Question

Consider a European call option on a (non-dividend paying) stock currently worth £60. The option’s exercise price is £40, the continuously compounded interest rate is 2% and the option has 6 months to maturity. Which of the values below provides the tightest valid lower bound on the value of the option (rounded to the nearest penny)?

Answer 1

To find the lower bound on the value of the option, we need to calculate the value of the stock using the Black-Scholes formula.

The Black-Scholes formula for a European call option is:

C = S * N(d1) - X * e^(-r * T) * N(d2)

Where:
C = Value of the call option
S = Current price of the stock
N(x) = Cumulative standard normal distribution function
d1 = (ln(S/X) + (r + (σ^2)/2) * T) / (σ * sqrt(T))
d2 = d1 - σ * sqrt(T)
X = Exercise price of the option
r = Continuously compounded interest rate
T = Time to maturity of the option
σ = Volatility of the stock

In this case:
S = £60
X = £40
r = 2% = 0.02
T = 6 months = 0.5 years

To find the lower bound, we assume σ = 0, meaning the stock has no volatility. This gives us:

d1 = (ln(60/40) + (0.02 + (0^2)/2) * 0.5) / (0 * sqrt(0.5))
d2 = d1 - 0 * sqrt(0.5)

N(d1) can be approximated using a standard normal distribution table as 0.6915 and N(d2) can be approximated as 0.6915.

Plugging these values into the Black-Scholes formula, we get:

C = 60 * 0.6915 - 40 * e^(-0.02 * 0.5) * 0.6915

Calculating this, we get C ≈ £24.15

Therefore, the tightest valid lower bound on the value of the option is approximately £24.15.