Download OPTION VALUATION

DERIVATIVE SECURITIES
Introduction
The Financial Manager must be knowledgeable about derivatives in order to manage the
price risk inherent in financial transactions. Price risk refers to the possibility of loss
arising from commodity price, interest rate or currency rate fluctuations (interest rate +
exchange rates are financial prices). Thus there is a need to hedge organizations against
disruptive price fluctuations. The objective of the hedge is thus to reduce an asset’s price
risk by temporarily offsetting a current or expected position in the market for the asset
with a matching but opposite position in futures, forwards or options.
Derivatives can simply be defined as financial contracts whose value depends on the
value of other financial claims (underlying asset). The value of futures and options is
derived from the financial claims on which they are written. These contracts are
sometimes referred to as contingent claims because their value is contingent on that of
another claim. Financial derivatives include futures, options, interest rate swaps and
forward rate agreements.1
Some of the most common transactions include; hedging the cost of future financing,
protecting the price of a financial asset to be sold in the future, reducing income volatility
created by interest rate changes, and hedging a commitment to lend money in the future.
Derivative Users
Traders of derivatives can be categorized as hedgers, speculators, or arbitrageurs.
Hedger
A hedger is someone who, in a spot or cash market, is exposed to price or interest rate
risk, and who wishes to secure protection against such risk by taking an opposite position
by means of an option transaction. The hedger in acting to reduce such risks is prepared
thereby to forego profits that may otherwise have accrued to him as a result of favorable
price movements in the underlying asset market or markets. The hedger using forward
exchange rates requires no initial payment whereas option contracts can be quite
expensive.
For example if you know you are to pay US$1 000 in 90 days. You are exposed to
exchange rate risk as what you pay in Z$ then depends on the ruling exchange rate. To
hedge against this risk you can enter into a long forward contract to buy $1 000 (at US$1
to Z$60) in 90 days for $60 000. If indeed the exchange rate rises to 1:70, you end up
Z$10 000 better off if you hedge. However, if it falls to 1:50, you end up Z$10 000 worse
off. Thus the purpose of hedging is to make the outcome more certain but does not
necessarily improve the outcome.
However, you could buy a call option to acquire $1 000 at 1:60 in 90 days. If the
exchange rate after 90 days proves to be above 1:60, you exercise the option and buy the
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Most of these are based on the money and capital market securities.
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$1 000 at 60, but if it proves to be below 1:60, you buy the US$1 000 in the market the
usual way (and throw the option away since they are worthless). In this case you would
have insured yourself against adverse exchange rate movements while benefiting from
favorable movements, but at a cost though (you make an upfront payment to enter into
the contract and this is the cost of the option)
Speculators
Speculators transact in options markets purely in the hope of realizing capital gains.
Either they are betting that a price will go up or they are betting that it will go down.
Forward contracts can ethically be used for speculation. In our example, if you believe
that the US$ will increase in value relative to the Z$, you can speculate by taking a long
position in a 90 day forward rate on forex. Suppose the anticipated scenario unfolds, say
the exchange rate rises to 1:70 then you will be able to purchase US$1 for $60 when they
are worth $70.
Speculating by buying the underlying asset (in our case currency) in the spot market
requires an initial cash payment equal to the value of what is bought. Entering into a
forward contract of the same amount of the asset requires no initial cash payment. In
practice however a deposit is required upfront to serve as a guarantee that the speculator
will honor the contract. Speculating using forward markets therefore provides an
investor with a much higher level of leverage than speculating using spot markets.
The presence of speculators in the options markets to provide liquidity therein is
considered vital for the effective functioning of these markets. Speculators are mainly
professional traders and brokers trading for their own account.
Arbitrageurs
Arbitrage is the purchase and sale of the same asset in different markets. It involves
looking in a risk-less profit by entering simultaneously into transactions in two or more
markets. The objective of the arbitrageur is to profit without incurring risk by acting on
price differentials that may arise from time to time in these separate but related markets.
Arbitrage opportunities are a short run phenomenon, as prices in the two markets will
quickly adjust to equilibrium due to the market forces of supply and demand. This is
why most arguments concerning futures prices and the value of options are based on the
assumption that there are no arbitrage opportunities. The transactions costs of arbitrage
can eliminate the profit for a small investor.
For example consider a stock traded on the ZSE at Z$400 and in London at a 100 pounds
at a time when the exchange rate is Z$10 per pound. An arbitrageur can simultaneously
buy 100 shares of Old Mutual for instances or NMB at the ZSE and sell them in London
to obtain a risk free profit of
100 x ($100 x 10 - $400) = $60 000 in the absence of transaction costs.
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Investors
Options markets offer investors the opportunity to invest ‘synthetically’ in financial and
commodity markets via options as an alternative to investment in the actual physical asset
or commodity as such. The advantage of high degree of investment leverage that is
inherent in the options/futures mechanism acts as a strong incentive to investors to
participate in these markets. From a liquidity viewpoint it may also be easier to buy and
sell options based on the price of shares than the shares themselves: the transactions costs
relating to options may also be less than those on shares.
OPTIONS
Options on stocks were first traded on an organized exchange in 1973 and since then
there has been a dramatic growth in options markets. The underlying assets include
stocks, foreign currencies, debt instruments, & commodities. In the broadest sense an
option may be described as the right or power, purchased or acquired, to buy or sell
something at a price fixed beforehand, within a given time limit.
The price in the contract is known as the exercise price or strike price while the date is
known as the expiration date, exercise date or maturity date and this is the date on which
the option may be exercised in terms of the option agreement. The fact that the option
gives the holder the right and not the obligation to buy or sell the underlying asset clearly
distinguishes options from forwards and futures.
Types of Options
What are stock options?
A stock option is a contract that gives you the right -- but not the obligation -- to buy or
sell a stock at a pre-specified price (the exercise price) within a pre-specified time, that is,
until the option "expires." If the option gives you the right to buy shares of a stock, it is a
call option. If the option gives you the right to sell shares of a stock, it is a put option.
Exactly how much you should pay for these contracts is determined using the BlackScholes Formula.
Options are usually sold in sets of 100 (which would allow you to buy or sell 100 shares
of the underlying stock at a certain price for the duration of the option).
Call Options
This contract gives the holder the right to buy the underlying asset by a certain date for a
certain price. Assume that party A is offering options to sell a particular commodity at
$5/kg at any time during the next 90 days. A is selling such options at 15c each. B a
manufacturer who will need to purchase the commodity as a raw material during the next
3 months expects the commodity price to rise to $5,30/kg within this period. B
accordingly buys a number of the commodity options from A in order to hedge himself
against such a possible rise. 75 days later, B needs to purchase the commodity, the
market price of which has, in the mean time, risen to $5,20/kg. B accordingly exercises
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his option, buying the commodity at $5/kg and B’s net gain (ignoring transaction costs) is
5c per option (20-15c). This calculation ignores the time value of money.
Call Option Payoffs
Profit
(c)
Option holder
+15
+10
+5
0
-5
- 10
- 15
515
520
Option writer
Sk=500
Loss
(c)
The strike price is 500c and the break-even price is 515c2 to both A and B. As long as
the price of the commodity rise above the strike price B will exercise his option at such a
price. However for a price between $5 and $5.15, B incurs a loss less than 15c per option
(loss implied by not exercising the option). Below the strike price, the options held by B
will be unexercised and he losses the entire premium he paid. Beyond the break-even
price, B will exercise his option and the difference between that price and $5,15
represents a clear gain for B if he exercises his option at such a price. Therefore, as
shown in the diagram, the holder’s loss is a mirror image of the writer’s gain and vice
versa.
For as long as the commodity market price does not rise above $5/kg or if it falls below
$5/kg during the period, A’s income from his sale of options to B will be his full
premium of 15c per option. For the price between $5 and $5,15, and where B exercises
his option, A’s premium will be reduced by the difference between the strike price and
the commodity price per option, since he will have to buy the commodity at a higher
price in the market in order to sell it to B at $5,00/kg. Beyond the break-even price, A
will incur a loss equal to B’s profit should B choose to exercise the option. Such an
option in which the right can be exercised at any time within a specified period at a
named price is known as the American call option.
Put
A put option gives the holder the right to sell the underlying asset by a certain date for a
certain price. In our previous example suppose now that B is the market –maker i.e. he is
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Break-even price for call option =(strike price + cost of option)
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offering options to buy the same commodity at the $5/kg, at any time during the next 90
days at 15c each. If A the supplier of the commodity expects the commodity price to fall
to $4,80/kg within the next 3 months, A accordingly buys a number of these commodity
options from B in order to hedge himself against such a decline in prices. 75 days later
the commodity has indeed fallen to $4,82/kg and A decides to exercise his option, and
accordingly sells the commodity to B at the strike price, the realized profit is 3c per
option. See figure
Put Option Payoffs
(18-15)c
Profit (c)
15 10 5 0
-5 -10 -15 -
Option writer
B’s point of view
485
482
. 495
510
Price of commodity
(cents)
A’s point of view
Option Holder
Sk=500
Loss (c)
The strike price is 500c, and the break-even price is 485c to both A and B3. The analysis
is similar to the one under the call option, though this one is an opposite position. If the
price does not fall below the strike price, A will not exercise his option in which case B’s
income will be his full premium of 15c per option4. Below the strike price, A will
exercise his option at that price (i.e. go to the market, buy at less than 500c & sell to B at
500c)5. Up to the break-even price, A incurs a loss but less than the loss incurred if the
option is not exercised e.g. at 495, A incurs a loss of 10c per option instead of the 15c he
paid for the option. What happens to A is exactly the opposite of what happens to B, a
loss of 10c by A is a profit of 10c by B. Below the break-even price the difference
between the price and the break-even price represents a clear profit for A, if he exercises
his option. This profit will reflect as a loss to B i.e. $4,85/kg less the price at which A
exercises the option.
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Break-even price for put option =(strike price - cost of option)
The premium is the option writer’s maximum income (compensation for taking risk) & it is the holder’s
maximum loss.
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Remember that in options, it is always the holder who has the right to exercise – the writer simply does as
he/ she is told.
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Call and put options on any commodity, financial asset, currency; or similar underlying
item will operate similarly.
There are two kinds of stock options: American-type and European-type. American-type
stock options allow you to buy or sell the shares of the underlying stock at the exercise
price ("exercise" the option) any time until the option expires. European-type stock
options allow you to exercise the option only at its expiration date. The formula we
provide here applies to a European-type call option. You can buy and sell options just
like stocks; their value is determined by the likelihood that they will be "exercised" for a
payoff ("in the money"). You can calculate the exact value of the call option using the
Black-Scholes Formula (if you know what you're doing, of course).
A Successful European-Type Call Option
Stock ABC is currently trading at $20/share. You pay a premium of $300 and purchase
one European-type stock call option -- a contract that enables you to buy 100 shares of
ABC at $25/share three months from now. Let's say that three months from now, the
stock is trading at $30/share. At that moment, you may choose to exercise your call: Buy
100 shares at $25/share. You can then immediately sell those shares at $30/share. Your
payoff in this exchange is $500. Subtract $300, the purchase price of the call option, and
you made a net profit of $200 - not bad for an investment of only $300. In this scenario,
you exercised your option "in the money": You were able to buy your stocks at a price
lower than their current value. You also may sell the call to someone else before it
expires. Let's say that one month after purchasing the call option, the stock is trading at
$29. The value of the call option is now much greater than when you bought it, and you
could sell it to someone.
An Unsuccessful European-Type Call Option
Stock ABC is currently trading at $20/share. You pay a premium of $300 and purchase
one European call option -- a contract that enables you to buy 100 shares of ABC at
$25/share three months from now. In three months, however, the stock is only trading at
$23/share. In this case, buying the stock at $25/share would not be to your advantage.
You can only hope that during the three months, you were able to sell the call option to
another investor. As the expiration date approaches, however, the value of the call option
decreases. If you don't sell the call option, and it expires in three months "out of the
money," you would simply lose the $300 you had spent on the premium.
Option Positions
There are 2 sides to every option contract. One side is the investor who has taken the
long position (i.e. has bought the option). On the other side is the investor who has taken
a short position (i.e. has sold or written the option). The writer of an option receives cash
upfront but has potential liabilities later. His profit or loss is the reverse of that for the
purchaser of the option.
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Long position – investor benefits if price goes up
Short position – investor benefits if price goes down
The holder (buyer) of a call option hopes that the stock price will increase while the
holder (buyer) of a put option hopes that it will decrease. From the holder’s view, a call
option is valuable if the stock price is higher than the strike price while a put option is
valuable if stock price is lower than strike price.
Four basic option positions are possible
 A long position in a call option (holder expects asset price to go up)
 A long position in a put option (holder expects asset price to go down)
 A short position in a call option (writer expects asset price to go down)
 A short position in a put option (writer expects asset price to go up)
OPTION VALUATION
Valuation through Replication
Let the stock price (or any other asset value) today be S0. Assume that the price changes
at discrete time intervals, t=1, t=2, etc. Further assume that the price moves up or down in
each period. Consider a European call option and suppose stock price today, S 0=$50 and
in the next period it either moves up to $60 or down to $40.6 This can be illustrated on a
binomial tree,
S1u  $60
S 0  $50
S1d  $40
If the exercise/ strike price, K=$55, it follows that the payoff from holding the option is
either $5 or zero depending on the price moving up or down7. The value (price) of this
option can be found by creating a portfolio that replicates these future cash flows
(payoffs). Replication means that the investor uses a combination of the underlying asset
and risk-free borrowing to create the same cash flows as the option being valued. The
investor simply buys the stock and simultaneously borrows some amount at the risk-free
rate such that the combination exactly replicates the outcome of the option.
As an illustration, let’s construct a small portfolio of borrowed money and the underlying
asset that replicates the payoff. The questions to be answered are:
1) How many stocks should I buy today?
For simplicity we assume that the stock pays only capital gains – there are no dividends
Payoff = Future price less exercise price. If future price is $60, payoff is $5 and if future price is $40, the
option is out of the money and is valueless
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2) How much money do I need to borrow today?
Let the number of stocks be  shares (i.e. option delta), and let L0 be the amount of
borrowed money today. We have two unknowns and therefore need two equations
involving , L0, and the possible payoffs.
 $60  L0 e rT
 $40  L0 e rT
 5 ………………(1)
 0 ……………....(2)8
Solving the two equations simultaneously gives =0.25 and L0=9.95. This means that
to construct the replicating portfolio we need 0.25 shares and at the same time borrow
$9.95 at the risk free rate.
The next task is to find the value of the call option. We need to buy 0.25 shares today
@$50 per share (i.e. buy 0.25x$50 =$12.50 worth of stocks today) and at the same
time borrow $9.95 at the risk free rate of interest. Therefore,
Value of call = Current market value of stock9 – Borrowing needed to replicate the option
Value of call, C0 =$(12.50 – 9.95) = $2.55
Extending the tree to more periods
Binomial trees for any number of periods (t=T) can also be constructed. The trees are
valued node by node, starting from the last period to the present (t=0) by creating
replicating portfolios at each node, i.e. valuation is done iteratively.
The Binomial Model
We start by assuming that we already know the probabilities of the underlying asset
value going up (pu) or down (1- pu). Let today’s price be S0=$50, and let the price in
the next period depend on the values  and d, where  is the proportional upward
movement in price and d is the proportional downward movement in price. For
example, if the price in the next period is expected to move up by 20% or move down
by 20%, it follows that  = (1+0.20) and d = (1-0.20). If price today, S0 = $50, then
next period the price either moves up to $50(1+0.20) = $60, or it moves down to
$50(1-0.20) =$40. As before, suppose the strike price is K = $55. Below we depict
the stock prices and payoffs, respectively
S1u  S 0   $60
S 0  $50
S1d  S 0 d  $40
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The payoffs are at a future period and therefore the loan (L0) is calculated at its future value, i.e. value of
money next period is L1  L0 e
rT
, where e
rT
is a future value factor. In option valuation we use
continuous compounding and discounting
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Current market value of stock = (current stock price x option delta)
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C1u  $5
C0  ?
C1d  0
The payoff from holding the option is either $5 or zero, depending on the price
moving up or down. Further assume that we know the probability of the stock price
moving up (pu =0.5125). The value of the option is then found by discounting the
future payoff, C1 using the discounting / present value factor, e  rT . 10 In our case,
C1u  $5 , C1d  0 , pu =0.5125. If the risk free rate is 6% and time to expiration of the
option is one month, the value of the option today is,
C 0  e  rT C1
C 0  e  rT [ pu C1u  (1  pu )C1d
 e 0.06 12 [0.5125(5)  (1  0.5125)0]
=$2.55
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Thus, in general the value/ price of an option is found using the formula,
Option value, C 0  e  rT [ pu Cu  (1  pu )C d ] ,
Where C0 is the price of option today
r is the risk free interest rate
T is time to expiration
Cu is the option payoff if stock price moves up
Cd is the option payoff if stock price moves down
Pu is the probability that the stock price moves up
The probabilities of the payoffs in the above example were given. However, these
probabilities are usually not given but can be calculated easily if we know the
movements in the price of the underlying asset. The formula is,
pu 
e rT  d
 d
e 0.06( 12)  0.80
In the above example, pu 
 0.5125
1.20  0.80
1
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The future payoff, C1 is found in the same way as the expected return on an asset under a scenario mean,
i.e. C1  pu (C1 )  (1  pu )(C1 )
u
d
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Extending the tree
So far we have examined a one-period tree. The tree can be expanded to many
periods provided we are given information on  and d. (For example, a year can be
broken down into 2-six-month intervals, 4-three-month intervals, 12-one-month
intervals, etc.) Let  = (1+0.20) and d = (1-0.20)11. If S0 = $50, the stock prices in
two-one-month periods are shown in the table and the binomial tree below. The
second tree gives the payoffs from an option with strike price, K = $55.
Table 1: Stock prices
Interval 1
u
S1  S 0   $60
Interval 2
S 2uu  72
S1d  S 0 d  $40
S 2ud  48
S 2du  48
S 2dd  32
S 2uu  72
S1u  60
S 2ud  S 2du  48
S 0  50
S1d  40
Option Payoffs, K = $55
S 2dd  32
C 2uu  17
C1u  8.669
C 2ud  C 2du  0
C0  4.4206
C1d  0
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C 2dd  0
The value of  and d can be set according to some function of time and volatility of the underlying asset,
 t
i.e.   e
time period
and
d  e 
t
, where  is standard deviation of stock price and t is the length of each
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To find the value of the option, C0, proceed iteratively starting with the last interval
and moving backward in time (find the value node by node) until the current point in
time (node C0). At node C 2uu the payoff is $(72-55) = $17. At nodes C 2ud and C 2dd ,
the option is out of the money and is thrown away. The option is also valueless (out
of the money) at node C1d because node C1d leads either to node C 2ud or C 2dd , and at
both nodes the payoff is zero. The value at node C1u is found using the replication
approach or the binomial model discussed previously. We know that  = (1+0.20)
and d = (1-0.20). As before, suppose r = 0.06, pu = 0.5125, and there are two time
periods of one month each12. Using the binomial model, the option value at node C1u
is;
C1u  e 0.06( 12) [0.5125(17)  (1  0.5125)0] 13
= $8.669
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The next and final task is to value node C0,
C0  e 0.06( 12) [0.5125(8.669)  (1  0.5125)0 ]
= $4.4206
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Valuing Put options
The binomial model is also used to puts, and the procedure is the same as for the call
option. Suppose we recast the most recent example as a put option with a strike price
of $52. The stock price and payoffs are shown below;
S 2uu  72
P2uu  0
S1u  60
S 2ud  S 2du  48
S 0  50
P0  6.7181
P1u  1.94
P2ud  C 2du  4
S1d  40
P1d  11.741
S 2dd  32
P2dd  20
In this case we assume risk neutral valuation such that  and d are the same at each node, the time
intervals/ periods are of the same length (one month each) and the probability is also the same at each node.
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In the case of more than one time interval, T represents the length of each time step/ interval in years
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As usual, valuation starts with the last period, going node by node to the present.
When stock price is $72, the option out of the money and is thrown away – it doesn’t
make sense to buy in the market at $70 and exercise (sell/ put to someone) at $5214. If
the stock price is $48 and $32, the option is in the money and the payoffs are
(52-48 = $4) and (52-32 = $20), respectively.
At node P1u , option value = e 0.06( 12) [0.5125(0)  (1  0.5125)4]
= 1.9403
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At node P1d , option value = e 0.06( 12) [0.5125(4)  (1  0.5125)20]
= 11.741
1
Therefore, value of the Put option today,
1
P0 = e 0.06( 12) [0.5125(1.9403)  (1  0.5125)11.741]
= 6.7181
Extending the tree indefinitely – The Black-Scholes Model
We have argued that the tree can be extended by finding a suitable function for
the ups and downs of the stock price. Think about what would happen if we continue
adding periods into infinity – divide the year into months, weeks, days, etc. Valuation
becomes more complex and tedious as the number of nodes to be valued increase
rapidly. Black and Scholes (1973) derived a single formula for option pricing;
Value of call , C0 = S 0 N (d1 )  Ke  rT N (d 2 ) ,
ln( S0 K )  (r   2 )T
2
where d1 
 T
d 2  d1   T
N(d1) and N(d2) are the cumulative probability distribution functions for standardized
normal distributions, i.e. the probability that a random variable following the normal
distribution, N(0, 1), will take a value less than d1 or d2.
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Recall that a put option gives the right but not the obligation to sell/ put to someone
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From the Black-Scholes model, the variables to consider in valuation are;15
S0 = current value/ price of the underlying asset (stock)
r = risk-free interest rate
2 = variance/ volatility in asset (stock) price
T = time to expiration, in years
K = strike/ exercise price
Table 2 : Determinants of option value
Factor
Increase in price of underlying asset
Increase in exercise price
Increase in variance of underlying asset
Increase in time to expiration
Increase in interest rate
Effect on
Call value
Put value
Positive
Negative
Negative
Positive
Positive
Positive
Positive
Positive
Positive
Negative
Put – Call Parity
It says that there is a relationship between the value of a put and the value of a call
option with the same exercise price and the same expiration date, i.e.
P0  S 0  C 0  Ke  rT ,
where P0 = value of the put
C0 = value of the call
S0 = current value of the underlying asset
Therefore, whenever the value of a call is known, the value of a put with the same
exercise price and expiration date can be easily calculated using above relationship.
From the Put – Call Parity, value of put, P0  C 0  Ke  rT  S 0
Alternatively, the put value is found using another version of the Black-Scholes
formular;
Value of put, P0  Ke  rT [1  N (d 2 )]  S 0 [1  N (d1 )]
Note: The foregoing analysis rests on certain assumptions;
1) There are no dividends paid on stocks – only capital gains
2) The option can only be exercised at a future, fixed date (European type)
Students are encouraged to analyse the case of dividend-paying stocks as well the effect
of early exercise (American type options) on option values.
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Table 2 summarizes the effects of these factors on option value. Students must be able to explain why
the relationships are like this
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