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)?
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.
To find the tightest valid lower bound on the value of the European call option, we can use the Black-Scholes formula. The formula for the value of a European call option is:
C = S*e^(rt)*N(d1) - X*e^(-rt)*N(d2)
Where:
C = Value of the call option
S = Current stock price
X = Exercise price
r = continuously compounded interest rate
t = time to maturity
N = cumulative standard normal distribution function
d1 = (ln(S/X) + (r + 0.5σ^2)t) / (σ*sqrt(t))
d2 = d1 - σ*sqrt(t)
Using the given values:
S = £60
X = £40
r = 2% = 0.02
t = 6 months = 0.5
σ (volatility) is not given in the question, so we cannot calculate the exact value.
To find the tightest valid lower bound, we can assume the volatility is zero. In this case, the option would be completely worthless since there is no uncertainty or price movement.
Plug in the values into the formula with σ = 0:
C = £60*e^(0.02*0.5)*N(d1) - £40*e^(-0.02*0.5)*N(d2)
= £60*e^(0.01)*N(d1) - £40*e^(-0.01)*N(d2)
= £60*e^(0.01)*1 - £40*e^(-0.01)*0
= £60*e^(0.01)
≈ £60 * 1.0100
≈ £60.60
Therefore, the tightest valid lower bound on the value of the option is £60.60.