Question

Find the limit of the Black-Scholes values of plain vanilla European call and put options as T → 0 and as T → ∞. You may assume q = 0.

Answer 1

European Call option value:

When T→ 0 and T→ ∞

The value of a European call option increases as the underlying asset’s price go up and decreases as the asset’s value goes down.

As T→ 0, the value of the call option will be equal to the expected value of the asset at expiry (at time T) less the strike price (X) i.e. c = E (S_T) - X

At any point in time the expected value of the option can't be less than X*e^-rt , where r is risk free rate, t = time.

The lower bond of a European call option will be c >= S₀ – X*e^-rt

The upper bound of a European call option will be c <= X*e^-rt

European Put option value:

When T→ 0 and T→ ∞

The value of a European put option decreases as the underlying asset’s price go up and increases as the asset’s value goes down.

As T→ 0, the value of the put option will be equal to the strike price less expected value of the asset at expiry (at time T) i.e. p = X-E (S_T)

The lower bond of a European put option will be p >= X*e^-rt –S₀

The upper bound of a European put option will be p <= X*e^-rt