Reconciling a Delta-Hedged Call Example’s Portfolio Values
Summary
The document explains how to reconcile the portfolio value changes in a textbook example of a financial institution that has written a European call and dynamically bought shares to hedge it. The share position at the later date is found by summing the shares purchased across the table. Multiplying that total by the stock price gives the market value of the shares held, which explains the stated ending share value.
To obtain the net change, the answer combines three components: the gain in share value, the loss on the short option, and the debit in the cash position. Their sum accounts for the small overall change reported in the example. The response supplies the relevant arithmetic and figures from the question, but the table itself is not included in the document, so the cumulative share count and cash debit cannot be independently reconstructed from the provided text. This is an accounting explanation of one example, rather than a general assessment of hedging performance or transaction costs.
Key ideas
- Sum the cumulative share purchases to determine the ending hedge position.
- Multiply shares held by the stock price to calculate their market value.
- Combine the share gain, option loss, and cash debit to reconcile the net change.
- The explanation depends on table values that are not reproduced in the document.
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Full text
# How to understand this example from Hull's book? # How to understand this example from Hull's book? I just started reading Hull's book, and I got stuck in an example where a financial institution has sold for $300,000 a European call option on 100,000 shares of a non-dividend-paying stock. - Stock price is 49 - Strike price is 50 - $r$ is 0.05, - vol is 20%, - $T$ is 0.3846 The example says > Initially, the value of the written option is 240,000. In the situation depicted in Table 18.2, the value of the option can be calculated as 414,500 in Week 9. Thus, the financial institution has lost 174,500 on its short option position. Its cash position, as measured by the cumulative cost, is 1,442,900 worse in Week 9 than in Week 0. The value of the shares held has increased from 2,557,800 to 4,171,100. The net effect of all this is that the value of the financial institution’s position has changed by only 4,100 between Week 0 and Week 9. table 18.2 is as follow, My question is: where did the numbers 4,171,100 and 4,100 come from? ## Answer by Alex C (score 1, accepted) https://quant.stackexchange.com/a/27588 If you compute the cumulative sum of the 'Shares Purchased' column you will find that in Week 9 the company owns a total of 78,700 shares. Each share is worth 53.00 (see 'Stock Price' column), so the value of the shares held in Week 9 is 78700*53 = 4,171,100. The increase in share value is 4,171,100-2,557,800 = 1,613,300 The loss in the option position is -174,500 The debit in the cash position is -1,442,900 (=4,000,700-2,557,800 in the 6th column) Combining these three elements (one positive and two negatives) we have 4100 dollars.
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