Question

Consider a stock which is currently selling at $4.5. The stock price will either go up to $5 + x with probability 0.5 or go down to $5 − x with probability 0.5 one period later. The one-period riskless rate of interest is 5%.

a) What are the market prices of at-the-money put options that expire at the end of the period when x is set equal to $0.5, $1, $1.5, $2, and $2.5, respectively?
b) Plot the one-period put option prices against the five possible values of x. And what is the reason for the pattern that you plot

Answer 1

Ans a) (i) If x= $0.5 then upmove (u)= $5.5 and downmove(d) = $4.5 and Riskless rate (Rf= 5%), Current stock price= $4.5

After 1 year, Put option price=  [(0 x 0.5)+ (0x 0.5)]/1.05

= 0

(ii) If x= $1 then upmove (u)= $6 and downmove(d) = $4 and Riskless rate (Rf= 5%), Current stock price= $4.5

After 1 year, Put option price=  [(0 x 0.5)+ (0.5x 0.5)]/1.05

= 0.24

(iii) If x= $1.5 then upmove (u)= $6.5 and downmove(d) = $3.5 and Riskless rate (Rf= 5%), Current stock price= $4.5

After 1 year, Put option price=  [(0 x 0.5)+ (1x 0.5)]/1.05

= 0.48

(iii) If x= $2 then upmove (u)= $7 and downmove(d) = $3 and Riskless rate (Rf= 5%), Current stock price= $4.5

After 1 year, Put option price=  [(0 x 0.5)+ (1.5x 0.5)]/1.05

= 0.71

(iv) If x= $2.5 then upmove (u)= $7.5 and downmove(d) = $2.5 and Riskless rate (Rf= 5%), Current stock price= $4.5

After 1 year, Put option price=  [(0 x 0.5)+ (2.5x 0.5)]/1.05

= 1.19

Ans b) Since strike price is fixed, only payoff is changing due to change is value of X, so the pattern shows straight line as seen below graph