How to understand this example from Hull's book?

Question

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 1

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.