Question

Helios' stock is currently selling for $82.00 a share but is expected to decrease to either $77.90 or increase to $90.20 a share over the next year. The risk-free rate is 3 percent. What is the current value of a 1-year call option with an exercise price of $82?

$4.25 |
||

$3.95 |
||

$3.65 |
||

$2.75 |
||

$0.00 |

Answer #1

Let S^{+} be the expected upward stock price in one
year, S^{-}, expected downward price and X be the exercise
price.

Call value in 1 year, if price increases, c^{+} =
Max(0,S^{+}-X) = Max(0,90.2-82) = $8.20

Call value in 1 year, if price goes down, c^{-} =
Max(0,S^{-}-X) = Max(0,77.90-82) = 0

Hedge ratio (h) =
(c^{+}-c^{-})/(S^{+}-S^{-})

= (8.20-0)/(90.2-77.9)

= 0.66667

According to no-arbitrage approach

c = hS + PV(-hS^{-}+c^{-})

Where, c is current value of call option

PV is present value factor

S is current stock price

c = 0.66667*82 + [1/(1+0.03)]*(-0.66667*77.90+0)

= 54.67-50.42

= 4.25

Therefore, the current value of 1 year call option is $4.25

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