Suppose the annually compounded risk-free rate is 5 percent for all maturities. A non-dividend-paying common stock is trading at $100. Suppose you are considering a European call option with a strike price of $110. What is the time to maturity of this option where the boundary condition begins to be nonzero?

To determine when the boundary condition of the European call option becomes nonzero, we need to find the time to maturity at which the option begins to have value, given that the risk-free rate $r = 5\%$, the current stock price $S_0 = 100$, and the strike price $K = 110$.

### Key Concepts

The boundary condition for a European call option becomes nonzero when the option starts to have intrinsic value. In other words, the price of the underlying asset $ S_t $ must be greater than the strike price $ K $ for the option to be in-the-money.

For a European call option, the payoff at maturity is:

\[
C(T) = \max(S_T - K, 0)
\]

So, the option will have a nonzero value when the stock price $ S_T $ is greater than the strike price $ K $.

### Approach

Using the risk-free rate and the assumption of no arbitrage, the stock price grows at the risk-free rate in a risk-neutral world. Therefore, the future stock price at time $ T $ can be expressed as:

\[
S_T = S_0 e^{rT}
\]

We want to find the time $ T $ at which the stock price reaches the strike price $ K = 110 $. Thus, we set the future stock price equal to the strike price:

\[
S_0 e^{rT} = K
\]

### Solving for $ T $

Substitute the given values:

\[
100 e^{0.05T} = 110
\]

Solve for $ T $:

\[
e^{0.05T} = \frac{110}{100} = 1.1
\]

Taking the natural logarithm of both sides:

\[
0.05T = \ln(1.1)
\]

\[
T = \frac{\ln(1.1)}{0.05}
\]

Now, let's calculate $ T $.The time to maturity $ T $ at which the boundary condition for the European call option begins to be nonzero is approximately 1.91 years.

Would you like more details, or do you have any questions?

Here are 5 related questions for further exploration:
1. How does the boundary condition change with different interest rates?
2. What happens if the strike price is lower than the current stock price?
3. How would the introduction of dividends affect the time to maturity?
4. What would the impact be if the call option were American instead of European?
5. How does volatility affect the option price over time?

**Tip:** Understanding the relationship between time, interest rates, and option payoff can help you in determining the appropriate time to exercise options optimally.

This problem explores the time to maturity of a European call option with a strike price of $110, where the boundary condition becomes nonzero. The solution involves key financial math concepts, such as exponential growth and logarithms, using the formula S_T = S_0 e^{rT}. Suitable for undergraduate students in finance and mathematics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation