Question

1. Zepo, Inc. stock is currently $80. The stock price will move up by 15% or down by 13% each period. The value of a two-period call option with an exercise price of $62 if the Rf is 4% per period is closes to:
   1. $19.17
   2. $22.99
   3. $27.11

Answer 1

Current strock price = S = $80

We will denote upmove by a '+' superscript and a downmove by a '-' superscript. Since a stock can move up by 15% and down by 13% each period,

At the end of the 1st period, there are two possibilities (stock moves up or down):

S⁺ = $80 x (1.15) = $92

S^- = $80 x (0.87) = $69.60

At the end of the 2nd period, there are three possibilities (a. stock which moved up in 1st period moves up in 2nd too, b. stock which moved up in 1st period moves down in 2nd or stock which moved down in 1st period moves up in 2nd (both lead to the same outcome at the end of 2nd period), c. stock which moved down in 1st period moves down in 2nd too ):

a. S⁺⁺ = $80 x (1.15) x (1.15) = $105.80

b. S^+- = $80 x (1.15) x (0.87) = $80.04

or S^-+ = $80 x (0.87) x (1.15) = $80.04 (as mentioned above, both of these lead to same outcome of $80.04)

c. S^-- = $80 x (0.87) x (0.87) = $60.552

At the call-option expiry at the end of 2 periods, the exercise price is $62 and since it is a call option (right to buy at $62), it will be exercised only if the stock value is higher than $62 i.e. in scenarios a. and b.

So, payouts for each scenario are as below (either there will be some benefit if the stock value is higher than $62 or there will be no benefit i.e. $0 payout):

a. Maximum between $0 and ($105.80 - $62) = Payout⁺⁺ = $43.80

b. Maximum between $0 and ($80.04 - $62) = Payout^+- = Payout^-+ = $18.04 [for both routes of upmove followed by downmove and vice-versa as explained above]

c. Maximum between $0 and ($60.552 - $62) = Payout^-- = $0

Before proceeding further, we need to calculate the risk-neutral probabily of an upmove and downmove.

Probability of upmove, P_U = ( 1 + Rf - Down-move factor) / (Up-move factor - Down-move factor)

= (1 + 0.04 - 0.87) / (1.15 - 0.87) = 0.607

Probability of downmove, P_D = 1 - P_U = 1 - 0.607 = 0.393

Basis all the above values, we will first calculate the price of the option at the end of year 1:

Option⁺ = (P_U x Payout⁺⁺ + P_D x Payout^+-) / (1+Rf)

Option^- = (P_U x Payout^-+ + P_D x Payout^--) / (1+Rf)

So, Option⁺ = (0.607 x $43.80 + 0.393 x $18.04) / (1.04) = $32.38

Option^- = (0.607 x $18.04 + 0.393 x $0) / (1.04) = $10.53

Using the above, we calculate the Option price today as:

Option value= (P_U x Option⁺ + P_D x Option^-) / (1+Rf) = (0.607 x $32.38 + 0.393 x $10.53) / (1.04) = $22.9

Answer. The value of 2-period call option with an exercise price of $62 is closest to b. $22.99