A European call is currently selling with strike $32, expiry in 11 months and current underlying asset value $37. The yearly volatility is 33% and the continuously compounding interest rate per year is 6%. What is a rational price for the call?
Question
A European call is currently selling with strike 37. The yearly volatility is 33% and the continuously compounding interest rate per year is 6%. What is a rational price for the call?
Solution
To calculate the price of a European call option, we can use the Black-Scholes-Merton model. The formula for a call option is:
C = S0 * N(d1) - X * e^(-rt) * N(d2)
where:
- C is the price of the call option
- S0 is the current price of the underlying asset
- N is the cumulative standard normal distribution function
- d1 and d2 are variables calculated as follows:
- d1 = [ln(S0/X) + (r + σ^2 / 2) * t] / (σ * sqrt(t))
- d2 = d1 - σ * sqrt(t)
- X is the strike price
- r is the risk-free interest rate
- t is the time to expiration in years
- σ is the volatility of the underlying asset
Given the values:
- S0 = $37
- X = $32
- r = 6% or 0.06
- t = 11/12 years (since it's 11 months)
- σ = 33% or 0.33
We can calculate d1 and d2 as follows:
- d1 = [ln(37/32) + (0.06 + 0.33^2 / 2) * (11/12)] / (0.33 * sqrt(11/12))
- d2 = d1 - 0.33 * sqrt(11/12)
Then we can substitute these values into the formula for C.
Note: N(d1) and N(d2) are values you would look up in a standard normal distribution table, or calculate using a statistical software or calculator.
This will give you the price of the call option.
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