whether the position offers arbitrage; reverse the trade if the pricing discrepancy has the opposite sign. The example outlines the method but leaves the final cash-flow calculation to the reader, and assumes the stated two outcomes and financing rate apply.
Key ideas
- The call’s delta is obtained from the change in option payoff divided by the change in stock price across states.
- A stock position can offset the call’s state-dependent payoff.
- Include the risk-free return on initial borrowing or investment when comparing cash flows.
- A hedged portfolio with a price inconsistent with its discounted payoff indicates an arbitrage opportunity.
Tags
Full text
# How to calculate riskless profit out of call options? # How to calculate riskless profit out of call options? I'm having trouble with working out a question that I can't currently ask my lecturer as they're away. Hoping for some help here with why the answer is (a). A stock price is currently \$40. It is known that at the end of two months it will be either \$36 or \$44. The riskfree interest rate is 6% p.a. with continuous compounding. If a two-month call option on the stock with strike \$42 is trading at \$1.20, how can you make a riskless profit? - a) Long 1 stock and short 4 call options - b) Short 1 stock and long 4 call options - c) Long 1 stock and short 1 call option - d) Short 1 stock and long 2 call options ## Answer by Magic is in the chain (score 0, accepted) https://quant.stackexchange.com/a/42101 The call in the up state has payoff of 2 and it has a payoff of zero in the down state. The stock is worth 44 and 36, in the two states, respectively. So the delta is 0.25=(2-0)/(44-36). So you can hedge one call with 0.25 stock units or 4 calls with one stock. You would then need to generate the cash flows of the portfolio, funding the stock purchase/sale at the risk free rate, to decide between a) and b). E.g., sell one unit of stock at 40 and invest the proceeds at risk free rate and buy 4 calls(1.2*4=4.8). This would have a net flow of -4.8 at time zero. Work out the cash flows of this portfolio in both up and down states (should be same in both states by definition as portfolio is hedged). Take the present value of this and compare to 4.8. If it is positive then that is arbitrage otherwise reverse the strategy.
Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.