Constructing a Conditional Arbitrage in a Three-Period Price Model

Summary

The document asks how to state an arbitrage strategy in a three-period market model, where the discounted risky asset can take different values over time. The proposed strategy waits until the first period and buys only if the asset is priced at 8, then sells in the next period. In the described outcomes, the price either rises to 12, producing a gain, or stays at 8, producing no profit or loss. The alternative answer describes the same conditional purchase rule more briefly.

The reasoning illustrates how a trading decision contingent on observed prices can create an arbitrage when it avoids losses while retaining a positive chance of profit. The document does not include the full model tree or probability details, so it is not possible to independently verify all possible paths, initial costs, or admissibility assumptions. Its conclusion depends on the stated next-period outcomes after the price reaches 8.

Key ideas

• A conditional strategy can wait for a specified price before entering a position.
• Buying at 8 and selling in the next period yields a gain if the price reaches 12 and no gain or loss if it remains at 8.
• The proposed arbitrage depends on the model's possible price paths after the purchase condition is met.
• A full arbitrage check also requires the complete market model and trading assumptions.

Tags

# Find arbitrage opportunity in the given market model


# Find arbitrage opportunity in the given market model












Consider the following 3-period-market-model:

The discounted price of the risky asset $S$:

How can I find an arbitrage opportunity in this model?

I know that there would be no arbitrage if we replace the first $8$ by something in $(8,12)$ or if we replace the second $8$ by something in $(5,8)$ but I don't know how I can explicitly state the arbitrage opportunity in the given market. So I'm looking for a portfolio which is an arbitrage opportunity.

## Answer by AFK (score 1)

https://quant.stackexchange.com/a/46241

The arbitrage strategy is: if the stock is at 8 at $t=1$ buy it else do nothing. Then sell it at $t=2$.

Either the stock has increased to 12 and you made a profit or it is still worth $8$ and your PnL is 0.

So you are guaranteed not to lose any money but you have a non zero probability of making money (equal to the probability of the stock price increasing to 12 conditional on it being 8 at $t=1$).

## Answer by dm63 (score 0)

https://quant.stackexchange.com/a/46234

Isn't it as simple as: 1) wait one period 2) if the stock is at 8, buy it. If the stock is at 2 , do nothing. This is an arbitrage strategy, since there is a positive probability of a risk free profit.

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