dominance theorems that bound the value of a call option

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10 Οκτ 2013 (πριν από 3 χρόνια και 6 μήνες)

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DOMINANCE THEOREMS THAT BOUND THE VALUE OF A CALL OPTION


The call option value is a function of
.

Fact

1
: Generally,
.

Fact

2
: The payoff to a call option at maturity,
T
, is

meaning t
hat
.

Fact 3:
, because even if
, stockholders have voting rights, while
option holders do not.


An American call on a non
-
dividend paying stock will not be exercised before the
expirati
on.


Proof: The present value of a bond with the par value equal to $1 is
. Note
that



Now consider the following two portfolios, A and B.


Portfolio A: long in one European call for

and
X

bonds for $
.

Portfolio B: one share of stock for
S
.


Then, given the value of these portfolios at maturity, it must be true that









If exercised, the value of an American call is

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Thus, the holder of an American call option can always do better by selling it in the
marketplace rather than exercising it prior to
expiration.


Result:


Premature exercise of an American call may occur if the underlying security pays
dividends (and if the option is inadequately protected against the dividend payment)


Proof. Construct the following two portfo
lios.


Portfolio A: One European call and

bonds.

Portfolio B: One share of stock.


Given the value of these portfolios at maturity, it is clear that







Put Call Parity


With European options,

With American options,

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DERIVATION OF THE OPTION PRICING FORMULA


THE BINOMIAL
APPROACH


Easier to understand while it provides solutions not only for

a closed
-
form European
option but also for the more difficult American option problem where numerical
simulations are necessary.


A.

One
-
Period Model


Assume that the probability the stock price will move upward is
q
. Assume also that

; and

. Then,


; and


Construct a risk
-
free hedge portfolio composed of one share of stock,
S
,
held long
and
m

shares of a call option

sold short
. Then, at the

end of the period,






We also know that




.



.


If
p
is the hed
ging probability, then
.


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Theorem


In equilibrium,

for a risk neutral investor.


Proof.


A risk neutral investor would require
. Solving for
q
,

.


Obviously, t
he
rate of return on a call =
.

Finally, w
e conclude that:


1.
;

2.

3.
Value of a call does not depend on risk preference.


B.

Two
-
Period Model






We know previously that the value of a call that started the end of the first period is





Substituting these into

gives

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Note (Re
balancing Portfolio):
; and


C.

Digression on Binomial Distribution


What is the probability,
, of observing two heads out of three coin flips?
Useful for answering this question is Pascal
’s triangle.


Number of Trials


Pascal's Triangle


T

= 0






1






T

= 1






1

1






T

= 2





1

2

1





T

= 3




1

3

3

1




T

= 4



1

4

6

4

1



Number of heads,
n

=


T

T
-
1,

…………………………,

T
-
T




Now,

from the Pascal’s
triangle. In fact,
. Therefore,




The binomial properties are:
; and


D.

The Complete Binomial Model


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General payoff of a call =
, where
T
= the total number of time
periods, and
n
= the number of upward movements in the stock price,
.




Let

be the positive integer that bounds those states of nature where
the option has a
nonnegative value. Then,




.


Note that


Therefore,


,
w
here



,

and

.


the

smallest nonnegative integer greater than

and if
, then
.

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= the complementary binomial probability that
. It refers to the
probability th
at the sum of
n
random variables, each of which can take on the value 1
with probability
p
and 0 with probability

will be greater than or equal to
.


E.

Extending It To Continuous Time

With a Fixed Option Life


L
et
T
be a fraction of a year and
n
be the
compounding frequency
in interval
T
.


;
; and


As
, it can be proven that

and

where
,
, and
;
.


Black Scholes Option Pricing Formula:



Notes: The up and down parameters can be adjusted as



;
; and




F.

Polynomial Approximation of

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,
where
;



and
;
;
;
;
;
; and
.


It is easy to verify that as the number of binomial trials increases, it appro
aches to the
Black and Scholes’ formula.


G.

Pricing American Put Options


Knowled
ge of put
-
call parity and the call option pricing formula is sufficient to value a
European put option.


Unfortunately, American put options can be exercised before maturity.

However,
.


Form a riskless hedge portfolio to buy a frac
tion of the risky assets and simultaneously
buy a put.






However,
.

Substituting
m

and solving it for the current put
price gives


, w
here
p

is the hedging probabilities.


To show in two
-
period context that it may be better to exercise the put early,

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It is possible that at the end of the first period,
, in which ca
se it is better to
exercise early.


Task: Write a computer program that uses an iterative technique starting with the set of
possible final payoffs in
n
periods and working backward can solve for the present value
of American puts, which should exceed the

value of European puts.


H.

Dividends, European Calls, American Calls, and Early Exercise


(1)

Dividend Yield,
, Model


The stock price rises to

at time 1, when a dividend is paid in the amount of
. The
ex
-
dividend stock price is then
. Similarly, when the stock price falls to
, the
ex
-
dividend stock price will be
. An up and down in the stock price at time 2 is
still
u
and

d
, but will change from the base of either

or
.


With American calls, we must see to see if

in time 1, in which case it is
better to exercise the call early. The binomial

will be replaced by
, for which
we compute the present value to value the American call at time 0.


(2)

Dollar Dividend,
D
,

Model


It is not possible now to recombine a tree. Consequently, with a non
-
recombining tree,
there
will be

stock prices with
n
compounding periods. This is contrasted to

stock price possibilities with a recombining tree.

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A trick to simplify the problem is to begin with the
stock price net of the
present
value of
future dividends. That is, if it is the stock price minus the present value of the dividends,
i.e.
, that follows the binomial process,

will rise to

or falls to

in time 1. In time 2, both

and

will move according to up and down parameters,
u
and
d
, in a recombining fashion.


With American calls, we must see if

from the binomial proc
ess, to
determine that it is better to exercise early.


I.

Applications


(1)

Equity as a Call Option


Assume that a firm has a risky debt and equity. Debt carries no coupon but has a face
value equal to
D
. Bond is secured by the firm’s assets, but bondholders c
annot force the
firm into bankruptcy until T. The firm pays no dividends.


Theorem


Equity is a call option written on the firm’s assets. Therefore,

at
maturity.


Earlier we learned that buying a stock and a put option are equival
ent to holding a bond
and a call
. At maturity
,


In the present context, then,
.



The market value of the firm
, where


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(2)

Value of Limited L
iability

or Stockholders’ Option to Default


The market value of debt

at maturity,
, where
S
is the value of call option on
the firm’s assets. Since
, it follows that
.

.


(3)

Comparison Between Secured Debt, Subordinated Debentures and Common Stocks


Assume that a firm has a secured debt in the amount of

and subordinated debentures
for
. Then,
.




Value of subordinated debts and common equities

, where




Market value of secured debts




Market value of equities
, where




Market value of subordinated debts


(4)

Call Options on Bonds


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Construct a perfectly hedged portfolio, i.e.







We also know that
.


Substituting
m
,




(5)

When the strike price is stochastic.


A stock exchange offer between two unlevered firms, in which firm A tenders for the
shares of firm B and the offer is good for
T

days.


, where

and
.


J.

Underlying Stochastic Process


1.

Black & Scholes

Wiener Process
:
, where
is a Wiener process.

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2.

Cox & Ross Jump Process:
.


= a continuous
-
time Poisson process,


= the intensity of the process,


= the jump amplitude

3.

Constant Elasticity of Variance

Process
:
;
.


K.

Derivation of Black
-
Scholes Option Pricing Formula


(1)

Hedged Portfolio


The fact that the value of a call is directly directed to the price of the underlying, i.e.
, means that one can form a perfectly hedged portfolio.

The value of

the portfolio
is
, and the change in the value of the portfolio is:




Hedging refers to a situation where the value of stocks held rises (falls) with a rise in
stock price, i.e.

, while

one loses (gains) from calls sold short, i.e.
, such that
.


If
hedge ratio
,
h
,

is
defined as
,

meaning that it is
the number of stocks
that
must be held long for

each call sold short. That is,

is
the number of stocks on which
the call must be sold short for each stock held long.


In fact, Black
-
Scholes model suggests
. Since
,
the hedged portf
olio must be rebalanced occasionally.

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(2)

Differential Equation Using Itô’s Lemma




(3)

Return on
Perfectly Hedged Portfolio


Theorem


Return on wealth committed to a hedged position in stocks and options is risk
-
free.


Proof.

Since
; and
, it follows that

. If
, borrow at

and invest in the hedged position. If
, short sell stock
s, i.e. +
S
, and buy options, i.e.


c
, thereby achieving a full
hedge on shorted stocks. Invest the difference,
, into risk
-
free assets.


Now,

and
.


Letting

an
d
,


Now,


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.


Boundary conditions:

at maturity; and


The solution is the Black Scholes Option Pricing Formula:



where
; and
.


L.

Interpretation of Black
-
Scholes Option Pricing Formula


1.


is the inverse of the hedge ratio, that is, for each share of stock, a riskless
hedge contai
ns

call options.

2.


is the probability that the option will finish in the money, i.e. that the option
will be exercised.

3.

As
, both


. Therefore,

4.


As
, both


. Therefore,

5.

As
,
, if
. However,
, if
. If
,
it is
also true that


. Therefore,
. In the
meant
ime, if
, it is also true that


.
Therefore,

6.

As
,
, if
. However,
, if
. If
, it is also true that


. Therefore,
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. In the meantime, if
,
it is also true that


. Therefore,

7.

If
, then
.


M.

Option Greeks


1.

Call Delta,

, where

Note 1


, as

Note 2


For in
-
the
-
money call options,
, as
. For the out
-
of
-
the
-
money call,

2.

Call Gamma,
, and

is the greatest, when
.
However, if either

or
,
.

3.


4.

Call Rho,

Note



5.

Call Vega,
, and

is the greatest, when
.
However, if either

or
,
.

6.

Call Theta,
;
and

is U
-
shaped.


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N.

Dividend Paying Stocks and Black
-
Scholes Option Pricing


1.

With known discrete dividends,
, where
t
is the time to the ex
-
dividend date.

2.

With known continuous dividend yield,
,
.


O.

American Call Options


Compute the minimum level of a dividend, which will make you exercise your option
early by using the Black
-
Scholes pricing formula.


P.

Estimating
Volatility


1.

Historical volatility

Let daily
. Then,

The annual volatility rate is


2.

Implied volatility


Write a program to solve for implied volatility.


Approximate rule (Brenner
-
Subramanyam):
, for at
-
the
-
money cal
l
options. Note that the call price is almost linear, if at
-
the
-
money.

We find that there exists term structure of implied volatility.

When the implied volatility is plotted against exercise price, the relationship is known
as the volatility smile or the
volatility skew.

Existence of the volatility smile implies that the Black
-
Scholes pricing model misses
the fact that options with higher implied volatilities are
options that are
more
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expensive
than

options with lower implied volatilities.

It is typical t
o observe that
calls with lower exercise price sell for more, while the implied volatility is the lowest.


3.

Newton
-
Raphson Iteration Techniques.


Write a program using equations in Page 187 in Don Chance.

The procedure is:


(a)

Begin with

(b)

Compute

per Black
-
Scholes’.

(c)

Compare

to the market price,
.

(d)

If
,
, where

is computed using

in Step (a)

(e)

Compute again for

to compare to
.

(f)

If
,
, where

is computed using

(g)

Same thing repeats.


4.

Exponentially w
eighted variance


5.

GARCH Techniques


Newton
-
Raphson Iteration


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Consider a function
. Suppose that one wants to find the value of
, which
gives

and
, one of the popu
lar numerical analysis problems. The
symbol

represents a mathematical function, whose value depends on the value of
.
In finance, we have such a problem when we find the internal rate of return, which will
res
ult in zero net present value.

















The Newton
-
Raphson iteration method is as follows.


1.

Pick an arbitrary value of
x
, such that
. Let’s say,
.

2.

Find the value of
, such
that
.

3.

Compute the value of

to see if
.

4.

If not, compute
, such that
.

5.

Compute the value of

to see if
.

x
0

x
1

r

x


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6.

Continue this process until we find the value of

such that
.


Q.

Put Option Pricing


Put call parity

.
Substituting the Black
-
Scholes call pricing
formula,



R.

Put Option Greeks


1.

Put Delta



2.

Put Gamma


, which is same as the call Gamma.

3.


4.

Put Rho,

5.

Put Vega,
, which is the same
expression as for the
call.

6.

Put Theta,