Question

Suppose that \(\$ 1,000\) is invested in European call options on a stock with current price \(S(0)=60\) dollars. The options expire after 6 months with strike price \(X=40\) dollars. Assume that \(\sigma=30 \%, r=8 \%\), and the expected logarithmic return on stock is \(12 \%\). Compute VaR after 6 months at \(95 \%\) confidence level. Find the final wealth if the stock price grows at the expected rate. Find the stock price level that will be exceeded with \(5 \%\) probability and compute the corresponding final payoff.

Short answer

Expected Stock Price: 63.71, Stock price exceeding with 5% probability: 37.30, Expected final wealth: 395.17, VaR Final Wealth: 0

Step-by-step solution

- Calculate the Expected Stock Price

Use the formula for expected stock price growth: \[ S(T) = S(0)e^{(\text{expected return} \times T)} \] Substitute the given values: \[ S(0) = 60, \text{ expected return} = 0.12, T = 0.5 \text{ years} \] \[ S(0.5) = 60 \times e^{(0.12 \times 0.5)} \] Calculate the exponent: \[ e^{0.06} \approx 1.06184 \] Therefore, the expected stock price: \[ S(0.5) = 60 \times 1.06184 \approx 63.71 \]

- Calculate the Stock Price Level Exceeding with 5% Probability

Use the formula for Value at Risk (VaR) with log-normal distribution: \[ \text{VaR} = S(0)e^{\big((\text{r} - \frac{1}{2}\text{σ}^2) \times T + \text{σ} \times \text{x}_{95\text{%}}\big)} \] Substitute the given values: \[ S(0)=60, r=0.08, \text{σ}=0.30, T=0.5 \text{ years} \] \[ \text{x}_{95\text{%}} \text{ is the z-score corresponding to 95% confidence level}\approx -1.645 \] \[ \text{VaR} = 60e^{\big((0.08-\frac{1}{2}0.3^2) \times 0.5 + 0.3\times(-1.645)\big)} \] Calculate the exponent: \[ 0.08-0.045=0.035 \] \[ (0.035 \times0.5) + 0.3 \times (-1.645) = 0.0175 - 0.4935 = -0.476 \] Therefore, \[ \text{VaR} = 60e^{-0.476} \approx 60 \times 0.6217 \approx 37.30 \].

- Calculate the Corresponding Final Payoff

The value of the call option: \[ \text{Call Option Payoff} = \text{max}(S(T) - X, 0) \] For expected stock price: \[ S(T) = 63.71, X = 40 \] So, \[ \text{Call Option Payoff} = \text{max}(63.71 - 40, 0) \approx 23.71 \] For the VaR case: \[ S(T) = 37.30, X = 40 \] So, \[ \text{Call Option Payoff} = \text{max}(37.30 - 40, 0) = 0 \]

- Calculate the Final Wealth

Determine final wealth by multiplying the payoff by initial investment divided by stock price: For expected stock price: \[ \text{Final Wealth} = \frac{1000}{60} \times 23.71 \approx 395.17 \] For the VaR case: \[ \text{Final Wealth} = \frac{1000}{60} \times 0 = 0 \]

Key concepts

European call options

A European call option is a financial contract that gives the owner the right, but not the obligation, to buy a specific stock at a predetermined price (known as the strike price) on a specific date. European options differ from American options because they can only be exercised at expiration, not before. In our exercise, the European call options have a strike price of \(40 and will expire in 6 months.

Investing in European options can be advantageous if you believe the stock price will rise above the strike price before expiration. In this example, with an initial stock price of \)60 and given parameters like volatility and interest rate, we can analyze potential profits and risks (e.g., calculating the payoff and final wealth).

Log-normal distribution

Understanding the log-normal distribution is crucial when dealing with stock prices. Stock prices are often modeled using a log-normal distribution because they can't be negative and tend to exhibit a compounding growth pattern.

This distribution is particularly useful when calculating Value at Risk (VaR), which measures the risk of loss on an investment. In our exercise, we assume that the stock price follows a log-normal distribution with volatility (σ) of 30% and use this in VaR calculations to find the underlying stock price level that will be exceeded with 5% probability.

Expected stock price

To calculate the expected stock price after a given period, we use the formula \ \ (S(T) = S(0)e^{(\text{expected return} \times T)}) where T is the time in years. For our exercise, the current stock price (S(0)) is \(60, the expected return is 12%, and the time period (T) is 0.5 years.

Using the formula, we multiply the current stock price by the exponent of the product of the expected return and time period: \ [S(0.5) = 60 \times e^{0.06 \times 0.5)} \ e^{0.06} \ This gives a final expected stock price of approximately \)63.71, indicating potential growth.

Final wealth calculation

Final wealth calculation involves determining the payoff from the options investment and then scaling it according to the initial investment. For our exercise, we find the payoff by subtracting the strike price from the expected stock price: \ [\text{Call Option Payoff} = \text{max}(S(T) - X, 0) \ where S(T) = 63.71 and X = 40, giving us a payoff of \(23.71] \ We then calculate final wealth using: \ [\text{Final Wealth} = \frac{1000}{60} \times 23.71 \ This results in a final wealth of approximately \)395.17, showing the profit gained from the investment given the expected stock price.