Chapter 16 – Option Valuation
1. We start by finding the value of Pu . From this point, the put can fall to an expiration-date
value of Puu = $0 (since at this point the stock price is uuS0 = $121) or rise to a final value
of Pud = $5.50 (since at this point the stock price is udS0 = $104.50, which is less than the
Buy 3 puts at price Pu 16.50 0.00
Total $121.00 $121.00
The portfolio must have a current market value equal to the present value of $121:
Next we find the value of Pd . From this point (at which dS0 = $95), the put can fall to an
H=Pdu −Pdd
duS0−ddS0
=$5 . 50−$19 .75
$104 .50−$90 .25 =−1 . 0
Thus, the following portfolio will be worth $110 at option expiration regardless of the
ultimate stock price:
16-1
Chapter 16 – Option Valuation
Finally, we solve for P using the values of Pu and Pd . From its initial value, the put can rise
Chapter 16 – Option Valuation
P u － P d
c. The goal is a portfolio with the same exposure to the stock as the hypothetical
protective put portfolio. Since the put’s hedge ratio is – .5, we want to hold (1 –
Total 100 110
This payoff is identical to that of the protective put portfolio. Thus, the stock
plus bills strategy replicates both the cost and payoff of the protective put.
4. u = exp(
√
Δt
); d = exp(–
√
Δt
)
1
b. 4 subperiods, each 3 months
5. u = 1.5 = exp(
√
Δt
) = exp(
√
1
) 
√
Δt
1
6. Given S0 = X when the put and the call are at-the-money, the relationship of put-call
parity, P = C – S0 + PV(X) can be written as: P = C – S0 + PV(S0).
16-3
7. We first calculate the risk neutral probability that the stock price will increase:
p =
1 + r f － d
u － d
=
1 + .1 － .8
1.2 － .8
= .75
1 + r f
1.1
1 + r f － d
u － d
1 + .05 － . 95
1.1 － .95
1 + r f
$ 10.25
1 + r f
Chapter 16 – Option Valuation
a. i. The combined portfolio will suffer a loss. The written calls will expire in the
money. The protective put purchased will expire worthless. Each short call will
b. i. The delta of the call will approach 1.0 as the stock goes deep into the money,
Answer:
a. i. The option price will decline.
ii. The option price will increase.
16-5
Chapter 16 – Option Valuation
CFA 3
Answer:
udS0 = $60.00 × 0.80 = $48
duS0 = $40 × 1.20 = $48
ddS0 = $40 × 0.80 = $32
a. The value of a call option at expiration is: Max(0, S – X)
Cuu = Max (0, $72 – $60) = $12
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Chapter 16 – Option Valuation
b. The value of a call option at expiration is: Max(0, X – S)
16-7
whole or part.