A One-Period Hedge for a Stock Position with Interest on Short Proceeds

Summary

The document resolves a calculation involving a stock position hedged with a quantity of shares, with the hedge chosen to make the future value equal across an up and a down outcome. The equality leads to a hedge ratio of one half. It then computes present value by discounting the common future value, including interest earned on cash raised through the short position.

The key conceptual point is that a short position creates cash proceeds whose interest must be included in the portfolio payoff. The author reports that an alternative setup reaches the same result, following an explanation that clarified the negative position value. The example uses a single-period setup and stated numerical inputs; it does not explain the broader derivation, market assumptions, transaction costs, or how the calculation generalizes to other price moves or instruments.

Key ideas

• The hedge quantity is chosen by equating portfolio future values across the two stock outcomes.
• The stated setup produces a hedge ratio of one half.
• Cash received from shorting stock contributes interest to the payoff calculation.
• Present value is found by discounting the common future value.
• The example is limited to the specific one-period assumptions and inputs described.

Tags

# Error on Paul Wilmott Section 5.2?


# Error on Paul Wilmott Section 5.2?












I gave this a long and hard thought because Paul Wilmott is a respected quant and I don't want to criticize his book, but am I correct in concluding that this section contains lots of errors? These are my observations and the specific section is at the bottom. I'm not trying to bash him, I'm just genuinely interested to master quantitative finance but couldn't get past this section because I think the calculations are wrong.

## Answer by ensabahnur (score 1)

https://quant.stackexchange.com/a/39785

I finally found my error,thanks dm63 for the explanation. I had a hard time imagining the negative position value and that it implies that I also get the interest from getting cash for the short..I used a slightly different approach but got the same result.

FV when stock goes up = FV when stock goes down

1 -1Δ+100Δ(r)=1Δ+100Δ(r)

1 -1Δ=1Δ

Δ=.5

PV= FV/(1+r)

PV= (.5+50(.1/252))/(1+(.1/252))

PV= 0.5196

Thanks for all the help guys! Really appreciate it.

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