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# L1.T4.4. Black-Scholes option model assumptions

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
AIM: List and describe the assumptions underlying the Black‐ Scholes‐ Merton option pricing model. Compute the value of a European option using the Black‐Scholes‐Merton model on a non‐dividend‐paying stock.

Questions:

4.1. Each of the following is an underlying assumption of the basic Black-Scholes option pricing model EXCEPT:
a. The stock price follows a geometric Brownian motion (GBM) which is a continuous process without jumps
b. The continuously compounded rate of return on the stock is normally distributed, such that the distribution of the future stock price is lognormal
c. The expected rate of return on the stock (mu) and volatility (sigma) are constant
d. The expected real-world (risky) rate of return on the stock is known and the value of the option is an increasing function of this rate of return

4.2. Each of the following is an underlying assumption of the Black-Scholes option pricing model EXCEPT:
a. There are no transactions costs or taxes. All securities are perfectly divisible.
b. There are no riskless arbitrage opportunities.
c. Short selling is NOT permitted
d. Security trading is continuous.

4.3 What is the Black-Scholes option price of a one-year European call option on a non-dividend-paying stock with a strike price of $40 when the current stock price is$40, the riskfree rate is 4%, and the volatility of the stock is 20% per annum? Note: please use the cumulative standard normal distribution table in your statistics text; but note the exam will either give you N(.) or the means to compute it.
a. $2.15 b.$3.97
c. $4.02 d.$5.28

4.4. A six-month European put option on a non-dividend-paying stock has a strike price of $100 when the current stock price is$100. The risk free rate is 4%. N(d1) = 0.57 and N(d2) = 0.48. What is the Black-Scholes price of the put option?
a. -$9.95 b.$6.96
c. $7.97 d.$8.33

4.5. A one-year European call option has a strike price of $10. The riskfree rate is 4% per annum. What is an estimate of the call price if the stock is$30; i.e., significantly in-the-money?
a. $18.80 b.$20.00
c. $20.39 d.$21.22

Answers:

Subscriber
4.1 a
4.2 c
4.3 b
4.4 b
4.5 c