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**Unformatted text preview: **e option approaches maturity, or t 0.
()( )
√ , when t approaches 0, σ√ approaches 0, hence d1 = d2
d1 approaches infinity, hence N(d1) and N(d2) approaches 1 – i.e. the entire area under the unit normal
distribution
e-rt approaches 1 Hence on the expiration date, the BS equation of C = SN(d1) - Xe-rt N(d2) will become max {0, S*-X}. In other
words, the value of the call is composed entirely of its intrinsic value. 20 Cheryl Mew FINS2624 – Portfolio Management Semester 1, 2011 B . HEDGE RATIO
N(d1) is also known as the hedge ratio. It represents the number of shares bought per call written required to
form a perfectly hedged portfolio.
The BS equation of C = SN(d1) - Xe-rt N(d2) suggests another equilibrium position at which no profitable risk free
opportunity exists. However, if c > C, at any point in time, we can:
1.
2.
3. write a “actual” call to obtain $c
borrow $ Xe-rt N(d2) and use the money to
buy N(d1) units of share which costs $ SN(d1) This means that the stock investment is partially financed by borrowing. By doing the above 3, we can form a
perfectly hedged portfolio (a portfolio with no market risk exposure to the stock) since we have effectively
closed off the short call (1) by the theoretical long call (2 and 3).
However, N(d1) depends on S and hence the hedge ratio is unlikely to remain constant. To stay perfectly
hedged, we must rebalance the composition of the synthetic long call position as follows: If N(d1) has gone up in value, borrow more money to purchase more shares
If N(d1) has gone down in value, sell some shares and use the proceeds to reduce the amount of risk
free borrowing C . DELTA
The delta of an option is defined as the partial derivative of the value of the option with respect to the price of
the underlying stock. By deriving C with respect to S, we get N(d1). Hence N(d1) can also be described as the
change in the value of a call given a small change in the value of the underlying stock.
This outcome is particularly useful for verifying that a...

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