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Unformatted text preview: erefore, rho is
positive (remember, rho for a call option is positive, because a call entails paying the fixed strike
price to receive the stock and a higher interest rate reduces the present value of the strike). For
stock prices smaller than $ 100, the put dominates and we know that the rho of a put is negative. 94 Chapter 12 The BlackScholes Formula Tables for 12.18
Inputs
Stock Price
Exercise Price
Volatility
Riskfree interest rate
Dividend Yield 50
60
40.000%
6.000%
3.000% Inputs
Stock Price
Exercise Price
Volatility
Riskfree interest rate
Dividend Yield 50
60
40.000%
6.000%
4.000% Inputs
Stock Price
Exercise Price
Volatility
Riskfree interest rate
Dividend Yield 50
60
40.000%
7.000%
3.000% Inputs
Stock Price
Exercise Price
Volatility
Riskfree interest rate
Dividend Yield 50
60
50.000%
6.000%
3.000% Option Price
Exercise at: Perpetual Options
Call
Put
26.35183 23.07471
317.3092 22.6908 Option Price
Exercise at: Perpetual Options
Call
Put
22.75128
23.82482
248.2475
21.75248 Option Price
Exercise at: Perpetual Options
Call
Put
27.10008
21.2744
334.9193
25.08067 Option Price
Exercise at: Perpetual Options
Call
Put
29.83555
27.62938
412.5475
17.45254 Question 12.18
a)
The price of the perpetual call option is $ 26.35. It should be exercised when the stock
price reaches the barrier of $ 317.31.
b)
The price of the perpetual call option is now $ 22.75. It should be exercised when the
stock price reaches the barrier of $ 248.25. The higher dividend yield makes it more costly to
forego the dividends and wait for an increase in the stock price before exercising the option.
Therefore, the option is worth less and it is optimal to exercise after a smaller increase in the
underlying stock price.
c)
The price of the perpetual call option is now $ 27.10. It should be exercised when the
stock price reaches the barrier of $ 334.92. The higher interest rate increases the value of the call
option and makes it attractive to wait a bit longer before you exercise the option, as you can 95 Part 3 Options continue to earn interest on the strike before you exercise. Therefore, the option is worth more
and it is only optimal to exercise after a larger increase in the underlying stock price.
d)
The price of the perpetual call option is $ 29.84. It should be exercised when the stock
price reaches the barrier of $ 412.55. Options love volatility. The chances of an even larger
increase in the stock price are high with a large standard deviation (and your risk is capped at the
downside). Therefore, the option is worth more and you wait longer until you forego the future
potential and exercise.
Question 12.20
a)
b) C(100, 90, 0.3, 0.08, 1, 0.05) = 17.6988
P(90, 100, 0.3, 0.05, 1, 0.08) = 17.6988 c)
The prices are equal. This is a result of the mathematical equivalence of the pricing
formulas. To see this, we need some algebra. We start from equation (12.3) of the text, the
formula for the European put option: ln S + r − δ + 0.5σ 2 T ln S + r − δ − 0.5σ 2 T K
K P (• ) = K × exp − rT × N − − S × exp − δ T × N −
σT
σT ( ) ( ) ( ) ( ) Now we replace:
K = S , r = δ , δ = r, S = K Then: K ln + (δ − r − 0.5σ 2 )T
S
= S × exp (− δT ) × N − σT K
S
Since ln = − ln S
K S ln − (δ − r − 0.5σ 2 )T
K
= S × exp (− δT ) × N σT K ln + (δ − r + 0.5σ 2 )T − K × exp (− rT ) × N − S σT S ln − (δ − r + 0.5σ 2 )T − K × exp (− rT ) × N K σT = S × exp (− δT ) × N (d 1 ) − K × exp (− rT ) × N (d 2 ) = C (• )
96 Chapter 13
MarketMaking and DeltaHedging
Question 13.2.
Using the Black Scholes formula we can solve for the put premium and the put’s delta: P = 1.9905
and = −0.4176. If we write this option, we will have a position that moves with the stock price.
This implies our delta hedge will require shorting 41.76 shares (receiving $41.76 (40) = $1670.4).
As before, we must look at the three components of the proﬁt. There will now be interest earned
since we are receiving both the option premium 199.05 as well as the 1670.40 on the short sale. This
$2369.45 will earn (rounding to the nearest penny) 2369.45 e.08/365 − 1 = .52 in interest. If the
stock falls to 39 we make 41.76 on our short sale and if the stock price rises to 40.5 we lose 20.88 on
our short sale. If the stock prices falls to 39 or rises to 40.5 the price of the put option we wrote will
be (using T = 90/365) P (39) = 2.4331 or P (40.5) = 1.7808. This implies our option position
will lose 243.31 − 199.05 = 44.26 if the stock falls by $1 and make 199.05 − 178.08 = 20.97 if
the stock rises by $0.50. Combining these results, our proﬁt will be
41.76 − 44.26 + .52 = −1.98 (1) −20.88 + 20.97 + .52 = .61. (2) if the stock price falls to $39 and Notice that, as in the case of the call option, the large change implies a loss and the small change
involves a proﬁt.
Question 13.4.
The 45strike put has a premium of 5.0824 and a delta of −.7185 and the 40strike put has a premium
of 1.9905...
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 Spring '14
 NguyenThiMaiLan
 Derivatives

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