Question

The stock’s price S is $100. After three months, it either goes up and gets multiplied by the factor U = 1.13847256, or it goes down and gets multiplied by the factor D = 0.88664332. — Options mature after T = 0.5 year and have a strike price of K = $105. — The continuously compounded risk-free interest rate r is 5 percent per year. — Today’s European call price is c and the put price is p. Call prices after one period are denoted by cU in the up node and cD in the down node. Call prices after two periods are denoted by cUD in the “up, and then down node” and so on. Put prices are similarly defined. Which set of arbitrage-free put prices (in dollars) is correct?

Group of answer choices

p = 2.00, pU = 0, and pD = 4.06

p = 15.03, pU = 4.06, and pD = 26.39

p = 8.41, pU = 0, and pD = 26.39

p = 8.41, pU = 2.00, and pD = 15.03

I need full explanation in detail not the answer

Answer 1

Solution:

Lets develop the put and the call two period binomial tree.

Given

Current stock price is $100  
Two three months period. So T for each period is 0.25  
u   1.13847256
d   0.88664332
Risk Free rate is 5% p.a. So r is 0.05  
Stike Price is $105. So K is $105  

So p value is computed using formula

P = e" - d u - d

p= =(EXP(0.05*0.25)- 0.88664332)/(1.13847256-0.88664332) =  0.50008

Option Value at each node is calculated using formula

f=e="[pfu + (1 - p)fa]

Using this the binomial tree for Call Option is as under:

CDEFGHIJ 12 14 <<- Stock Value « - Option Value 129.61 24.61 |<-=MAX(H14-105,0) 113.85 12.16 в к 100.94 20 21 22 <-=MAX(H20-105,0) 100 A 6.00 23 24 88.66 C 25 78.61 <-=MAX(H26-105,0)

Note: Stock value at Node B = Stock value at Node A * u; Stock value at Node C = Stock value at Node A * d .. similar for entire tree.

Call value are node B = EXP(-1*0.05*0.25)*(0.5*24.61+(1-0.5)*0).. similar for other nodes

Put Option binomial tree is as under:

C D E F G | 43 129.61 0 <-=MAX(105-H43,0) 44 46 113.85 2.00 48 100.94 4.06 <-=MAX(105-H49,0) 50 100 A 8.41 51 52 88.66 C 15.03 78.61 26.39 <-=MAX(105-H55,0) 56

Please check the notations: These two are probable answers:

p = 8.41, pU = 0, and pD = 26.39; p = 8.41, pU = 2.00, and pD = 15.03.

I think this is the answer as per your notations: p = 8.41, pU = 2.00, and pD = 15.03.

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