CHAPTER 15
The Black-Scholes-Merton Model
Practice Questions
既然是什么意思Problem 15.1.
What does the Black –Scholes –Merton stock option pricing model assume about the probability distribution of the stock price in one year? What does it assume about the
probability distribution of the continuously compounded rate of return on the stock during the year?
The Black –Scholes –Merton option pricing model assumes that the probability distribution of the stock price in 1 year (or at any other future time) is lognormal. It assumes that the
continuously compounded rate of return on the stock during the year is normally distributed.
Problem 15.2.
The volatility of a stock price is 30% per annum. What is the standard deviation of the percentage price
change in one trading day?
The standard deviation of the percentage price change in time t ∆
is where σ is the volatility. In this problem 03
=.
and, assuming 252 trading days in one year, 12520004t ∆=/=. so that 00019=.=. or 1.9%.
Problem 15.3.
Explain the principle of risk-neutral valuation.
The price of an option or other derivative when expresd in terms of the price of the
underlying stock is independent of risk preferences. Options therefore have the same value in a risk-neutral world as they do in the real world. We may therefore assume that the world is risk neutral for the purpos of valuing options. This simplifies the analysis. In a risk-neutral world all curities have an expected return equal to risk-free interest rate. Also, in a
risk-neutral world, the appropriate discount rate to u for expected future cash flows is the risk-free interest rate.
Problem 15.4.
Calculate the price of a three-month European put option on a non-dividend-paying stock with a strike price of $50 when the current stock price is $50, the risk-free interest rate is 10% per annum, and the volatility is 30% per annum.
In this ca 050S =, 50K =, 01r
=., 03σ
=., 025T =., and
12102417000917
d d d ==.=-.=.
The European put price is
0102550(00917)50(02417)N e N -.⨯.-.--.
0102550046345004045237e -.⨯.=⨯.-⨯.=.
or $2.37.
Problem 15.5.
What difference does it make to your calculations in Problem 15.4 if a dividend of $1.50 is expected in two months?
In this ca we must subtract the prent value of the dividend from the stock price before using Black –Scholes-Merton. Hence the appropriate value of 0S is
01667010501504852S e -.⨯.=-.=.
As before 50K =, 01r =., 03σ=., and 025T =.. In this ca
12100414001086
d d d ==.=-.=-. Th
e European put price is
0102550(01086)4852(00414)N e N -.⨯..-.-.
010255005432485204835303e -.⨯.=⨯.-.⨯.=.
or $3.03.
Problem 15.6.
What is implied volatility? How can it be calculated?
The implied volatility is the volatility that makes the Black –Scholes-Merton price of an option equal to its market price. The implied volatility is calculated using an iterative
缺月挂疏桐procedure. A simple approach is the following. Suppo we have two volatilities one too high (i.e., giving an option price greater than the market price) and the other too low (i.e., giving an option price lower than the market price). By testing the volatility that is half way between the two, we get a new too-high volatility or a new too-low volatility. If we arch initially for two volatilities, one too high and the other too low we can u this procedure repeatedly to bict the range and converge on the correct implied volatility. Other more sophisticated approaches (e.g., involving the Newton-Raphson procedure) are ud in practice.
智利火玫瑰Problem 15.7.
A stock price is currently $40. Assume that the expected return from the stock is 15% and its volatility is 25%. What is the probability distribution for the rate of return (with continuous compounding) earned over a two-year period?
In this ca 015=.μ and 025=.σ. From equation (15.7) the probability distribution for the rate of return over a two-year period with continuous compounding is:
⎪⎪⎭
⎫ ⎝⎛-ϕ225.0,225.015.022
<,
)03125.0,11875.0(ϕ
The expected value of the return is 11.875% per annum and the standard deviation is 17.7% per annum.
Problem 15.8.
A stock price follows geometric Brownian motion with an expected return of 16% and a volatility of 35%. The current price is $38.
a) What is the probability that a European call option on the stock with an exerci price
of $40 and a maturity date in six months will be exercid?
b) What is the probability that a European put option on the stock with the same exerci
price and maturity will be exercid?
a) The required probability is the probability of the stock price being above $40 in six
months time. Suppo that the stock price in six months is T S
⎥⎦
⎤⎢⎣⎡⨯⎪⎪⎭⎫ ⎝⎛-+5.035.0,5.0235.016.038ln ~ln 22ϕT S
问题分析报告i.e.,
()
2247.0,687.3~ln ϕT S
Since ln 403689=., we require the probability of ln(S T )>3.689. This is 3689368711(0008)0247N N .-.⎛⎫-=-. ⎪.⎝⎭
Since N(0.008) = 0.5032, the required probability is 0.4968.
b) In this ca the required probability is the probability of the stock price being less than
$40 in six months time. It is
10496805032-.=.
Problem 15.9.
Using the notation in the chapter, prove that a 95% confidence interval for T S is between
22(2)196(2)19600and T T S e S e -/-.-/+.μσ
μσ
From equation (15.3):
⎥⎦
⎤⎢⎣⎡⎪⎪⎭⎫ ⎝⎛-+T T S S T 220,2ln ~ln σσμϕ射手座与双鱼座
95% confidence intervals for ln T S are therefore
2
0ln ()1962S T +--.σμand
公选
20ln ()1962
生姜治什么S T +-+.σμσ95% confidence intervals for T S are therefore
2200ln (2)196ln (2)196and S T S T e e +-/-.+-/+.μσ
μσi.e.
22(2)196(2)19600and T T S e S e -/-.-/+.μσμσ
Problem 15.10.
A portfolio manager announces that the average of the returns realized in each of the last 10 years is 20% per annum. In what respect is this statement misleading?
This problem relates to the material in Section 15.3. The statement is misleading in that a
certain sum of money, say $1000, when invested for 10 years in the fund would have realized a return (with annual compounding) of less than 20% per annum.
The average of the returns realized in each year is always greater than the return per annum (with annual compounding) realized over 10 years. The first is an arithmetic average of the returns in each year; the cond is a geometric average of the returns.
Problem 15.11.
Assume that a non-dividend-paying stock has an expected return of μ and a volatility of σ. An innovative financial institution has just announced that it will trade a derivative that pays off a dollar amount equal to ln S T at time T where T S denotes the values of the stock price at time T .
a) U risk-neutral valuation to calculate the price of the derivative at time t in term of
the stock price, S, at time t
b) Confirm that your price satisfies the differential equation (15.16)
a) At time t , the expected value of ln S T is from equation (15.3)
2ln (/2)()S T t μσ+--
In a risk-neutral world the expected value of ln S T is therefore
2ln (/2)()S r T t σ+--
Using risk-neutral valuation the value of the derivative at time t is
提高安全意识()2[ln (/2)()]r T t e S r T t σ--+--
b) If
()2[ln (/2)()]r T t f e S r T t σ--=+-- then
()
()2()2()
2()
22
[ln (/2)()]/2r T t r T t r T t r T t f re S r T t e r t
f e S S
f e S S σσ--------∂=+----∂∂=∂∂=-∂
The left-hand side of the Black-Scholes-Merton differential equation is
()222()2ln (/2)()(/2)/2ln (/2)()r T t r T t e r S r r T t r r e r S r r T t rf
σσσσ----⎡⎤+----+-⎣⎦⎡⎤=+--⎣⎦=
Hence the differential equation is satisfied.
Problem 15.12.
Consider a derivative that pays off n T S at time T where T S is the stock price at that time.
When the stock pays no dividends and its price follows geometric Brownian motion, it can be shown that its price at time t ()t T ≤ has the form
()n h t T S ,
where S is the stock price at time t and h is a function only of t and T .
(a) By substituting into the Black –Scholes –Merton partial differential equation derive an ordinary differential equation satisfied by ()h t T ,.
(b) What is the boundary condition for the differential equation for ()h t T ,?
(c) Show that
2[05(1)(1)]()
()n n r n T t h t T e
σ.-+--,= where r is the risk-free interest rate and σ is the stock price volatility.
If ()()n G S t h t T S ,=, then n t G t h S ∂/∂=, 1n G S hnS -∂/∂=, and 222(1)n G S hn n S -∂/∂=- where t h h t =∂/∂. Substituting into the Black –Scholes –Merton differential equation we obtain 21(1)2
t h rhn hn n rh σ++-=
The derivative is worth n S when t T =. The boundary condition for this differential equation is therefore ()1h T T ,=
The equation
2[05(1)(1)]()()n n r n T t h t T e σ.-+--,=
satisfies the boundary condition since it collaps to 1h = when t T =. It can also be shown that it sati
sfies the differential equation in (a). Alternatively we can solve the differential equation in (a) directly. The differential equation can be written 21(1)(1)2
t h r n n n h σ=---- The solution to this is
21ln [(1)(1)]()2
h r n n n T t σ=-+-- or
2[05(1)(1)]()()n n r n T t h t T e σ.-+--,=
Problem 15.13.
What is the price of a European call option on a non-dividend-paying stock when the stock price is $52, the strike price is $50, the risk-free interest rate is 12% per annum, the volatility is 30% per annum, and the time to maturity is three months?
In this ca 052S =, 50K =, 012r =., 030=.σ and 025T =..
