# Contracts Anderson Ucla

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```					Chapter 10: Options on Risky Assets.

=================================================================
INTRODUCTION
==================

An option is a contract that gives its holder the right to buy (or sell) an asset at a
predetermined price for a given period. Theoretical developments in option
pricing models (OPM) have been quickly adopted for use in rapidly expanding
new option markets. Option pricing principles have also been extended to the
valuation of equity, risky debt, shares of dual funds, to futures contracts, and to
mergers.

=================================================================

05ba953b-1afb-4675-b169-a3e8e6ef1684.xls
=================================================================================
QUESTIONS
====================
1) Given the following key input parameter values for the Black-Scholes option pricing model, what
would be the value of the call?

s (instantaneous variance) =
2
S (stock price) =           \$25                                         0.09
X (exercise price) =        \$20       RF (risk-free rate) =              6%
T (time in years) =            4

a) How does the value of the call change as you increase the stock price in increments of \$5
to \$50?
b) If the premium on the call is defined as the value of the call minus the quantity (S-X), what
happens to the size of the premium as the stock price goes from \$25 to \$50, while the
exercise price remains at \$20?
c) How would a graph of the call price look in relation to a line which simply represents S-X?
View this graphically. What happens to the size of the distance between the call price line
as compared with the (S-X) line?

2) Answer the same questions for a call whose maturity is 9 years rather than 4 years.

3) Going back to the original conditions in Question 1, does the value of the call increase or
decrease as each of the key input parameters increases? For example, successively increase
each of the variables by 50 percent and see whether the call price increases or decreases.

4) Go back to the original conditions in Question 1, and think of S as being the market value of the
firm of \$25 million, and X as the face value of the debt at \$20 million. The T in this case is
interpreted as years to maturity of the debt. Now what corresponds to the value of the call is the
market value of equity. What is the market value of equity under the original conditions? Also,
what is the market value of the debt, as compared with its face value?

5) Assume that one firm is engaged in a merger with a firm of exactly equal size. Because the
cash flow streams of the two firms are not perfectly correlated, the variance of the combined
cash flow is reduced. What effect would this have on the value of the equity of the combined
firm? To test this, go back to the initial conditions, double the value of the firm, double the face
value of the debt, hold everything else the same, and decrease the variance to .06. Then, is the
equity value more or less than doubled?

6) Continue to explore option pricing by changing any assumptions used in the Black-Scholes
model.
=================================================================================

05ba953b-1afb-4675-b169-a3e8e6ef1684.xls
==========================================================================
SUMMARY OF ASSUMPTIONS AND RESULTS
==========================================

Assumptions (Black-Scholes Model):

S=       \$25.00    = Stock price
X=       \$20.00    = Exercise price
T=          4.00   = Time to maturity (years)
s2 =      0.0900    = Instantaneous variance
s=       0.3000    = Standard deviation
RF =         6.0%   = Risk-free rate

Results:

Value of call =     \$10.73

==========================================================================

05ba953b-1afb-4675-b169-a3e8e6ef1684.xls
==================================================================================
CALCULATION OF UNIT NORMAL VARIABLES (d)
==========================================

ln (S X ) + RF T 1
d1 =                   + s T
s T         2
=        ln( 25.00 / 20.00 ) + 6.0% * 4.00    +   0.5 * 0.3000 * SQRT( 4.00 )
0.3000 * SQRT( 4.00 )

=    1.07191

d 2 = d1 - s T

=    1.0719 – 0.3000 * SQRT( 4.00 )

=    0.47191

==================================================================================

05ba953b-1afb-4675-b169-a3e8e6ef1684.xls
=========================================================================
CALCULATION OF CUMULATIVE PROBABILITIES AND CALL VALUE
==========================================================

Cumulative probabilities:

N (d1 )            =         0.8581

N (d2 )            =         0.6815

Call value:

C = S N (d1 ) - Xe - RF T N (d 2 )
=       25.00 * 0.8581 – 20.00 EXP( –6.0% * 4.00 ) * 0.6815

=       \$10.73

=========================================================================

05ba953b-1afb-4675-b169-a3e8e6ef1684.xls
S     MAX(0,S-X)     d1        d2       N(d1)   N(d2)    Call Price
\$0       0           —         —        —        —         —
\$2       0        -3.1376   -3.7376   0.0009   0.0001     \$0.00
\$4       0        -1.9824   -2.5824   0.0237   0.0049     \$0.02
\$6       0        -1.3066   -1.9066   0.0957   0.0283     \$0.13
\$8       0        -0.8272   -1.4272   0.2041   0.0768     \$0.42
\$10       0        -0.4552   -1.0552   0.3245   0.1457     \$0.95
\$12       0        -0.1514   -0.7514   0.4398   0.2262     \$1.72
\$14       0         0.1055   -0.4945   0.5420   0.3105     \$2.70
\$16       0         0.3281   -0.2719   0.6286   0.3928     \$3.88
\$18       0         0.5244   -0.0756   0.7000   0.4699     \$5.21
\$20       0         0.7000    0.1000   0.7580   0.5398     \$6.67
\$22       2         0.8589    0.2589   0.8048   0.6021     \$8.23
\$24       4         1.0039    0.4039   0.8423   0.6568     \$9.88
\$26       6         1.1373    0.5373   0.8723   0.7045    \$11.60
\$28       8         1.2608    0.6608   0.8963   0.7456    \$13.37
\$30      10         1.3758    0.7758   0.9156   0.7811    \$15.18
\$32      12         1.4833    0.8833   0.9310   0.8115    \$17.03
\$34      14         1.5844    0.9844   0.9434   0.8375    \$18.90
\$36      16         1.6796    1.0796   0.9535   0.8598    \$20.80
\$38      18         1.7698    1.1698   0.9616   0.8790    \$22.71
\$40      20         1.8552    1.2552   0.9682   0.8953    \$24.64
\$42      22         1.9366    1.3366   0.9736   0.9093    \$26.59
\$44      24         2.0141    1.4141   0.9780   0.9213    \$28.54
\$46      26         2.0882    1.4882   0.9816   0.9316    \$30.50
\$48      28         2.1591    1.5591   0.9846   0.9405    \$32.46
\$50      30         2.2272    1.6272   0.9870   0.9481    \$34.43
\$52      32         2.2925    1.6925   0.9891   0.9547    \$36.41
\$54      34         2.3554    1.7554   0.9907   0.9604    \$38.39
\$56      36         2.4160    1.8160   0.9922   0.9653    \$40.37
\$58      38         2.4745    1.8745   0.9933   0.9696    \$42.36
\$60      40         2.5310    1.9310   0.9943   0.9733    \$44.35
\$62      42         2.5857    1.9857   0.9951   0.9765    \$46.34
\$64      44         2.6386    2.0386   0.9958   0.9793    \$48.33
\$66      46         2.6899    2.0899   0.9964   0.9817    \$50.32
\$68      48         2.7396    2.1396   0.9969   0.9838    \$52.31
\$70      50         2.7879    2.1879   0.9973   0.9857    \$54.31

05ba953b-1afb-4675-b169-a3e8e6ef1684.xls
CALL AND STOCK PRICE

\$80

\$70

\$60

\$50

\$40
Call Price

\$30

\$20

\$10

\$0
\$0   \$10   \$20   \$30            \$40                \$50          \$60         \$70         \$80

-\$10
Stock Price

Stock Price         Call Price         S-X

05ba953b-1afb-4675-b169-a3e8e6ef1684.xls

```
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