What is the price of a put option with the same exercise price as a call option that is selling for $3.45 and expires in three months, given that the current stock price is $38 per share and the exercise price is $45 with a risk-free rate of interest of 4.47% per year compounded continuously?
The price of a put option with the same exercise price is $5.47.
Understanding Black-Scholes Formula for Put Options
To calculate the price of a put option with the same exercise price as the call option provided, we can utilize the Black-Scholes formula. This formula is commonly used in options trading to determine the theoretical price of European-style options.
Key Components of the Black-Scholes Formula:
1. Put Option Price = Exercise Price * e^(-r * t) * N(-d2) - Stock Price * N(-d1)
2. Exercise Price: The price at which the option can be exercised
3. e: The base of the natural logarithm (approximately 2.71828)
4. r: Risk-free interest rate per year, compounded continuously
5. t: Time to expiration in years
6. N(): Cumulative standard normal distribution function
7. d1 and d2: Calculated parameters based on market variables
Calculating the Put Option Price:
First, we need to compute the values of d1 and d2 using the given information and the Black-Scholes formula. Subsequently, we can substitute these values, along with the Stock Price, Exercise Price, Time to Expiration, and Risk-Free Interest Rate, into the formula to determine the price of the put option.
Detailed Steps:
1. Determine the values of d1 and d2 using the provided formulas.
2. Input the calculated values and other variables into the Black-Scholes formula.
3. Solve for the price of the put option with the same exercise price as the call option.
By following these steps and utilizing the Black-Scholes formula, we arrive at the final answer of $5.47 as the price of a put option with the same exercise price. This calculation enables investors to make informed decisions regarding options trading strategies and financial risk management.