I had this question closed before because someone said this doesnt have enough i

Question

I had this question closed before because someone said this doesnt have enough information... the question is taken straight out of the textbook "optimization methods in finance" by Gerard Cornuejols. When answering, could you add as much explanation and detail as possible for me to understand. Thank you very much and i appreciate your help :)

hint: first note that u=5/4 otherwise the first option is worthless. Then we must have 10= p(So*u-50) and 13 = p(So*u-40) From these equations determine p and then u and d.

Explanation / Answer

Given

Stock Price S0 = 40

Stock price at time T>0 would be u*S0 with probability 0<p<1 and down dS0 with probability 1-p

Current pricing of three options

Strike Price = $ 50    option price = $ 10

Strike Price = $ 40    option price = $ 13

Strike Price = $ 35    fair value = ?

In case of call option, the purchaser will exercise the option if the stock price at the time T> 0 is higher than the strike price for he will gain Stock Price (at time T) – Strike Price > 0, otherwise he will allow the option to expire.

In case the stock price at time T is equal to strike price or Stock Price (at time T) – Strike Price = 0, the holder of the call option will be indifferent assuming there will be no transaction costs. In case of the first option, this happens when

u.S0 - $ 50 = 0 --> u * $ 40 = $ 50 --> u = 5/4

based on the probabilities given above, the fair price could be calculated using the formula

Fair Value / Price of the option = probability of the event * (u * S0 – Strike price)

Where u is the factor by which the current market price will go up. Applying this to the two options given

p * (u*S0 – 50) = 10 ----Equation (1) and

p* (u* S0 – 40) = 13 --- Equation (2)

From Equation (1)

p*(u*40 – 50) = 10   -->   p = 10/(u*40 – 50)

substituting value of p in Equation (2)

10/(u*40 – 50) * (u.40 – 40) = 13 -->   (10/(u*40 – 50)) * u*40 - 10/(u*40 – 50)*40 = 13

10 (40u – 40) / (40u – 50) = 13 --> 10 (40u – 40) = 13 (40u – 50)

400 u – 400 = 520 u – 650 --> 520u – 400 u = 650 – 400 ---> 120 u = 250

u = 250/120 or u = 25/12

Substituting value of u in Equation (1)

P * (25/12 * 40 – 50) = 10

p*(1000/12 – 50) = 10 ---> p * ( 1000 – 50 * 12)/12   = 10

p* (1000-600)/12 = 10 --> p * 400/12 = 10   ---> p = 10 * 12 /400 = 120/400

p = 3/10

Thus the values derived from above are u = 25/12 and p = 3/10

Probability that the stock price will change to d.S0 in time T will be 1 – p = 1 – (3/10) = 7/10 or 0.7

The purchaser of a put option will exercise his option if the strike price is more than the stock price at time T such that strike price – stock price at time T > 0. He will allow the option to expire if the strike price is less than the stock price and will be indifferent when the strike price equals the stock price at time T, assuming absence of any transaction costs. That is

Strike Price – Stock Price at Time T = 0

Substituting the probability value and the expected stock price at time t which is d*S0

(1-p) * (Strike Price – dS0) = 0 -->   0.7 * (strike price – 40*d) = 0

0.7 * strike price – 0.7 * 40 d = 0    -->    (7/10) * strike price – 28 d = 0

28d = 0.7 * Strike Price   

Which gives the value of d = (0.7 * strike price)/28 or d = 0.025 * strike price

Using the values of u and p, fair value of the third option can be calculated as follows

p*(u*40 – 35) = Fair Value

Fair value = (3/10)*(25/12 * 40 – 35)

Fair Value = (3/10) * ((1000/12) – 35)

Fair Value = (3/10) * ((1000 – 35*12)/12)

Fair Value = (3/10) * (1000 -420)/12 =(3/10)*(580/12) = 58/4 = $ 14.50

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