No-Arbitrage Interest-Rate Bounds in a Two-State Stock Model

Summary

The document considers a one-period market with a risk-free account earning rate r and a stock priced at 1 initially, with possible terminal prices of 2 and 1/2. It asks which rates avoid arbitrage and whether adding a terminal stock price of 1 changes the answer. The proposed approach combines a position in the risk-free account with an offsetting stock position, then checks the portfolio payoff in each state.

The text concludes that rates between -0.5 and 1 avoid arbitrage and says an additional state with stock price 1 does not alter the conclusion. However, the displayed payoff expressions and signs are inconsistent with the described positions, so the derivation does not reliably establish those bounds. The intended lesson is to test whether a zero-cost portfolio has nonnegative payoffs in every state and a positive payoff in at least one; the specific conclusion should be independently checked using consistent trading conventions.

Key ideas

• A one-period market is tested for arbitrage by examining zero-cost portfolios across all possible terminal states.
• The example uses a risk-free account and a stock with two possible terminal prices.
• Adding another possible stock price requires checking the portfolio payoff in that state as well.
• The stated rate bounds are not supported by a consistent derivation in the document.

Tags

# No arbitrage opportunities and interest rate


# No arbitrage opportunities and interest rate












Consider a financial market with one single period, with interest rate r and one stock S. Suppose that $S_0 =1$ and, for n=1, $S_1$ can take two different values: 2, 1/2.

For which values of r the market is viable? viable means (free of arbitrage opportunities).

What if $S_1$ can also take the value 1.

Author's Solution

We want to calculate the values of r such that there is an arbitrage opportunity. We take a portfolio with zero initial value $V_0=0$ Then we invest the amount q in the stock without risk, we have to invest −q in the risky stock (q can be negative or positive). We calculate the value of this portfolio in time 2.

$V_1(\omega_1)=q(r-1)$.......(1)

$V_1(\omega_2)=q(r+1/2)$........(2)

So, if r > 1 there is an arbitrage oppportunity taking q positive (money in the bank account and short position in the risky stock) and if r < −1/2 we have an arbitrage opportunity with q positive (borrowing money and investing in the risky stock). The situation does not change if $S_1$ take the value 1.

My query: I want to know,Now if r<-1/2 how there is an arbitrage opportunity?Would any member explain me in his reply?

That means to have a viable market, $r$ must be between -0.5 To 1

## Answer by ram123 (score -2)

https://quant.stackexchange.com/a/17060

$$ \mu = u(\sigma^2) $$ diffentiating $$ diff(u(\sigma^2)^2) = 2u(\sigma^2)u'(\sigma^2) =2\mu(\sigma^2)u'(\sigma^2) $$

both answer are right

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.