Q-29.

Following Hull, a riskless portfolio consists of long delta (d) shares + short one option.

If the stock moves up, value of the riskless portfolio = 13 \times \text{delta} - \3 $ loss on the written call option; and

If the stock moves down, value of the riskless portfolio = $7 \times \text{delta}$. Setting them equal (i.e., riskless payoff):

13 \times d - \3 = $7 \times d$, and$6d = 3$, so$d = 0.5$.

If delta (d) = 0.5, then value of portfolio today is:

\10 \times 0.5 - f = 5 - f = $3.5 \times e^{-1%}$, such that

f = 5 - \3.5 \times e^{-1%} = $1.53483$

Notationally,

• $u = \frac{13}{10} = 1.3$; $d = \frac{7}{10} = 0.7$

• $p = \frac{e^{r \times \Delta t} - d}{u - d} = 0.51675$

• f = e^{-rT} \times (0.51675 \times \3 + 0) = $1.53483 $