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$6`d = 3$, 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\%} = \$1.53`483$

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) = \$1.53`483$