The Price Of An American Call On A Non-dividend-paying Stock Is 4. The Stock Price Is 31, The Strike Price Is 30, And The Expiration Date Is In Three Months. The Risk-free Interest Rate Is 8% P.a. Continuously Compounded. The Lower Bound For The Price
The Price of an American Call Option: A Comprehensive Analysis
In the world of finance, options are a type of derivative that gives the holder the right, but not the obligation, to buy or sell an underlying asset at a predetermined price. American call options, in particular, are a popular type of option that allows the holder to buy an underlying stock at a specified strike price before or at the expiration date. In this article, we will delve into the world of American call options and explore the factors that affect their price.
The Given Parameters
Let's start by examining the given parameters of the problem:
- The price of the American call option is $4.
- The stock price is $31.
- The strike price is $30.
- The expiration date is in three months.
- The risk-free interest rate is 8% per annum, continuously compounded.
The Lower Bound for the Price
The problem asks us to find the lower bound for the price of the American call option. To do this, we need to understand the factors that affect the price of an American call option. The price of an American call option is determined by the following factors:
- The stock price
- The strike price
- The time to expiration
- The risk-free interest rate
- The volatility of the underlying stock
The Black-Scholes Model
The Black-Scholes model is a widely used mathematical model for pricing options. It takes into account the following factors:
- The stock price
- The strike price
- The time to expiration
- The risk-free interest rate
- The volatility of the underlying stock
The Black-Scholes model is based on the following assumptions:
- The stock price follows a geometric Brownian motion.
- The risk-free interest rate is constant.
- The volatility of the underlying stock is constant.
The Lower Bound for the Price of an American Call Option
To find the lower bound for the price of the American call option, we need to use the Black-Scholes model. The lower bound for the price of an American call option is given by the following formula:
C(S, t) ≥ e^(-r(T-t)) * [S * N(d1) - K * N(d2)]
where:
- C(S, t) is the price of the American call option at time t.
- S is the stock price.
- K is the strike price.
- r is the risk-free interest rate.
- T is the time to expiration.
- N(d1) and N(d2) are the cumulative distribution functions of the standard normal distribution.
Calculating the Lower Bound
Now that we have the formula for the lower bound, let's calculate it using the given parameters:
- S = $31
- K = $30
- r = 8% per annum, continuously compounded = 0.08
- T = 3 months = 0.25 years
- t = 0 (since we are calculating the lower bound at time 0)
First, we need to calculate the values of d1 and d2:
d1 = (ln(S/K) + (r + σ^2/2) * T) / (σ * sqrt(T)) d2 = d1 - σ * sqrt(T)
where σ is the volatility of the underlying stock. Since we don't know the value of σ, we will assume it to be 0.2 (this is a reasonable assumption for a stock with a moderate level of volatility).
Now, we can calculate the values of d1 and d2:
d1 = (ln(31/30) + (0.08 + 0.2^2/2) * 0.25) / (0.2 * sqrt(0.25)) = 0.129 d2 = 0.129 - 0.2 * sqrt(0.25) = -0.071
Next, we need to calculate the values of N(d1) and N(d2):
N(d1) = 0.559 N(d2) = 0.469
Now, we can calculate the lower bound for the price of the American call option:
C(S, t) ≥ e^(-r(T-t)) * [S * N(d1) - K * N(d2)] = e^(-0.08 * 0.25) * [31 * 0.559 - 30 * 0.469] = $3.43
In this article, we have explored the factors that affect the price of an American call option. We have used the Black-Scholes model to calculate the lower bound for the price of the American call option. The lower bound for the price of the American call option is $3.43. This means that the price of the American call option cannot be less than $3.43.
- Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81(3), 637-654.
- Merton, R. C. (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science, 4(1), 141-183.
- Hull, J. C. (2018). Options, futures, and other derivatives. Pearson Education.
- Wilmott, P. (2006). Paul Wilmott on quantitative finance. John Wiley & Sons.
The Price of an American Call Option: A Comprehensive Q&A
In our previous article, we explored the factors that affect the price of an American call option and calculated the lower bound for the price of the American call option using the Black-Scholes model. In this article, we will answer some of the most frequently asked questions about American call options.
Q: What is an American call option?
A: An American call option is a type of option that gives the holder the right, but not the obligation, to buy an underlying stock at a specified strike price before or at the expiration date.
Q: What are the key factors that affect the price of an American call option?
A: The key factors that affect the price of an American call option are:
- The stock price
- The strike price
- The time to expiration
- The risk-free interest rate
- The volatility of the underlying stock
Q: What is the Black-Scholes model?
A: The Black-Scholes model is a widely used mathematical model for pricing options. It takes into account the following factors:
- The stock price
- The strike price
- The time to expiration
- The risk-free interest rate
- The volatility of the underlying stock
Q: How does the Black-Scholes model calculate the price of an American call option?
A: The Black-Scholes model calculates the price of an American call option using the following formula:
C(S, t) ≥ e^(-r(T-t)) * [S * N(d1) - K * N(d2)]
where:
- C(S, t) is the price of the American call option at time t.
- S is the stock price.
- K is the strike price.
- r is the risk-free interest rate.
- T is the time to expiration.
- N(d1) and N(d2) are the cumulative distribution functions of the standard normal distribution.
Q: What is the lower bound for the price of an American call option?
A: The lower bound for the price of an American call option is given by the Black-Scholes model. The lower bound for the price of an American call option is $3.43.
Q: Can the price of an American call option be less than the lower bound?
A: No, the price of an American call option cannot be less than the lower bound.
Q: What is the difference between an American call option and a European call option?
A: The main difference between an American call option and a European call option is that an American call option can be exercised before or at the expiration date, while a European call option can only be exercised at the expiration date.
Q: What is the advantage of an American call option over a European call option?
A: The advantage of an American call option over a European call option is that it can be exercised before or at the expiration date, which can provide more flexibility to the holder.
Q: What is the disadvantage of an American call option over a European call option?
A: The disadvantage of an American call option over a European call option is that it can be more expensive to buy and sell.
In this article, we have answered some of the most frequently asked questions about American call options. We have also provided a comprehensive overview of the factors that affect the price of an American call option and the Black-Scholes model for pricing options.
- Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81(3), 637-654.
- Merton, R. C. (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science, 4(1), 141-183.
- Hull, J. C. (2018). Options, futures, and other derivatives. Pearson Education.
- Wilmott, P. (2006). Paul Wilmott on quantitative finance. John Wiley & Sons.