Answer to Question #285322 in Financial Math for Red

Question

A stock with a current price of $300, can go either up by 20%, or down by 15%. You have the opportunity to buy an American call option on this stock for $65, which matures at the end of one period, with an exercise price of 275.

a-      Please draw the trees of the stock and option values

b-      What is the price of the call option today if the risk free rate is 8%.

c-      What is your decision for this investment opportunity? Explain your answer

Answer 1

Solution:

Given:

Spot price, $S =300$

$\begin{aligned} u &=1.20 \\ d &=0.85 \end{aligned}$

American call option

Two period.

Exercise, x=275

Risk free rate, r=8%

Solution:

Risk neutral probability $=\frac{(1+r)-d}{u-d}$

$\begin{aligned} &=\frac{(1+8 \%)-0.85}{1.20-0.85}=0.6571 \\ &up\ move=65.71\% \\ &down\ move=34.29 \% \end{aligned}$

$\begin{aligned} \text { up move } &=62.22 \% \\ \text { down move } &=37.78 \% \end{aligned}$

(a) Binomial tree for american call:

$\begin{gathered} 120.36=\frac{(193.75 \times 62.22 \%)+(25 \times 37.78 \%)}{(1+8 \%)} \\ v_{S} \\ 375-275=100 \end{gathered}$

So, we select 120.36

(2)

$\begin{aligned} 14 . 40 &=\frac{(25 \times 62.22 \%)}{(1+8 \%)} \\ & \text { vs } \\ 240-275 &=\text { out of the money } \end{aligned}$

Se, we select 14.40.

(3)

$\begin{gathered} 74.38=\frac{(120.36 \times 62.22 \%)+(14.40 \times 37.78 \%)}{(1+8 \%)} \\ \text { VS } \\ 300-275=25 \end{gathered}$

So we select 74.38.

(b) Price of the call option =74.38

(c) Given American coll option price =$ 65

Calculated American calloption price = $74.38

As given American call option price and calculated American call option price are diffecent, there is a scope of profit from this investment.

Buy the American call option al lower of the above prices.