Why Arbitrage Forces Riskless Assets to Share a Price
Summary
The document explains why a portfolio with a certain future payoff must earn the risk-free rate when markets exclude arbitrage. The argument is based on comparing two riskless assets that deliver the same amount at the same maturity but have different prices today.
An investor can buy the cheaper asset and sell the more expensive one. The trade produces a positive cash flow at inception, while the equal maturity payoffs cancel with certainty, creating an arbitrage. This simple replication argument underlies the binomial option pricing principle and also appears in riskless-portfolio derivations such as the Black–Scholes framework. The explanation assumes that the assets’ future payoffs are truly identical and that the long and short positions can be established without frictions or constraints; those market assumptions are not examined in the answer.
Key ideas
- Assets with identical certain payoffs at maturity should have the same price today in an arbitrage-free market.
- Buying the cheaper asset and shorting the more expensive one creates an immediate gain when future cash flows cancel.
- The no-arbitrage condition explains why a riskless portfolio earns the risk-free rate.
- The same reasoning supports replication arguments in option pricing models.
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Full text
# Why must a riskless portfolio earn the risk-free rate? # Why must a riskless portfolio earn the risk-free rate? In Options, Futures and Other Derivatives when Hull introduces the risk-neutral approach to pricing European options in the one-step binomial model, he claims that > Riskless portfolio must, in the absence of arbitrage opportunities, earn the risk-free rate of interest. By riskless portfolio, he means a portfolio with totally predictable payoff. He then uses this argument to give the correct current price of the option which makes arbitrage impossible. Now I understand why the risk-neutral price of the call option is the only arbitrage-free price. If the call were overpriced an arbitrageur would long a replicating portfolio (which easily exists in this model by some elementary linear algebra) and short the call and if it were underpriced he or she would do the opposite. In fact this is just the simplest version of the universal principle that the arbitrage-free value of any replicable contingent claim is just the discounted expectation of its payoff under the risk neutral probability measure. But I just don't understand Hull's principle I quoted above. There should be an obvious arbitrage opportunity if the return on a riskless portfolio doesn't match the risk-free rate, but since I have very poor finance intuition (I'm from math background), I can't construct one by myself. Forgive my ignorance but I still hope someone can help, thanks. ## Answer by LocalVolatility (score 8, accepted) https://quant.stackexchange.com/a/32782 If you imagine you have two risk-less assets that have a unit payoff at maturity $V_1(T) = V_2(T) = 1$ but their present value is not equal, e.g. $V_1(t) < V_2(t)$. You buy the cheaper, sell the more expensive, have a strictly positive cash-flow today and at maturity the cash-flows cancel out with certainty. This is a free lunch arbitrage. The same argument is used in the Black-Scholes PDE derivation where you construct a locally risk-free portfolio.
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