The Black-Scholes Model Chapter 13

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The Black-ScholesModelChapter 13
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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
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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).
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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)
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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)
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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

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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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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
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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
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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 ---
--- --- --- ---
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• 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

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• 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) ?

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• 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

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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

• 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 Inputs

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

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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.

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The inputs1. r continuously compounded
2. R simple. ? r ln1R.3. Ra Rb n
the number of days to the T-bill
maturity.
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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

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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.

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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 uis

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Historical Volatility

• The estimate of the Historical Volatility is

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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

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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.
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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
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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
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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.