solve ....c). Let S0 = 100,K = 120,σ = 0.3,r = 0.08 and δ = 0. Let VC(0,T) denote the Black-Scholes European call price for the maturity T. i). Does the limit of VC(0,T) as T → ∞ exist? If it does, what is it? ii).Now, set δ = 0.001 and let VC(0,T,δ) denote the Black-Scholes European call price for the maturity T. Again, how does VC(0,T,δ) behave as T → ∞? iii). Interpret in a sentence or two the differences, if there are any, between your answers to questions in ((i). and (ii).

To solve the problem using the Black-Scholes formula for European call options, let's analyze the two cases step by step.

### Given Parameters:
- \( S_0 = 100 \) (initial stock price)
- \( K = 120 \) (strike price)
- \( \sigma = 0.3 \) (volatility)
- \( r = 0.08 \) (risk-free interest rate)
- \( \delta = 0 \) or \( \delta = 0.001 \) (dividend yield)

### i) Behavior of \( V_C(0,T) \) as \( T \to \infty \) with \( \delta = 0 \)

Using the Black-Scholes formula for a European call option:
\[
V_C(0,T) = S_0 N(d_1) - K e^{-rT} N(d_2)
\]
where:
\[
d_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2)T}{\sigma \sqrt{T}}
\]
\[
d_2 = d_1 - \sigma \sqrt{T}
\]
As \( T \to \infty \):
- \( d_1 \) and \( d_2 \) approach \( +\infty \) if \( S_0 > K \) (the option is likely to be exercised).
- Therefore, \( N(d_1) \to 1 \) and \( N(d_2) \to 1 \).

Thus:
\[
\lim_{T \to \infty} V_C(0,T) = S_0 - 0 = S_0 = 100
\]

### ii) Behavior of \( V_C(0,T, \delta) \) as \( T \to \infty \) with \( \delta = 0.001 \)

Using the same formula, but now with \( \delta > 0 \):
\[
V_C(0,T,\delta) = S_0 e^{-\delta T} N(d_1) - K e^{-rT} N(d_2)
\]
As \( T \to \infty \):
- The term \( S_0 e^{-\delta T} \to 0 \).
- The term \( K e^{-rT} \to 0 \) as well.

Thus, the limit can be expressed as:
\[
\lim_{T \to \infty} V_C(0,T, \delta) = 0
\]

### iii) Interpretation of the Differences

The key difference between the two cases is that, when the dividend yield \( \delta \) is zero, the value of the call option converges to the initial stock price as maturity approaches infinity. However, when a positive dividend yield is introduced, the value of the call option converges to zero, reflecting the dilution of value due to dividends paid out to shareholders.

### Summary of Results:
1. **With \( \delta = 0 \)**: \( \lim_{T \to \infty} V_C(0,T) = 100 \).
2. **With \( \delta = 0.001 \)**: \( \lim_{T \to \infty} V_C(0,T,\delta) = 0 \).

Would you like more details on the Black-Scholes model or any specific calculations? Here are some questions for further exploration:

1. How does the Black-Scholes model handle options with different expiration periods?
2. What impact do changes in volatility have on option pricing?
3. How do dividends influence the pricing of American vs. European options?
4. What are the implications of using a risk-free interest rate in the model?
5. How can numerical methods be used to approximate option prices in more complex scenarios? 

**Tip**: Always consider how changes in market conditions affect the parameters in the Black-Scholes model for accurate pricing.

Explore the behavior of Black-Scholes European call prices as maturity approaches infinity, with specific parameters: S0 = 100, K = 120, σ = 0.3, r = 0.08, and dividend yields of δ = 0 and δ = 0.001. Learn the differences in limit behaviors and the impact of dividends on option pricing.

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