The Black-Scholes Model Chapter 12

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The Black-ScholesModelChapter 12
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Valuation

• 1. Calculate the expected payoff from the option
• 2. Discount at the risk-free rate

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Probability distribution
ST
CFST-K
K
K
St
CF0
t
T
time
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Pricing an European Call The BlackScholes
model ct PV Calls Expected cash flow ct
PVE (max0, ST K). ct e-r(T-t)maxE (0,
ST K).
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Pricing an European Call The BlackScholes
model ct e-r(T-t) 0lST?KPr(ST?K) E
(ST K)lSTgtKPr(STgtK) ct e-r(T-t) E (ST
K)lSTgtKProb(STgtK)
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Pricing an European Call The BlackScholes
model Again ct e-r(T-t)E (ST
K)lSTgtKProb(STgtK) ct e -r(T-t)E
(ST)lSTgtKProb(ST gt K) Ke-r(T-t)Prob(ST gt
K)
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The Stock Price Assumption (Sec. 12.1

• In a short period of time of length ?t, the
  return on the stock is normally distributed
• Consider a stock whose price is S
• where µ is expected return and s is volatility

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The Lognormal Property

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

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The Lognormal Distribution

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The BlackScholes formula(Sec. 12.7) Pricing a
European call ct StN(d1) Ke-r(T-t)
N(d2) Where 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.
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The BlackScholes formula(Sec. 12.7) Pricing a
European put pt Ke-r(T-t) N(- d2) StN(-
d1). Where 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.
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INTEL Thursday, September 21. 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 ---
--- --- --- ---
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• S 61.48
• K 65
• JAN T-t 121/365 .3315yrs
• R 4.82?r ln1.0482 .047
• 48.28
• or, for the actual calculation ? .4828

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• d1ln(61.48/65) .047 .5(.48282).3315/(.4828
  )?(.3315)
• -.0052
• d2 d1 (.4828)?(.3315) -.2832
• N(d1) .49792 N(d2) .388484
• c 61.48(.49792) 65e-(.047)(.3315)(.388484)
• c 5.751
• p 5.751 -61.48 65e-(.047)(.3315)
• p 8.266

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• S 61.48
• K 90
• JAN T-t 121/365 .3315yrs
• R 4.82?r ln1.0482 .047
• 48.28
• or, for the actual calculation ? .4828

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• d1ln(61.48/90) .047 .5(.48282).3315/(.4828
  )?(.3315)
• -1.1759
• d2 d1 (.4828)?(.3315) -1.4539
• N(d1) N(-1.17) - .59N(-1.17) N(-1.18)
• .1210 - .59.1210 - .1190
• .11982.
• N(d2) N(-1.45) - .39N(-1.45) N(-1.46)
• .0735 - .39.0735 - .0721
• .072954.

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c 61.48(.11982) 90e-(.047)(.3315)(.072954) c
7.3665336 6.464353433 c .90 Estimate
the put value p .90 - 61.48
90e-(.047)(.3315) p 28.03
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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
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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)
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The Inputs for the Black and Scholes formula

• 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

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The stock price

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

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The time to expiration

• 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.

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The interest rateThe input depends on the
information one has. The rate must be with
continuous compounding. Thus,1. annual rate,
r, with continuous compounding Input r.
2. Annual rate, R, with annual compounding
First calculate r ln1R Input r.
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The interest rate3. Ra Rb n the number of
days to the T-bill maturity.
Input r.
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Input Ra and Rb online.wsj.com Markets Market
data bonds
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The Volatility (VOL)(Sec. 12.3)

• 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

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Historical Volatility(Sec. 12.4)

• Take n1observed prices S0, S1, . . . , Sn at
  the end of the i-th time interval, i0,1,n.
• 2. ? is the length of time interval in years.
• If the time interval between i and i1 is
• one week, then ? 1/52. If the time
• interval is one day, then ? 1/365.

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Historical Volatility

• Calculate the continuously compounded return in
  each time interval as

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Historical Volatility

• Calculate the standard deviation of the
• ris

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Historical Volatility

• The estimate of the
• Historical Volatility is

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Implied Volatility (VOL) (Sec. 12.9)

• 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

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Calculating the Implied Vollatility 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
Market St K r T-t and Market ct The
solution yields Implied Vol.
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Dividend adjustment(12.10) 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. Assume that it is paid out
continuously during the year and use Where
St is the current asset market price
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Dividend adjustment Case 2. It is known that the
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