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Unformatted text preview: portfolio with N(d1) shares, 1 short call and Xe-rt N(d2)
borrowing is a perfectly hedged portfolio. Say if N(d1) = 0.5, then for every $1 increase in stock, there is 50c
increase in the short call. So if the stock goes up by $1, then having: N(d1) = 0.5 units of stock will result in a gain of $0.5
1 short call will result in a loss of -1*$0.5 = $0.5 Thus the loss in the call is offset completely by the gain in the stock and there is no change to the value of the
overall portfolio despite the change in the value of the stock.
In summary, delta reflects the price sensitivity of the call with respect to a change in price of the underlying
stock. The deeper the call is in the money, the larger the delta and the more the call is exposed to uncertain
stock price movements. 21 Cheryl Mew FINS2624 – Portfolio Management Semester 1, 2011 C OMMON FACTORS AFFECTING THE VALUES OF CALLS AND PUT OPTIONS
There are certain factors which affect the values of calls and put options: Current Stock Price (S) - There is a positive (negative) relationship between the premium of a call
(put) and the underlying stock price. The larger (smaller) the stock price, the larger the intrinsic value
of a call (put).
Exercise Price (X) - There is a negative (positive) relationship between the premium of a call (put) and
the exercise price. The larger (smaller) the exercise price, the smaller the intrinsic value of a call (put).
Risk Free Rate (r) –
o C and r are directly related - C = SN(d1) - Xe-rt N(d2) C + Xe-rt N(d2) = SN(d1), as this shows
that the higher the r, the less money we need to deposit to get $X in the future
o P and r are inversely related – P - Xe-rt N(-d2) = -SN(-d1), as this shows that the higher the r,
the less money that we get at the beginning and still have to return $X in the future
Time to maturity (t) and stock volatility (σ)
o Generally, the time value and stock volatility is directly related to the time value of the
o For the relationship between C and t, as explained above, the longer the time to maturity,
the less we need to set aside for the deposit and the more valuable is the call option.
o In the case of P and t, the longer the time to maturity, the less we can borrow now and less
valuable is the put option. Hence the relationship between P and t is mixed, positive in the
context of option time value, but negative in the context of the present value concept. 22...
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