The document discusses the Black-Scholes option pricing model (BSOPM) and its assumptions. It outlines the derivation of the BSOPM, specifying the process the stock price follows and constructing a riskless portfolio to hedge the option. It then presents the BSOPM formula and defines the terms. The document also discusses the standard normal distribution curve used in the formula and provides an example calculation. Finally, it covers extensions such as dividends, the relationship between the BSOPM and binomial option pricing model, and estimating volatility.

1. Chapter 18
Continuous Time Option Pricing Models
• Assumptions of the Black-Scholes Option Pricing Model (BSOPM):
– No taxes
– No transactions costs
– Unrestricted short-selling of stock, with full use of short-sale proceeds
– Shares are infinitely divisible
– Constant riskless interest rate for borrowing/lending
– No dividends
– European options (or American calls on non-dividend paying stocks)
– Continuous trading
– The stock price evolves via a specific ‘process’ through time (more on
this later….)
©David

2. Derivation of the BSOPM
• Specify a ‘process’ that the stock price will follow (i.e., all
possible “paths”)
• Construct a riskless portfolio of:
– Long Call
– Short ∆ Shares
– (or, long ∆ Shares and write one call, or long 1 share and write
1/∆ calls)
• As delta changes (because time passes and/or S changes), one
must maintain this risk-free portfolio over time
• This is accomplished by purchasing or selling the appropriate
number of shares.
©David

3. The BSOPM Formula
C = SN( d1 ) − Ke −rTN( d2 )
where N(di) = the cumulative standard normal distribution
function, evaluated at di, and:
ln(S/K) + (r + σ 2 /2)T
d1 =
σ T
d2 = d1 − σ T
N(-di) = 1-N(di)
©David

4. The Standard Normal Curve
• The standard normal curve is a member of the family of normal curves
with μ = 0.0 and σ = 1.0. The X-axis on a standard normal curve is often
relabeled and called ‘Z’ scores.
• The area under the curve equals 1.0.
• The cumulative probability measures the area to the left of a value of Z.
E.g., N(0) = Prob(Z) < 0.0 = 0.50, because the normal distribution is
symmetric.
©David

5. The Standard Normal Curve
The area between Z-scores of -1.00 and +1.00 is 0.68 or 68%.
Note also that the area What is N(1)?
to the left of Z = -1 What is N(-1)?
equals the area to the
right of Z = 1. This
holds for any Z.
The area between Z-scores of -1.96 and +1.96 is 0.95 or 95%.
What is the area
in each tail? What
is N(1.96)? What
is N(-1.96)?
©David

6. Cumulative Normal Table
(Pages 560-561)
Cumulative Probability for the Standard Normal Distribution
2nd digit of Z
z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
-1.3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823
1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177
N(-d) = 1-N(d) N(-1.34) = 1-N(1.34)
©David

7. Example Calculation of the BSOPM Value
• S = $92
• K = $95
• T = 50 days (50/365 year = 0.137 year)
• r = 7% (per annum)
• σ = 35% (per annum)
What is the value of the call?
©David

8. Solving for the Call Price, I.
• Calculate the PV of the Strike Price:
Ke-rT = 95e(-0.07)(50/365) = (95)(0.9905) = $94.093.
• Calculate d1 and d2:
ln(S/K) + (r + σ 2 /2)T ln(92/95) + (0.07 + 0.1225/2)0 .137
d1 = =
σ T 0.35 0.137
ln(0.96842 ) + 0.01798 − 0.03209 + 0.01798
= = = − 0.1089
(0.35)(0.3701) 0.12955
d2 = d1 – σT.5 = -0.1089 – (0.35)(0.137).5 = -0.2385

9. Solving for N(d1) and N(d2)
• Choices:
– Standard Normal Probability Tables (pp. 560-561)
– Excel Function NORMSDIST
– N(-0.1089) = 0.4566; N(-0.2385) = 0.4058.
– There are several approximations; e.g. the one in fn 8 (pg.
549) is accurate to 0.01 if 0 < d < 2.20:
N(d) ~ 0.5 + (d)(4.4-d)/10
N(0.1089) ~ 0.5 + (0.1089)(4.4-0.1089)/10 = 0.5467
Thus, N(-0.1089) ~ 0.4533
©David

10. Solving for the Call Value
C = S N(d1) – Ke-rT N(d2)
= (92)(0.4566) – (94.0934)(0.4058)
= $3.8307.
Applying Put-Call Parity, the put price is:
P = C – S + Ke-rT = 3.8307 – 92 + 94.0934 = $5.92.
©David

11. Lognormal Distribution
• The BSOPM assumes that the stock price follows a “Geometric
Brownian Motion” (see
http://www.stat.umn.edu/~charlie/Stoch/brown.html for a depiction of
Brownian Motion).
• In turn, this implies that the distribution of the returns of the stock,
at any future date, will be “lognormally” distributed.
• Lognormal returns are realistic for two reasons:
– if returns are lognormally distributed, then the lowest possible return in
any period is -100%.
– lognormal returns distributions are "positively skewed," that is, skewed to
the right.
• Thus, a realistic depiction of a stock's returns distribution would have
a minimum return of -100% and a maximum return well beyond
100%. This is particularly important if T is long.
©David Dubofsky and 18-11
Thomas W. Miller, Jr.

12. The Lognormal Distribution
E ( ST ) = S0 e µT
2 2 µT σ 2T
var ( ST ) = S0 e (e − 1)
©David

13. BSOPM Properties and Questions
• What happens to the Black-Scholes call price when the call gets
deep-deep-deep in the money?
• How about the corresponding put price in the case above? [Can
you verify this using put-call parity?]
• Suppose σ gets very, very close to zero. What happens to the
call price? What happens to the put price?
Hint: Suppose the stock price will not change from time 0 to time T.
How much are you willing to pay for an out of the money option?
An in the money option?
©David

14. Volatility
• Volatility is the key to pricing options.
• Believing that an option is undervalued is tantamount to
believing that the volatility of the rate of return on the stock will
be less than what the market believes.
• The volatility is the standard deviation of the continuously
compounded rate of return of the stock, per year.
©David

15. Estimating Volatility from
Historical Data
1. Take observations S0, S1, . . . , Sn at intervals of t (fractional
years); e.g., τ = 1/52 if we are dealing with weeks; τ = 1/12 if we
are dealing with months.
2. Define the continuously compounded return as:
 S   S + div 
ri =ln i  ; on an ex-div day: ri = ln i
S   S 

 i− 
1  i−1 
3. Calculate the average rate of return
4. Calculate the standard deviation, σ , of the ri’s
5. Annualize the computed σ (see next slide).
©David

16. Annualizing Volatility
• Volatility is usually much greater when the market is open (i.e. the asset
is trading) than when it is closed.
• For this reason, when valuing options, time is usually measured in
“trading days” not calendar days.
• The “general convention” is to use 252 trading days per year. What is
most important is to be consistent.
σ annual = σ per trading day × 252
σ annual = σ weekly × 52
σ annual = σ monthly × 12
Note that σyr > σmo > σweek > σday
©David

17. Implied Volatility
• The implied volatility of an option is the volatility for which the
BSOPM value equals the market price.
• The is a one-to-one correspondence between prices and
implied volatilities.
• Traders and brokers often quote implied volatilities rather than
dollar prices.
• Note that the volatility is assumed to be the same across strikes,
but it often is not. In practice, there is a “volatility smile”, where
implied volatility is often “u-shaped” when plotted as a function
of the strike price.
©David

18. Using Observed Call (or Put) Option Prices
to Estimate Implied Volatility
• Take the observed option price as given.
• Plug C, S, K, r, T into the BSOPM (or other model).
• Solving σ (this is the tricky part) requires either an iterative
technique, or using one of several approximations.
• The Iterative Way:
– Plug S, K, r, T, and σ into the BSOPM.
ˆ
– Calculate σ ˆ (theoretical value)
– ˆ
Compare c (theoretical) to C (the actual call price).
– If they are really close, stop. If not, change the value of
and start over again.
©David

19. Volatility Smiles
In this table, the option premium column is the average bid-ask price for June 2000 S&P 500 Index call and put options at the
close of trading on May 16, 2000. The May 16, 2000 closing S&P 500 Index level was 1466.04, a riskless interest rate of 5.75%,
an estimated dividend yield of 1.5%, and T = 0.08493 year.
call price put price
Strike Call price Call IV with σ = 0.22 Put Price Put IV with σ = 0.22
1225 1.5625 0.336 0.054
1250 2.34375 0.3283 0.153
1275 2.625 0.3021 0.39
1300 3.8125 0.292 0.904
1325 5.75 0.2855 1.919
1350 129.5 0.2834 124.335 8.25 0.2762 3.753
1375 107.625 0.2689 102.51 11.375 0.2641 6.809
1400 86.625 0.2534 82.353 14.875 0.2463 11.533
1425 67.625 0.2423 64.288 20.0625 0.2315 18.35
1450 50.6875 0.2324 48.648 29 0.2285 27.592
1475 35.625 0.2201 35.612 38.625 0.2152 39.437
1500 22.5 0.2036 25.178 50.875 0.2016 53.885
1525 14 0.1982 17.173 66.75 0.1923 70.762
1550 7.875 0.1912 11.291 85.375 0.1826 89.761
1575 4.375 0.1896 7.154 106.875 0.1788 110.505
1600 2.28125 0.1881 4.367 129.5 0.1668 132.6
1625 1.21875 0.1897 2.569 153.375 0.1512 155.684
1650 0.625 0.1912 1.457
©David

20. Graphing Implied Volatility versus Strike (S = 1466):
Implied Volatility Volatility Smiles
0.4
0.3
Call IV
0.2
Put IV
0.1
0
1200 1400 1600
Strike Price
2
S S
IV = α + β1ln  + β 2 ln  + ε
X X
Using the call data:
α= 2.17; β1 = -4.47; β2 = 2.52; R2 = 0.988
©David

21. Dividends
• European options on dividend-paying stocks are valued by substituting the
stock price less the present value of dividends into Black-Scholes.
* -rT
C = S 0 N(d1 ) - Ke N(d 2 )
Where: S* = S – PV(divs) between today and T.
• Also use S* when computing d1 and d2.
• Only dividends with ex-dividend dates during life of option should be
included.
• The “dividend” should be the expected reduction in the stock price
expected.
• For continuous dividends, use S* = Se-dT, where d is the annual constant
dividend yield, and T is in years.
©David

22. The BOPM and the BSOPM, I.
• Note the analogous structures of the BOPM and the
BSOPM:
C = SΔ + B (0< Δ<1; B < 0)
C = SN(d1) – Ke-rTN(d2)
• Δ = N(d1)
• B = -Ke-rTN(d2)
©David

23. The BOPM and the BSOPM, II.
• The BOPM actually becomes the BSOPM as the number of
periods approaches ∞, and the length of each period
approaches 0.
• In addition, there is a relationship between u and d, and σ, so
that the stock will follow a Geometric Brownian Motion. If you
carve T years into n periods, then:
u = eσ T/n
−1
d = e −σ T/n − 1
and, the probability of an uptick, q, must equal
μ T
q = 0.5 + 0.5
σ n
©David

24. The BOPM and the BSOPM: An Example
• T = 7 months = 0.5833, and n = 7.
• Let μ = 14% and σ = 0.40
• Then choosing the following u, d, and q will make the stock follow
the “right” process for making the BOPM and BSOPM consistent:
u = e 0.40 0.0833
− 1 = 0.1224
d = e −0.40 0.833 − 1 = −0.10905
and, the probability of an uptick, q, must equal
0.14
q = 0.5 + 0.5 0.0833 = 0.5505
0.40
©David

25. Generalizing the BSOPM
• It is important to understand that many option pricing models are
related.
• For example, you will see that many exotic options (see chap. 20)
use parts of the BSOPM.
• The BSOPM itself, however, can be generalized to encapsulate
several important pricing models.
• All that is needed is to change things a bit by adding a new term,
denoted by b.
©David

26. The Generalized Model Can Price
European Options on:
1. Non-dividend paying stocks [Black and Scholes (1973)]
2. Options on stocks (or stock indices) that pay a continuous dividend
[Merton (1973)]
3. Currency options [Garman and Kohlhagen (1983)]
4. Options on futures [Black (1976)]
©David

27. The Generalized Formulae are:
Cgen = Se ( b-r ) T
N ( d1 ) - Ke -rTN ( d2 )
Pgen = Ke -rTN ( -d2 ) - Se( N ( -d1 )
b-r ) T
ln(S/K) + (b + σ 2 /2)T
d1 =
σ T
ln(S/K) + (b − σ 2 /2)T
d2 = = d1 − σ T
σ T
©David

28. By Altering the ‘b’ term in These Equations,
Four Option-Pricing Models Emerge.
By Setting: Yields this European Option Pricing Model:
b=r Stock option model, i.e., the BSOPM
b=r-d Stock option model with continuous
dividend yield, d (see section 18.5.2)
b = r – rf Currency option model where rf is the
foreign risk-free rate
b=0 Futures option model
©David

29. American Options
• It is computationally difficult to value an American
option (a call on a dividend paying stock, or any put).
• Methods:
– Pseudo-American Model (Sect. 18.10.1)
– BOPM (Sects. 17.3.3, 17.4)
– Numerical methods
– Approximations
• Of course, computer programs are most often used.
©David

30. Some Extra Slides on this Material
• Note: In some chapters, we try to include some extra slides in
an effort to allow for a deeper (or different) treatment of the
material in the chapter.
• If you have created some slides that you would like to share with
the community of educators that use our book, please send
them to us!
©David

31. The Stock Price Process, I.
• The percent change in the stock prices does not
depend on the price of the stock.
• Over a small interval, the size of the change in the
stock price is small (i.e., no ‘jumps’)
• Over a single period, there are only TWO possible
outcomes.
©David

32. The Stock Price Process, II.
• Consider a stock whose current price is S
• In a short period of time of length ∆t, the change in the stock price is
assumed to be normal with mean of µ S ∆t and standard deviation,
∀ µ is expected return and σ is volatility
σS Δt
©David

33. That is, the Black-Scholes-Merton model
assumes that the stock price, S, follows a
Geometric Brownian motion through time:
dS =µ σ
Sdt + Sdz
dS
=µ σ
dt + dz
S
 σ2 
dlnS =  μ −

dt + σdz
 2 
©David

34. The Discrete-Time Process
∆S
= µ∆t + σ∆z
S
∆S St +∆t − St S 
= = ln  t +∆t ÷
S St  St 
∆t = a unit of time
z = a normally distributed random variable
with a mean of zero, and a variance of t
∆z = is distributed normally, with E(∆z) = 0, and VAR(∆z)=∆t
©David

35. Example:
• Suppose ∆t is one day.
• A stock has an expected return of µ = 0.0005 per day.
– NB: (1.0005)365 – 1 = 0.20016, 20%
• The standard deviation of the stock's daily return distribution is
0.0261725
– NB: This is a variance of 0.000685
– Annualized Variance: (365)(0.000685) = 0.250025
– Annualized STDEV:
• (0.250025)0.5 = 0.500025, or 50%
• (0.0261725)(365)0.5 = 0.500025, or 50%
©David

36. Example, II.
• Then, the return generating process is such that
each day, the return consists of:
– a non-stochastic component, 0.0005 or 0.05%
– a random component consisting of:
• The stock's daily standard deviation times the realization
of ∆z,
∀ ∆z is drawn from a normal probability distribution with a
mean of zero and a variance of one.
• Then, we can create the following table:
©David

37. The First 10 Days of a Stochastic
Process Creating Stock Returns:
NON‑ STOCHASTIC STOCHASTIC S(T)=S(T‑ 1)[1.0+R]
TREND PRICE COMPONENT R = µ∆t + σ∆z
DAY ∆z S(T)=S(T‑ 1)[1.00050] (0.0261725)∆z ∆t = 1 DAY
0 1.000000 1.000000
1 ‑ 2.48007 1.000500 ‑ 0.064910 0.935590
2 ‑ 0.87537 1.001000 ‑ 0.022911
0.914623
3 ‑ 0.80587 1.001501 ‑ 0.021092 0.895789
4 ‑ 1.03927 1.002002 ‑ 0.027200 0.871871
5 0.10523 1.002503 0.002754 0.874709
6 0.66993 1.003004 0.017534 0.890483
7 ‑ 0.21137 1.003505 ‑ 0.005532 0.886002
8 2.19733 1.004007 0.057510 0.937398
9 ‑ 0.82807 1.004509 ‑ 0.021673 0.917551
10 0.58783 1.005011 0.015385 0.932126
©David

38. The Result (Multiplying by 100):
A Graph of a Stock Price Following a
Geometric Brownian Motion Process
110.000
Price (times 100)
100.000
90.000
80.000
1 6 11 16 21 26 31 36 41 46 51 56 61
Day
©David

39. Risk-Neutral Valuation
• The variable µ does not appear in the Black-Scholes equation.
• The equation is independent of all variables affected by risk preference.
• The solution to the differential equation is therefore the same in a risk-
free world as it is in the real world.
• This leads to the principle of risk-neutral valuation.
Applying Risk-Neutral Valuation:
1. Assume that the expected return from the stock price is the risk-free
rate.
2. Calculate the expected payoff from the option.
3. Discount at the risk-free rate.
©David

40. Two Simple and Accurate Approximations
for Estimating Implied Volatility
1. Brenner-Subramanyam formula. Suppose:
S = Ke − rT
then,
2π C
σ≈ ×
T S
©David

41. 2. Corrado and Miller Quadratic formula.
 S − Ke −rT 
C − ÷
8π  2 
If S > Ke −rT : σ≈ ×
T 3S − Ke −rT
 S − Ke −rT 
C − ÷
8π  2 
If S ≤ Ke −rT : σ≈ ×
T 3Ke −rT −S
©David

42. Let’s Have a Horse Race
• Data: S = 54, K = 55, C = 1.4375, 29 days to expiration (T=29/365), r =
3%. We can verify:
• The actual ISD is 30.06%.
• Brenner-Subramanyam estimated ISD is 23.67%.
• Corrado-Miller estimated ISD is 30.09%.
©David