Question

Call options with an exercise price of $125 and one year to expiration are available. The market price of the underlying stock is currently $120, but this market price is expected to either decrease to $110 or increase to $130 in a year's time. Assume the risk-free rate is 6%. What is the value of the option?

Answer 1

Current Market Price (So) = $ 120

Risk free Rate (i) = 6%

Expected Price in 1 Year

S(upward) = $ 130

S(downward) = $ 110

Exercise Price = $ 125

Fair Future Price = So * (1+i)^n

= $ 120*(1+0.06)

= $ 127.2

Let the Probability of attaining Upward price($ 130) at the time of Expiry = "P"

Then,

($ 130 * P) + ($ 110 * (1 - P)) = $ 127.2

(130-110) * P = 127.2 - 110

20 P = 17.2

P = 0.86

Therefore P(Downward) = 1- 0.86

P(Downward) = 0.14

Therefore, Price of Call Option =

= [(0.86 * ($ 130 - 125)) + (0.14 * 0)] / (1 + 0.1)

= $ 4.3

Alternatively, if Continous Compounding is used,

Current Market Price (So) = $ 120

Risk free Rate (i) = 6%

Expected Price in 1 Year

S(upward) = $ 130

S(downward) = $ 110

Exercise Price = $ 125

Fair Future Price = So * e^rt

= $ 120* e^0.06

= $ 120* 1.0618

= $ 127.42

Let the Probability of attaining Upward price($ 130) at the time of Expiry = "P"

Then,

($ 130 * P) + ($ 110 * (1 - P)) = $ 127.42

(130-110) * P = 127.42 - 110

20 P = 17.42

P = 0.87

Therefore P(Downward) = 1- 0.87

P(Downward) = 0.13

Therefore, Price of Call Option =

= [(0.87 * ($ 130 - 125)) + (0.13 * 0)] / e^0.1

= $ 4.355