No-Arbitrage Bounds for a Hedged Option Portfolio
Summary
The document considers why a riskless portfolio formed from an option and a hedge in the underlying must earn the risk-free rate in a binomial pricing model. The answer frames the argument as a contradiction: assume an arbitrage exists, then show that neither buying nor selling the relevant portfolio can provide a riskless profit. In the model, the bond and stock are the primitive assets, and a dynamic allocation between them can hedge contingent claims such as options.
This response offers a high-level explanation of the no-arbitrage logic rather than walking through the algebra for each of the question’s option payoff cases. It does not spell out the hedge ratios, initial cash flows, or the conditions needed for the replication argument. Its scope is the binomial model and European options as posed in the question; it is a conceptual starting point, not a full proof or a discussion of transaction costs, funding constraints, or market frictions.
Key ideas
- In the binomial model, the bond and stock are the primitive assets.
- Dynamic allocations to those assets can hedge contingent claims.
- The no-arbitrage argument proceeds by ruling out profitable buying and selling strategies.
- The answer gives a conceptual outline rather than case-by-case hedge calculations.
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# Arbitrage when risk-free portfolio earns less than riskless portfolio # Arbitrage when risk-free portfolio earns less than riskless portfolio I'm currently reading Paul Wilmott's excellent book on option pricing. Near the beginning, he constructs a risk-free portfolio using an option, and a short on the underlying to hedge the risk. I'm specifically interested in European options. A no-arbitrage argument follows: - If this portfolio earns more than the risk free rate: borrow money at the risk-free rate, buy the portfolio, and make money off the arbitrage. I've scoured the internet, but couldn't find an explanation for the second argument, which I have a hard time grasping! By shorting the portfolio, we short an option, and "short a short", meaning we go long on the stock. So, when we short the portfolio, we might even have to spend additional money, if shorting the option didn't give enough money to buy the stock. This segment focuses on the binomial model, so I've tried separating this to 3 cases: - When in both the up and down state the option is worth more than 0. In this case, the arbitrage relies on buying the amount of stock that can be had by exercising the option. I have a hard time finding arguments to why in this case the option should be worth more than the stock at the period before expiration. - When in both the up and down state the option is worth 0. I understand this case, the option is worth 0 at the turn before expiration, and the hedging is a degenerate case (longing 0 stocks). - When in the up state the option is worth > 0, and in the down state the option is worth = 0. Like in case 1, I can't find a good argument. As you can see, I'm out of answers. I don't even understand why a risk-less portfolio must earn the risk-free rate. Anyone has a clue? ## Answer by Bob Jansen (score 2) https://quant.stackexchange.com/a/44111 Collecting some of the comments as it's getting too long. - The binomial model only assumes properties of two assets: the bond and the stock. In the model, it's possible to hedge contingent claims (i.e. options) using a dynamic allocation to the bond and the stock. - "where the return on a portfolio of an option + shorts on the underlying is less than the risk-free rate", in the model these don't exist. The proof works by contradiction. Suppose there is some arbitrage: either it's possible to earn money by buying or selling the portfolio. Now show that both are impossible. Hence, arbitrage is not possible.
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