Black Scholes Lec 14

Document Sample

```					                                                                             Normal (Gaussian) Distribution

0.25                                                                                          1.1
1

Probability Density
0.2                                                                                          0.9
0.8

Cumulative
Probability
0.15                                                                                          0.7
0.6
?                          0.5
0.1                                                                                          0.4
0.3
0.05                                                                                          0.2
0.1
0                                                                                        0
2        3.6   5.2    6.8    8.4       10 11.6 13.2 14.8 16.4 18

The Black-Scholes Model
... pricing options and calculating
Greeks
(c) 2006-2008, Gary R. Evans. May be used for non-profit educational uses only without permission of the author.

The Standard Normal Distribution
which will be useful
0.45                                       1.1

0.4                                       1

Dist                           0.35
83.65%
0.9
0.8
Cumulative Probability
Probability Density

σ t
0.3
0.7
0.25                                       0.6
0.2                                       0.5
Dist
0.4
0.15       σ t
0.3
0.1
0.2
0.05                                       0.1
0                                        0
-4           -3         -2     -1          0          1      2         3   4

Dividing a normal distribution with mean 0 by its standard deviation
produces the standard normal distribution, where we can describe the
probability of a number being X standard deviations away from its mean.
Shown is the probability of a value being less that +1 SD.

1
Comparing the Normal and Lognormal
distribution
( x− µ )           2
Equation 1 shows the normal probability
1   −
f ( x) =      e 2σ
2
1.                                                            density function (pdf). Assume that the mean
σ 2π                                       is zero and the standard deviation is 0.10

( ln ( x )− µ )2       Equation 2 shows the lognormal
g ( x; µ x , σ x ) =
1               −
2.                                                    e            2σ 2              pdf of x, where µ and have the
xσ 2π                                              same value as in Equation 1.
Compare 2 to 1.
µ +σ 2
2
3.        Meang = e                       = e 0.005 = 1005
.        Equation 3 is the mean and
equation 4 is the standard
deviation of the lognormal

(e              )
σ2
distribution. Equivalent values
SDg =                 − 1 e2 µ +σ           = 0100753 are calculated.
2
4.                                                       .
Note the highlighted differences.

About the relationship between the Normal
Thanks to Ben and Lognormal Distributions
Preskill '08

If x is a random variable with a normal distribution, the y = e x has a lognormal
distribution. Likewise, if we consider a multiple of x;

If        x ≈ N ( µ ,σ 2 )                       then                           β x ≈ N ( βµ , β 2σ 2 )

β e x ≈ LNN ⎛ β e
µ +σ 2
(               )
, β 2 eσ − 1 e 2 µ + σ ⎞
2

⎜                                                                     ⎟
2             2
and
⎝                                                                     ⎠

So if in our case we assume                         r ≈ N (0,010)
.                         then this expression below

Si = S0 e ≈ LNN ⎛ S0 e
σ
(        )
, S0 2 eσ − 1 e 2 σ ⎞
2

⎜                                                                     ⎟
2         2
r                                          2
will be useful:
⎝                                                                     ⎠

2
An assumed normal distribution for a stock
continuous growth rate
4.5

4

3.5

3

2.5

2

1.5

1

0.5

0

-3 .5      -2 .6 2 5   -1 .7 5       -0 .8 7 5       0      0 .8 7 5   1 .7 5    2 .6 2 5    3 .5

This represents the assumed normal distribution for a the log continuous
growth rate for a stock that has a sigma (volatility) of 0.10, between the
ranges of +/- 3.5 sigma. Let's assume that this stock has a spot value of
100. What will happen if we translate these growth rates to stock prices?

The resulting transformed log-normal
distribution of stock prices
Note: We did not calculate
4.5       this with EXCEL's
LOGNORMDIST function,
Si = 100e ri
4        although we could have.
3.5

3                                                                For example, the lower bound:
2.5
- 3.5 SD                                                          70 = 100e −0.35
2

1.5
(70)
3.5 SD
1

µ +σ 2
2                                (142)
0.5
M s = 100e                     = 100.50
0

70             80          90             100            110        120         130         140

These are the same data, whereas the abscissa is transformed through the
original distribution from growth rates to stock prices based and made
symmetric. This distribution is log-normal. 100 is the Median.

3
4.5

4

3.5

3
The two compared next
2.5

2
to each other. They
1.5
represent the same data.
1                                                                     Both show the
0.5                                                                     distribution between the
0                                                                     range of +/- 3.5 standard
-3.5   -2.625   -1.75   -0.875   0     0.875   1.75    2.625    3.5   deviations from the
mean, assuming a
4.5
standard deviation of
4

3.5
0.10 with a mean
3
(starting) stock value of
2.5                                                                     100.
2
The top graph is
1.5

1
normally distributed and
0.5
the bottom graph is log-
0                                                                      normally distributed
70       80       90       100       110   120        130      140

How Black-Scholes works ...
The Black-Scholes model is used to price European options
(which assumes that they must be held to expiration) and related
custom derivatives. It takes into account that you have the option
of investing in an asset earning the risk-free interest rate.
It acknowledges that the option price is purely a function of the
volatility of the stock's price (the higher the volatility the higher
Black-Scholes treats a call option as a forward contract to deliver
stock at a contractual price, which is, of course, the strike price.
In our treatment, let's consider in-the-money call options of non-
dividend paying stocks.

4
The Essence of the Black-Scholes Approach
• Only volatility matters, the mu (drift) is not important.
• The option's premium will suffer from time decay as we
approach expiration (Theta in the European model).
• The stock's underlying volatility contributes to the option's
• The sensitivity of the option to a change in the stock's
value (Delta) and the rate of that sensitivity (Gamma) is
important [these variables are represented mathematically
in the Black-Scholes DE, next lecture].
• Option values arise from arbitrage opportunities in a world
where you have a risk-free choice.

The Black-Scholes Model: European Options

C = SN (d1 ) − Ke
(
− r t 365   ) N (d )
2

C = theoretical call premium        ln( S / K ) + (r + σ 2 / 2)(t 365)
S = current stock price       d1 =
σ t
N = cumulative standard
normal probability dist.       ln( S / K ) + (r − σ 2 / 2)(t 365)
d2 =
t = days until expiration                          σ t
K = option strike price
r = risk free interest rate   d 2 = d1 − σ t
σ = daily stock volatility
Note: Hull's version (13.20) uses annual volatility. Note the difference.

5
Memo: Algebraic Estimation of CNPD
Where        x≥ 0     or NORMSDIST(d1 or d2,TRUE)

N ( x ) = 1 − (a1k + a2 k 2 + a3 k 3 ) N '( x ) where

N '( x ) =
1 − x2 2
e                and         a1 = 0.4361836
2π                              a2 = − 01201676
.
1
k=                                            a3 = 0.9372980
1 + αx
α = 0.33267                                   Where x < 0
(otm calls and itm puts)
Provides values accurate to 0.0002.
Note that this is easily programmable.
N ( x) = 1 − N (− x)

Breaking this down ...

C = SN (d1 ) − Ke
(
− r t 365   ) N (d )
2

This term discounts the price of the stock at which you will have the
right to buy it (the strike price) back to its present value using the
risk-free interest rate. Let's assume in the next slide that r = 0.

ln( S / K ) + (r + σ 2 / 2)t
d1 =
σ t

Dividing by this term (the standard deviation of stock's daily
volatility adjusted for time) turns the distribution into a standard
normal distribution with a standard deviation of 1.

6
C = S × N (d1 ) − K × N (d 2 )                       (assuming r to be 0)

... and some more
This term, our x of two slides ago, represents the spread in continuous
growth terms between the stock price and the strike price, and when
normalized by the denominator, the spread as the number of standard
deviations. For example, if S = 110 and K = 100 and volatility = 10%, then
this terms equals 9.5%, or about one standard deviation. x > 0 for itm calls
and otm puts and x < 0 for otm calls and itm puts.

ln( S / K ) + (σ 2 / 2)(t 365)                 ln( S / K ) − (σ 2 / 2)(t 365)
d1 =                                           d2 =
σ t                                             σ t

This term has the effect of                   This term has the effect of centering
removing the bias from continuous             the transformed lognormal
compounding.                                  distribution on the median price (we
will see in a later example where
K=100).

Using the Black-Scholes Model
There are variations of the Black-Scholes model that prices for dividend payments
(within the option period). See Hull section 13.12 to see how that is done (easy to
understand). However, because of what is said below, you really can't use Black-
Scholes to estimate values of options for dividend-paying American stocks
There is no easy estimator for American options prices, but as Hull points out in
chapter 9 section 9.5, with the exception of exercising a call option just prior to an ex-
dividend date, "it is never optimal to exercise an American call option on a non-
dividend paying stock before the expiration date."
The Black-Scholes model can be used to estimate "implied volatility". To do this,
however, given an actual option value, you have to iterate to find the volatility solution
(see Hull's discussion of this in 13.12). This procedure is easy to program and not very
time-consuming in even an Excel version of the model.
For those of you interest in another elegant implied volatility model, see Hull's
discussion of the IVF model in 26.3. There you will see a role played by delta and
vega, but again you would have to iterate to get the value of the sensitivity of the call to
the strike price.

7
An example ...
Consider an itm option with 4 days to expiration. The strike
price is 100 and the price of the stock is 110 and the stock
has an daily volatility of 0.05. (The stock is about one
standard deviation in the money). Assume an interest rate of
0.

ln(110 / 100) + (0.025 / 2).011
d1 =                                   = 0.9532
0.05 4
d 2 = d1 − 0.05 4 = 0.8532

C = 110 N (d1 ) − 100 N (d 2 ) = 10.95

What effectively we are doing ...
Stock Price times
cumulative density.                                      less                 Strike Price times
cumulative density.

0.45                                  1.1
0.4                                  1
Shown                                                                                               0.9
for an                                                        0.35
83.65%              0.8
Cumulative Probability
Probability Density

itm call.                                                      0.3
0.7
0.25                                  0.6
0.2                                  0.5
0.4
0.15
0.3
0.1
0.2
0.05                                  0.1
0                                   0
-4     -3    -2     -1          0      1      2      3     4

8
Perfectly hedging my zero-sum game
Suppose that I am in a business that has to make a single \$100 bet every day. I
place the bet and then a RNG pulls a number from a normal distribution
(shown on the next slide) that has mean value of \$100, a standard deviation of
\$30, subject to the limits of +/- 3.5 standard deviations. I want to escape the
risk of this bet and pass it on to two risk takers.
I offer one of them to pay me for the right to accept any outcome up to \$100
and the other to pay me for the right to accept any outcome above \$100.
This offer is subject to the condition that the two payments sum to \$100 and
that each of the other two players is also playing a zero-sum game, which
implies that the expected value of their bet equals the amount that they pay
me.
Effectively, I will have written an at-the-money call and put with the strike
price of \$100.
How much will each pay?

0.014

0.012                Put                                  Call
0.01

0.008                                                                 How much will
you sell these
0.006
for?
0.004

0.002

0

-5   16     37       58       79   100   121   142       163   184   205

May be useful
to know ...
d2 =
( )− σ
ln S K
d1 =
( )+ σ
ln S K
σ        2                          σ         2

9
The Greeks as BS formulas
∂ Pc
∆c =         = N ( d1 )
∂ Ps                         ... when you build a Black-
Scholes model you can
easily include these Greek
∂ Pp
= N ( d1 ) − 1
estimators.
∆p=
∂ Ps

∂ 2 Po N ′ (d1 )                              e− d 2
2

Γ =       =                             N ′(d ) =
∂ Ps2   S0 Τ             where                  2π

... continued

∂ Pc S0 N ′ (d1 )σ
Θc =       =              − rKe − rΤ N (d 2 )
∂t     2 T

∂ Pp S0 N ′ (d1 )σ
Θp=        =              + rKe − rΤ N (d 2 )
∂t     2 T

∂ Pc
= S 0 Τ N ′ ( d1 )
e− d 2
2

V=                                   where      N ′(d ) =
∂σ s                                                     2π

10
The Black Scholes Merton DE
The Black Scholes differential equation is derived algebraically from Ito's
Lemma and is developed by Hull in Section 13.6. This equation must be
satisfied by any option that is a function of the price of the stock.

∂c      ∂ c 1 2 2 ∂ 2c
+ rS    + σ S       = rc
∂t      ∂S 2      ∂S2
The same equation expressed in terms of the Greeks is shown below and
clearly shows the role of the Greeks in determining an options value.

1 2 2
Θ + rS∆ +           σ S Γ = rc
2

11

```
DOCUMENT INFO
Shared By:
Categories:
Stats:
 views: 15 posted: 7/9/2010 language: English pages: 11
How are you planning on using Docstoc?