1. Cost of a Call Option with Strike Price 0:

A call option gives the holder the right, but not the obligation, to buy the underlying asset at a specified strike price on or before a specified expiry date.

Intuitive Argument:

• Strike Price = 0: The strike price being 0 means that the option holder can buy the underlying asset for free at expiration. This is a very favorable situation, as the option holder essentially gains the asset without any cost.
• Therefore, the cost of such a call option would be driven by the current price of the underlying asset, since the holder has the potential to buy the asset at no cost. As the price of the underlying asset rises, the value of the call option would increase.

Black-Scholes Formula for a Call Option:

The Black-Scholes formula for the price $C$ of a European call option is given by:

$C = S_0 N(d_1) - K e^{-rT} N(d_2)$ where:

• $S_0$ = Current stock price
• $K$ = Strike price
• $r$ = Risk-free interest rate
• $T$ = Time to expiration
• $N(x)$ = Cumulative distribution function of the standard normal distribution
• $d_1 = \frac{\ln(S_0 / K) + (r + \frac{\sigma^2}{2})T}{\sigma \sqrt{T}}$
• $d_2 = d_1 - \sigma \sqrt{T}$

For a call option with strike price 0 ($K = 0$), the formula simplifies as follows:

$C = S_0 N(d_1)$ where $d_1$ becomes:

$d_1 = \frac{\ln(S_0 / 0) + (r + \frac{\sigma^2}{2})T}{\sigma \sqrt{T}} = \infty$ Since $\ln(S_0 / 0)$ goes to infinity, $d_1 \to \infty$, and thus:

$N(d_1) \to 1$

So, the price of the call option becomes:

$C = S_0$

Thus, the cost of the call option with a strike price of 0 is exactly the current price of the underlying asset. This matches the intuition that the holder can buy the asset for free, so the option's value is simply the price of the asset.

2. Effect of Expiry Date Increasing to ∞:

When the expiry date $T$ increases to infinity, the situation changes:

• As $T \to \infty$, the time value of the option also becomes very high because there is always a nonzero probability that the asset price will rise above the current price $S_0$.
• In this case, the call option behaves more like a long position in the underlying asset because the option holder has the potential to buy the asset at no cost at any point in the future. Thus, as the time to expiry increases, the option becomes almost identical to owning the asset outright.
• As time approaches infinity, the option value approaches the current price of the asset, as the possibility of a future favorable price movement becomes nearly certain.

Conclusion:

• The cost of a call option with a strike price of 0 is exactly the current price of the underlying asset ($C = S_0$).
• As the expiry date increases to infinity, the value of the call option approaches the value of the underlying asset itself because the holder effectively gets the asset for free at any time in the future.

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Would you like further details or clarifications on any part of this? Here are some related questions:

1. How does the value of a call option change as the strike price increases?
2. How would the Black-Scholes formula behave for an option with a very high strike price?
3. What impact does the volatility of the underlying asset have on the cost of a call option?
4. How would the option's price change if we had a dividend-paying stock instead of a non-dividend stock?
5. How does the time to maturity impact the pricing of options in the Black-Scholes model?

Tip: Always remember that the time value of an option is significantly influenced by the time to expiration and volatility. As time increases, the option's potential for favorable price movements grows, which increases its value.