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The Black-Scholes Model Chapter 13

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Title: The Black-Scholes Model Chapter 13


1
The Black-ScholesModelChapter 13
2
Pricing an European Call The BlackScholes
model Assumptions 1. European options. 2. The
underlying stock does not pay dividends during
the options life.
ct Max0, ST K
t T
Time line
3
Pricing an European Call The BlackScholes
model ct NPVExpected cash flows ct NPVE
(max0, ST K). ct e-r(T-t)maxE (0, ST
K).
4
Pricing an European Call The BlackScholes
model Ct e-r(T-t)0?ST?KPr(ST?K) E
(STK)?STgtKPr(STgtK) ct e-r(T-t)E
(STK)?STgtKProb(STgtK)
5
Pricing an European Call The BlackScholes
model ct e-r(T-t)E (ST)?STgtKProb(ST gt
K) e-r(T-t)KProb(ST gt K)
6
The Stock Price Assumption(p282)
  • In a short period of time of length ?t the
    return on the stock is normally distributed
  • Consider a stock whose price is S

7
The Lognormal Property
  • It follows from this assumption that
  • Since the logarithm of ST is normal, ST is
    lognormally distributed

8
The Lognormal Distribution

9
BlackScholes formula ct StN(d1)
Ke-r(T-t) N(d2) d1 ln(St/K) (r .5?2)(T
t)/?v(T t) d2 d1 - ?v(T t), N(d) is the
cumulative standard normal distribution. All the
parameters are annualized and continuous in time
10
pt Ke-r(T-t) N(- d2) StN(- d1). d1
ln(St/K) (r .5?2)(T t)/?v(T t) d2 d1 -
?v(T t), N(d) is the cumulative standard
normal distribution. All the parameters are
annualized and continuous in time
11
INTEL Thursday, September 21, 2000. S
61.48 CALLS - LAST PUTS - LAST K
OCT NOV JAN APR OCT NOV JAN APR 40
22 --- 23 --- --- --- 0.56
--- 50 12 --- --- --- 0.63
--- --- --- 55 8.13 --- 11.5
--- 1.25 --- 3.63 --- 60 4.75
--- 8.75 --- 2.88 4 5.75
--- 65 2.50 3.88 5.75 8.63 6.00 6.63
8.38 10 70 0.94 --- 3.88 ---
9.25 --- 11.25 --- 75 0.31 ---
--- 5.13 13.38 --- --- 16.79 80
--- --- 1.63 --- --- ---
--- --- 90 --- --- 0.81 ---
--- --- --- ---
12
  • INTEL Thursday, September 21, 2000. S 61.48
  • S 61.48 K 65 JAN T-t 121/365
    .3315yrs
  • R 4.82?r ln1.0482 .047. ? 48.28
  • d1ln(61.48/65) .047 .5(.48282).3315/(.4828
    )?(.3315)
  • -.0052
  • d2 d1 (.4828)?(.3315) -.2832
  • N(d1) .4979 N(d2) .3885
  • c 61.48(.4979) 65e-(.047)(.3315)(.3885)
    5.75
  • p 5.75 -61.48 65e-(.047)(.3315) 8.26

13
  • INTEL Thursday, September 21, 2000. S 61.48
  • S 61.48 K 70 JAN T-t 121/365
    .3315yrs
  • R 4.82?r ln1.0482 .047. ? 48.28.
  • d1ln(61.48/70) .047 .5(.48282).3315/(.4828
    )?(.3315)
  • -.2718.
  • d2 d1 (.4828)?(.3315) -.5498
  • Next, we follow the extrapolations suggested in
    the text
  • N(-.2718) ?
  • N(-.5498) ?

14
  • N(d1) N(-.2718) N(-.27) - .18N(-.27)
    N(-.28)
  • .3936 -.18.3936 -.3897 .392274
  • N(d2) N(-.5498) N(-.54) - .98N(-.54)
    N(-.55)
  • .2946 - .98.2946 - .2912 .291268
  • ct StN(d1) Ke-r(T-t)N(d2)
  • c 61.48(.392274) 65e-(.047)(.3315)(.292268)
    4.04
  • pt Ke-r(T-t)N(- d2) StN(- d1)
  • BUT
  • Employing the put-call parity for European
    options on a non dividend paying stock, we have
  • p 4.04 - 61.48 70e-(.047)(.3315)
    11.81

15
Black and Scholes prices satisfy the put-call
parity for European options on a non dividend
paying stock ct pt St - Ke-r(T-t)
. Substituting the BlackScholes values ct
StN(d1) Ke-r(T-t) N(d2) pt Ke-r(T-t) N(- d2)
StN(- d1). into the put-call parity yields
16
ct pt StN(d1) Ke-r(T-t) N(d2) -
Ke-r(T-t) N(- d2) StN(- d1) ct - pt
StN(d1) Ke-r(T-t) N(d2) - Ke-r(T-t) 1 -
N(d2) St1 - N(d1) ct - pt St Ke-r(T-t)
17
The Inputs
  • St The current stock price
  • K The strike price
  • T t The years remaining to expiration
  • r The annual, continuously compounded
    risk-free rate
  • ? The annual SD of the returns on the
    underlying asset

18
The Inputs
  • St The current stock price
  • Bid price?
  • Ask price?
  • Usually mid spread

19
The Inputs
  • T t The years remaining to expiration
  • Black and Scholes continuous markets
  • ? 1 year 365 days.
  • Real world the markets are open for trading
    only 252 days.
  • ? 1 year 252 days.

20
The inputs1. r continuously compounded
2. R simple. ? r ln1R.3. Ra Rb n
the number of days to the T-bill
maturity.
21
The Volatility (VOL)
  • The volatility of an asset is the standard
    deviation of the continuously compounded rate of
    return in 1 year
  • As an approximation it is the standard deviation
    of the percentage change in the asset price in 1
    year

22
Historical Volatility (page 286-9)
  • Take n1observed prices S0, S1, . . . , Sn at
    the end of the i-th time interval, i0,1,n.
  • ? is the length of time interval in years.
  • For example, if the time interval between i and
    i1 is one week,
  • then ? 1/52, If the time interval is one day,
    then ? 1/365.

23
Historical Volatility
  • Calculate the continuously compounded return in
    each time interval as

24
Historical Volatility
  • Calculate the standard deviation of the uis

25
Historical Volatility
  • The estimate of the Historical Volatility is

26
Implied Volatility (VOL) (p.300)
  • The implied volatility of an option is the
    volatility for which the Black-Scholes price
    equals the market price
  • There is a one-to-one correspondence between
    prices and implied volatilities
  • Traders and brokers often quote implied
    volatilities rather than dollar prices

27
ct StN(d1) Ke-r(T-t) N(d2) d1 ln(St/K)
(r .5?2)(T t)/?v(T t), d2 d1 - ?v(T
t). Inputs St K r T-t and ct The solution
yields Implied Vol.
28
Dividend adjustments (p.301) 1. European
options When the underlying asset pays dividends
the adjustment of the Black and Scholes formula
depends on the information at hand. Case 1. The
annual dividend payout ratio, q, is known.
Use Where St is the current asset market price
29
Dividend adjustment 2. European options Case
2. It is known that the underlying asset will
pay a series of cash dividends, Di, on dates
ti during the options life. Use Where St
is the current asset market price
30
Dividend adjustment
  • American options
  • T the option expiration date.
  • tn date of the last dividend payment before T.
  • Blacks Approximation cMax1,2
  • Compute the dividend adjusted European price.
  • Compute the dividend adjusted European price with
    expiration at tn
  • i.e., without the last dividend.
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