Proving the Binomial No-Arbitrage Condition Between Risky and Safe Returns
Summary
The document works through the one-period condition that the risk-free return must lie strictly between the risky security’s down-state and up-state returns to avoid arbitrage. The questioner first considers shorting the risky asset and investing the proceeds, but notices that this trade is not guaranteed to profit when the risk-free return is below the down-state return. The accepted answer completes the argument by choosing the trade direction according to which return dominates in every state.
If even the down-state stock return exceeds the safe return, borrowing and buying the stock produces a positive payoff in either outcome. If the safe return exceeds even the up-state stock return, shorting the stock and investing the proceeds yields a positive payoff in both outcomes. Thus, returns outside the interval permit a zero-net-investment opportunity with nonnegative state payoffs and a positive gain. This argument assumes the stated two-outcome model and frictionless borrowing, lending, and short selling; it does not discuss transaction costs or market constraints.
Key ideas
- The no-arbitrage condition places the risk-free return strictly between the risky asset’s down and up returns.
- When the down-state return exceeds the safe return, borrow and buy the risky asset.
- When the safe return exceeds the up-state return, short the risky asset and invest the proceeds.
- Each dominating-return trade begins with no net investment and profits across the modeled states.
- The proof assumes borrowing, lending, and short selling are available without frictions.
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Full text
# arbitrage proof question # arbitrage proof question prove the condition $D<R<U$ is equivalent to the absence of arbitrage: R = risk free investment rate of return. U and D are returns corresponding to the upward/downward price movements of a risky security. The basic idea of arbitrage is to buy low, sell high with no risk or initial investment So Assume R >= U or R<=D (contrapositive) At time 0 1) We short the security with price S (get +S) 2) We invest the security into the risk free investment (-S) Balance at time 0 = 0 At time 1 1) We get our risk free investment back (+S(1+R)) 2) We buy back one share of the stock. There are two scenarios a) Stock price goes up: (-S(1+U)) Profit = S(1+R)-S(1+U) >= 0 because R>=U (profit is always non negative, there is arbitrage) b) Stock price goes down: (-S(1+D)) Profit = S(1+R)-S(1+D) <= 0 because R<=D (from assumption), but this shows profit can be negative here-there is no sure risk free profit arbitrage! so i'm stuck here. ## Answer by Quantuple (score 1, accepted) https://quant.stackexchange.com/a/25636 You're half way there. When $ R < D \ (< U) $, the return of the stock dominates the risk-free return in all states of the world. To benefit from that, just borrow cash and invest in the stock. At $t=0$ this requires no net investment: borrowing cash means your account is credited $S_0$, while subsequently buying the stock suggests it is debited $S_0$. At the end of a period you owe the bank interests on the cash which has been lent to you $S_0 (1+R) $ but being long stock you have a position now worth $S_0 (1+D) $ or $S_0(1+U) $ if you were to sell which is always more than the interests owed, hence positive profit in all states of the world, hence free lunch. When $ (D <) \ U < R $ the risk-free return dominates the stock return in all states of the world. To benefit from that, just short sell the stock and invest in the risk-free money market account. At $t=0$ this requires no net investment (just deposit the proceeds of the short sell on the risk-free account). At the end of the period you receive interests $S_0 (1+R) $ while being short stock costs you $S_0 (1+D) $ or $S_0(1+U) $ (since you sold without possessing the stock before hand, you need to buy to it before you can give it to your counterpaty) which is always less that what you earned, hence positive profit in all states of the world, hence free lunch.
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