The Black-Scholes Option Pricing Model and Option Greeks

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FINA 6220Professor Andrew Chen

• The Black-Scholes Option Pricing Model and Option
  Greeks
• Lecture Note 4

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Outline

• The Black-Scholes Analysis
• Assumptions
• Call and put prices
• Using the Models in Practice
• Model Inputs
• Adjusting for Dividends
• Pricing American call options with Dividends
• The Greeks

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The Black-Scholes Model

• Assumptions
• 1. European options
• 2. Stock is log-normally distributed with mean
  µ?t and standard deviation s(?t)½
• 3. No dividends
• 4. Constant risk-free interest rate
• 5. No frictions in the market place
• Deriving the Black-Scholes Model
• Form a risk-less portfolio by long ? shares and
  short one call option

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The Black-Scholes Model

• Consider the following portfolio Long ? shares
  and short one call option.
• V is the current Market value of the portfolio V
  ?S C
• Let ?C represent the amount that the value of the
  call option will change if the price of the stock
  moves by a tiny amount ?S
• The change in the value of the portfolio, for a
  tiny change in the value of the stock will be
  ?V??S ?C
• To ensure the portfolio remains a risk-less
  hedge, finding the value of ? sets ?V0

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The B-S Call and Put Formulas

• where
• N(x) is the cumulative normal distribution
  function, i.e., the probability of observing a
  value less than x when drawing randomly from a
  standard normal distribution (zero mean and unit
  variance).

(4.3)
(4.4)
(4.5)
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The B-S Call and Put Formulas
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B-S Call Option Formula

• Comments
• 5 Variables
• Interpretations
• European vs. American calls
• Delta of a call option
• Measures the change in value of he option for a
  1 change in the stock price.

(4.6)
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B-S Put Option Formula

• Combining the B-S call option equation with the
  put-call parity relation, leads easily to a
  comparable B-S formula for a European put on a
  non-dividend-paying stock.
• Put Delta

(4.7)
(4.8)
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B-S Put Option Formula
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Black-Scholes Model in Practice

• Computational Issues
• Example S46, K45, r5, ?30, and T6 months.
• From standard normal distribution tables, we get

N(d1) 0.62835
N(d2) 0.54595
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Black-Scholes Model in Practice

• Value of Call Option
• C (46)(0.62835) (43.889)(0.54595) 4.94
• Using DerivaGem with 99 time steps to value the
  American call option produces an option price of
  4.94.
• Thus, the values of European and American call
  options are the same when the underlying stock
  does not pay dividends during the life of the
  option.

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Black-Scholes Model in Practice

• Value of a European Put Option
• The Put price is
• P (43.889)(0.45405) (46)(0.37165) 2.83
• Using DerivaGem with 99 time steps to value the
  American put option produces an option price of
  2.91.
• Thus, the Black-Scholes European put option
  pricing model underprices the American put.

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Estimating Volatility from Past Prices

• Called the HSD (Historical Standard Deviation)
• Step 1 Take the natural logarithms of the prices
• Step 2 Compute the changes in logarithms. There
  will be N changes
• Step 3 Compute the mean of the changes
• Step 4 Compute the N deviations from the mean.
  Square these deviations and sum them up, ie.,
  compute Sum.
• Step 5 The estimate of the daily variance is
• Annualize the volatility
• Step 6 The volatility ? is the square root of s2.

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Estimating Volatility from Past Prices

Closing
Change Squared Date
Price
ln(Price) in ln(P) Deviation 01
Feb 03 115.00 4.745 02 Feb 03 112.39
4.722 -0.023 0.000196 03 Feb
03 111.13 4.711 -0.011 0.000004 04 Feb
03 105.25 4.656 -0.055 0.002116 05 Feb
03 101.25 4.618 -0.038 0.000841 08 Feb
03 101.94 4.624 0.006 0.000225 09 Feb
03 95.94 4.564 -0.060 0.002601 10 Feb
03 98.56 4.591 0.027 0.001296 11 Feb
03 104.88 4.653 0.062 0.005041 12 Feb
03 99.06 4.596 -0.057 0.002304 16 Feb
03 99.06 4.596 0.000 0.000081 17 Feb
03 95.13 4.555 -0.041 0.001024 18 Feb
03 96.19 4.566 0.011 0.000400 19 Feb
03 97.13 4.576 0.010 0.000361 22 Feb
03 102.06 4.626 0.050 0.003481 23 Feb
03 102.94 4.634 0.008 0.000289 24 Feb
03 99.94 4.605 -0.029 0.000400 25 Feb
03 98.50 4.590 -0.015 0.000036 26 Feb
03 97.81 4.583 -0.007 0.000004 Sum -0.162
0.020700 Mean -0.009
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Estimating Volatility from Past Prices

• Example (continued)
• Daily Variance Sum/(18-1) 0.001218
• Annual Variance Daily Variance ? 252
  0.306936
• Annual Standard Deviation 0.554 
• The estimate of volatility for XYZ stock is ?
  0.554.

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Implied Volatility or Implied Standard Deviation

• where C is the observed price of call option,
  c(s) is the model price of the option, and solve
  for s.
• Why implied volatilities are different?
• Non-Simultaneity of prices
• Bid-Ask Prices
• Model Mis-specification
• Using implied volatility in practice
• Volatility smile

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Adjusting B/S Model for dividends

• To adjust the B/S model for dividends, subtract
  the present value of the dividends during the
  life of the option from the current stock price.
• Example S 46, K 45, r 5, s 30, T 6
  months, and two dividends of 0.35 with ex-dates
  at the end of the 1st and 4th months.
• PV(D) 0.35e-0.05(1/12) 0.35e-0.05(4/12)
  0.6928
• S S PV(D) 46 0.6928 45.3072
• Using S for the current stock price, the
  DerivaGem software gives C 4.52
• and P 3.10. Notice that without dividends, C
  4.94 and P 2.83

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Black-Scholes Option Values

• Sample Black-Scholes Option Values (S100)

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Black-Scholes Option Values

• Sample Black-Scholes Option Values (S100)

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Black-Scholes Option Values

• Sample Black-Scholes Option Values (S100)

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The Option Greeks

• Delta (d)
• Essential in managing risk
• Hedge portfolio
• Short(long) option is hedged by buying(shorting)
  delta shares of underlying stock.
• Delta Neutral
• A position that is made riskless (for small price
  changes)

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The Option Greeks (example)

• Suppose we wanted to hedge a long position in the
  3 month 105 strike price call option when the
  volatility was 0.15 and the interest rate was 5
  percent.
• The options delta is 0.328, so we could short
  0.328 shares of the underlying asset for each
  option.
• For time spread of 105 strike calls, we could
  sell (0.328/0.157) 2.089 1-month for each
  3-month call long.
• For money spread for 3-month calls, we could sell
  (0.328/0.581) 0.565 100-strike call for each
  105-strike call long.

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The Option Greeks (example)

• Example (continued)
• To illustrate the bear-money-spread above,
  suppose the stock falls by 1 and we have a delta
  neutral position that is long 100 of the
  105-strike calls and short 57 of the 100-strike
  calls.
• The value of the long position should drop by
  about
• 100 x (-1) x 0.33 -33
• The short position should make a profit of about
• (-57) x (-1) x 0.58 33.06
• So the hedged position should have very low risk.

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Delta Neutral Strategies

• Value of Options Position or Portfolio
• Delta Neutral Position implies that
• That is,
• Therefore,

V N1C1 N2C2
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Portfolio Property of Option Deltas

• The delta of a portfolio of options is the sum of
  the deltas of the individual option positions.
• This allows one to summarize the price
  sensitivity of even a very complex portfolio of
  options based on a given underlying stock in a
  single number, the delta. (Of course, the delta
  of a portfolio of options is not limited to be
  less than 1.0.)
• Example Suppose a portfolio of options on a
  given underlying stock has a delta of 5000. A 1
  increase (decrease) in the underlying stock will
  produce a 5,000 increase (decrease) in the
  option position.

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Lambda (?)

• Delta gives the dollar change in the option value
  caused by a one dollar change in the price of the
  underlying stock.
• Lambda of an option is the percentage change in
  the option value due to a one percentage point
  increase in the underlying asset.

(4.10A)
Lambda (call) ?C ??(S/C)
Lambda (put) ?P ??(S/P)
(4.10B)
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Lambda (?) (S 100)
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Lambda (?) (S 100)
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Option Standard Deviation (s)

• Since s is the standard deviation of the
  underlying stock, then the standard deviation
  (SD) of a call is equal to the lambda of the call
  times the standard deviation of the stock.
• The standard deviation of a put is equal to the
  absolute value of lambda of the put time the
  standard deviation of the stock.

(4.12A)
(4.12B)
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Option Systematic Risk (beta)

• The systematic risk of a call is equal to the
  lambda of the call times the systematic risk of
  stock, and the systematic risk of a put is equal
  to the lambda of the put times the systematic
  risk of the stock.

(4.13A)
(4.13B)
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Example

• Compute the beta and SD of the 3 month 100 strike
  price call option when the stock price volatility
  is 30 and the stock beta is 0.95. (See 4-27)
• Compute the beta and SD of the 1 month 100 strike
  price put option when the stock price volatility
  is 30 and the stock beta is 0.95. (See 4-28)

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Theta (T)

• Theta refers to the rate of time decay for an
  option.
• Wasting assets
• Theta measures the rate at which the value
  decays.
• A common way of expressing decay is simply as the
  dollar loss in the option value over the next day
  if the underlying stock remains at the same
  price.

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Theta (T) (S 100)
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Theta (T) (S 100)
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Vega (v)

• Vega measures volatility sensitivity
• Since volatility is such an important determinant
  of option value, many options are quite sensitive
  to a change in volatility.
• Some people adopt a frequently used alternative
  term for the dollar change in option value caused
  by a one percentage point increase in volatility,
  the Greek letter kappa, written ?.
• Example

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Vega (v) (S 100)
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Vega (v) (S 100)
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Rho

• The final parameter in the option formula is the
  riskless interest rate.
• The change in the option value for a one
  percentage point increase in the interest rate is
  known as rho.
• The time value for a call option comes partly
  from the interest that can be earned by investing
  the strike price from the present to the
  expiration date.
• The higher the interest rate, the greater the
  calls time value, other things equal hence rho
  is positive for a call.
• The opposite is true for a put, since the put
  holder loses interest while waiting until option
  maturity to receive the strike price.

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Rho (S 100)
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Rho (S 100)
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