Bounding a Piecewise-Linear Payoff with Stock and Cash

Summary

The exercise asks for the no-arbitrage upper and lower bounds on a one-year claim linked to a non-dividend-paying stock. The payoff is zero below a lower threshold, rises linearly across a range of stock prices, and is capped above the upper threshold. With the current stock price specified and interest rates assumed to be zero, the stated bounds are zero and 100/6.

The answer explains the upper-bound construction: find a portfolio consisting of a quantity of stock and a cash amount whose value at expiry is at least the claim's payoff for every possible stock price. The current cost of any such portfolio is an upper bound, and minimizing that cost gives the tightest bound. The response identifies a candidate portfolio with a stock holding of one-sixth and no cash. The exchange only sketches the upper-bound argument and does not derive the lower bound or verify the candidate across all price regions, so those steps would need to be checked separately.

Key ideas

• An upper bound can be obtained from a stock-and-cash portfolio that dominates the payoff at every terminal stock price.
• The current cost of any dominating portfolio bounds the claim's price from above.
• The tightest upper bound comes from minimizing the cost over all such portfolios.
• The answer proposes a stock holding of one-sixth and zero cash for the stated payoff.

Tags

# Optimal Upper and Lower Bounds


# Optimal Upper and Lower Bounds












For the following exercise:

Give optimal upper and lower bounds on the price today for a product that pays a function of the spot price, $S$, of a non-dividend paying stock one year from now, there are no interest rates and the spot is $100$, when the pay-off is $0$ below $80$, increases linearly from $0$ at $80$ to $20$ at $120$ and then it is constant at $20$ above $120$

The answer is supposed to be $0$ and $100/6$

But I am not understanding how the upper bound is defined.

## Answer by Mark Joshi (score 1, accepted)

https://quant.stackexchange.com/a/15469

you have to find $\alpha$ and $\beta$ so that

$$ \alpha S_1 + \beta $$ is greater than or equal to the pay-off everywhere. Any such values gives an upper bound of $$ \alpha S_0 + \beta $$ Now try to find the smallest value of that. I am guessing that the answer is $\alpha = 1/6$ and $\beta=0.$

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