In evaluating options on a stock, a two-step stock price tree and a correspondin
ID: 2755314 • Letter: I
Question
In evaluating options on a stock, a two-step stock price tree and a corresponding put price tree have been constructed. Both trees are shown below. The time to expiration of the option is 0.5 years (each step, or t, is 0.25 years) and the risk-free rate is 1%. Assume that all options in the problem are European.
What is the strike price of the put option
What are the values of Puu and Pdd
What is the risk-neutral probability? (Hint: you need to calculate u and d used in the formula from the stock price tree. d = 1/u)
What are the values of Pu, Pd and P0 respectively (Hint: you can calculate Pu and Pd first and use the values of Pu and Pd to get P0)
Price a call with a strike price of $50 using the stock tree (Hint: you can use the formula for two-step tree directly, meaning there is no need to get Cu or Cd. Just get C0 directly)
Stock Price Tree
Put Price Tree
t = 0
t = 0.25
t = 0.5
t = 0
t = 0.25
t = 0.5
64.20127
Puu
56.65742
Pu
50
50
P0
Pud = 5
44.12485
Pd
38.94004
Pdd
Stock Price Tree
Put Price Tree
t = 0
t = 0.25
t = 0.5
t = 0
t = 0.25
t = 0.5
64.20127
Puu
56.65742
Pu
50
50
P0
Pud = 5
44.12485
Pd
38.94004
Pdd
Explanation / Answer
Answer No. 1 Strike Price of put option
Pay off put option at (t = 0.5) (Pud)= 5
or,Strike Price -Expected Market Price = 5
or,Strike Price - 50 = 5
Striket Price = 50+5 = 55
Where
Expected Market Price = 50 , (See Stock Price tree)
Answer no. 2 Value of Puu and Pdd
Value of puu = Strike Price - Expected price of put option at (t=0.5 )= 55 - 64.20127 = NIL
Value of Pdd = Strike Price - Expected price of Put option at (t=0.5) = 55- 38.94004 = 16.05996
Note : Pay off can not be nagative.
Answer No 3. Calculation of risk neutral Probablity
u = Expected price at t =0.25 / Spot price = 56.65742/50 = 1.1331484
d = 1/u = 1/ 1.1331484 = 0.882497
a = e^0.01*0.25= 1.002503128
Probablity of High price = (a - d) / (u-d) = 0.120006128 / 0.2506514 = 0. 4788
Probablity of low price = 1- 0.4788 = 0.5212
Answer no . 4 . Value of Pu , Pd , and Po
Value of Pu = Expected Pay off of put option at t =0.5 / e^ 0.01*0.25
= (0* 0.4788 + 5 * 0.5212 ) / 1.0025250318= 2.5994
Value of Pd = Expected pay off of put option at t =0.5 / e^0.01*0.25
=( 5 * 0.4788 + 16.05996 * 0.5212) / 1.0025250318 = 10.7373
Value of Po =Expected pay off at time t=0.25
= (2.5994*0.4788+ 10.7373 * 0.5212)/ e^0.01*0.25 = 6.8236
Answer no .5 Value of Call option
Payoff of call option at time (t= 0.5) (Cuu) = 64.20127 -50 = 14.20127
Pay of of call option at time (t=0.5)(Cud) = 50-50 = nil
Pay off of call optio at time (t=0.5 )(Cdd) = 38.94004 -50 = nil
Value of call option at time (t=0.25) (Cu) = ( 0.4788*14.20127 + 0.5212*0) / 1.0025250318 = 6.7824
Value of call option at time ( t= 0.25) (Cd) =( 0*0.4788+ 0* 5212)/1.0025250318 =0
Value of call option at Co = (6.7824*0.4788+0*0.5212)/ 1.0025250318 = 3.239