You plan to value a call option on a non-dividend-paying stock. The current stock value is𝑆0 = 50, the strike price of the option is 𝐾 = 53, the time to maturity is 𝑇 = 1. Supposethe risk-free interest rate is 5%.1) (10 points) Suppose the implied volatility for 𝐾/𝑆0 = 1 is 3%, implied volatilityfor 𝐾/𝑆0 = 1.05 is 3.5%, and implied volatility for 𝐾/𝑆0 = 1.1 is 4%.Calculate the option value based on BSM model.
Question
You plan to value a call option on a non-dividend-paying stock. The current stock value is𝑆0 = 50, the strike price of the option is 𝐾 = 53, the time to maturity is 𝑇 = 1. Supposethe risk-free interest rate is 5%.1) (10 points) Suppose the implied volatility for 𝐾/𝑆0 = 1 is 3%, implied volatilityfor 𝐾/𝑆0 = 1.05 is 3.5%, and implied volatility for 𝐾/𝑆0 = 1.1 is 4%.Calculate the option value based on BSM model.
Solution
To calculate the option value based on the Black-Scholes-Merton (BSM) model, we first need to determine the implied volatility. Given that the strike price (K) to the current stock value (S0) ratio is 53/50 = 1.06, we can interpolate the implied volatility between 3.5% (for K/S0 = 1.05) and 4% (for K/S0 = 1.1).
Let's denote the implied volatility as σ. We can use linear interpolation to estimate σ:
σ = σ(K/S0 = 1.05) + [(σ(K/S0 = 1.1) - σ(K/S0 = 1.05)) / (1.1 - 1.05)] * (1.06 - 1.05)
Substituting the given values:
σ = 0.035 + [(0.04 - 0.035) / (1.1 - 1.05)] * (1.06 - 1.05) σ = 0.035 + [0.005 / 0.05] * 0.01 σ = 0.035 + 0.001 = 0.036 or 3.6%
Now we can use the BSM model to calculate the option value. The BSM model is given by:
C = S0 * N(d1) - K * e^(-rT) * N(d2)
where:
d1 = [ln(S0/K) + (r + σ^2 / 2) * T] / (σ * sqrt(T)) d2 = d1 - σ * sqrt(T)
N(.) is the cumulative distribution function of the standard normal distribution.
Let's calculate d1 and d2:
d1 = [ln(50/53) + (0.05 + 0.036^2 / 2) * 1] / (0.036 * sqrt(1)) = -0.0603 d2 = d1 - 0.036 * sqrt(1) = -0.0966
Now we can calculate the call option value:
C = 50 * N(-0.0603) - 53 * e^(-0.05*1) * N(-0.0966)
Using the standard normal distribution table or a calculator, we find that N(-0.0603) = 0.4761 and N(-0.0966) = 0.4616.
Substituting these values:
C = 50 * 0.4761 - 53 * e^(-0.05) * 0.4616 = 23.805 - 50.194 = -26.389
Since the value of a call option cannot be negative, the option is worthless and its value is 0.
Similar Questions
When the non-dividend paying stock price is $20, the strike price is $20, the risk-free rate is 6%, the volatility is 20% and the time to maturity is 3 months which of the following is the price of a European call option on the stock? A.19.7N(0.1) –20N(0.2)B.20N(0.1) –19.7N(0.2)C.20N(0.2) –19.7N(0.1)D.19.7N(0.2) –20N(0.1)
Consider an option on a non-dividend-paying stock when the stock price is $30, the exercise price is $29, the risk-free interest rate is 5% per annum, the volatility is 25% per annum, and the time to maturity is four months. (a) What is the price of the option if it is a European call? (1 mark) (b) What is the price of the option if it is an American call? (1 mark) (c) What is the price of the option if it is a European put? (1 mark) (d) Verify that put–call parity holds. (1 mark)
When ABC was trading at $52 per share, you paid $6.40 for a call option (for one share) on the stock of ABC with a strike price of $50, and six months until maturity. After six months, the share price of ABC is $54.10.What is the value of the call option at expiration? Do not include the $ sign and answer to the nearest $0.01.
Consider a world in which some stock, S, can either go up by 25% or down by 20% in one year and noother outcomes are possible. The continuously compounded risk-free interest, r, is 5.5% and the current priceof the stock, S0, is $100.1. What are the possible stock values in one year’s time, ST ?2. What are the possible payoffs of a European call option written on stock S with a strike price, X, of$100 and time-to-expiration of 1 year, T = 1 ?3. Suppose you want to form a portfolio, P , consisting of short on one call option and long on somenumber, ∆, of the stock, such that the portfolio value in one year’s time, PT , does not depend on thevalue of the stock, ST . What would be the appropriate value of ∆, also called the hedge ratio or delta?4. What would be the (certain) portfolio value in one year’s time, PT ?5. What is the arbitrage-free value of the portfolio today, P0 ?6. What is the premium of the call option today, c0, if there is no arbitrage opportunity?7. Define p = (erT − d) /(u − d), and call this the risk-neutral probability that the stock price increases.What is the value of p ?8. What is the expected value of the stock in one year’s time, E (ST ), under the risk-neutral probabilities?9. At what continuous rate would the stock price have to grow to end up at the expected value?10. What would be the expected value of the call option in one year’s time, E (cT ), under the risk-neutralprobabilities?11. At what continuous rate would the call price have to grow to end up at the expected value?
Derive the call option price based on BSM model.Suppose you have already proved that𝐸[max(𝑉 − 𝐾, 0)] = 𝐸(𝑉)𝑁(𝑑1) − 𝐾𝑁(𝑑2)for a lognormally distributed 𝑉, where𝑑1 = ln[𝐸(𝑉)/𝐾] + 𝑤2/2𝑤𝑑2 = ln[𝐸(𝑉)/𝐾] − 𝑤2/2𝑤and 𝑤 is the standard deviation of 𝑙𝑛𝑉.
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.