Part-XI
The Binomial Option Pricing Model
Recap of Futures Pricing
• Futures and forward contracts were fairly
easy to value.
• All that we had to do was to identify a
relationship that would preclude both cash
and carry as well as reverse cash and carry
arbitrage.
Recap (Cont…)
• The ease of this result was due to the fact
that a futures/forward contract imposes an
obligation on both the parties to the
agreement.
• Options however are relatively more
complex.
• This is because one party has a right while
the other party has an obligation.
Recap (Cont…)
• Thus as far as the holder of the option is
concerned, he may or may not choose to
exercise his right.
• In the case of European options, the
decision to exercise would depend on
whether ST > X in the case of calls, or
whether ST < X in the case of puts.
Recap (Cont…)
• Consequently we are concerned with the
odds of exercise and the expected payoff at
expiration.
• For American options the issue is even more
complex, for the holder has the right to
exercise at any point in time.
Recap (Cont…)
• Consequently, for options, it matters not
only as to where the stock price is currently,
but also as to how it is expected to evolve.
• Hence, in order to value an option, we have
to postulate a process for the price of the
underlying asset.
• The eventual pricing formula is a function
of the process that is assumed.
Recap (Cont…)
• Some processes will lead to precise
mathematical solutions.
• These are called closed-form solutions.
• In other cases, all that we will get is a
partial differential equation, which has to be
solved by numerical approximation
techniques.
The Binomial Model
• The first model that we will study is called
the Binomial Option Pricing Model
(BOPM).
• This model assumes that given a value for
the stock price, at the end of the next period,
the price can either be up by X% or down
by Y%.
Binomial Model (Cont…)
• Since the stock price can take on only one
of two possible prices subsequently, the
name Binomial is used to describe the
process.
• We will first study the model using a single
time period.
• That is, we will assume that we are at time
T-1, and that the option will expire at T.
The One Period Model
• Let the current stock price be S0.
• At the end of the period, the price ST can be
The One Period Model (Cont…)
One Period (Cont…)
• Y in this case is obviously a negative
number.
• The stock price tree may be depicted as
follows.
The Stock Price Tree
Binomial Model (Cont…)
• Now consider a European call option.
• We will denote the exercise price by E,
since X has already been used to denote an
up movement for the stock price.
• In the case of the Binomial model, we
always start with the expiration time of the
option, since the payoffs at expiration are
readily identifiable.
Binomial (Cont…)
• We will then work backwards.
• Let us denote the call value if the upper
stock price is reached by Cu, and the call
value if the lower stock price is reached by
Cd.
• Cu = Max[0, uS0 – E]
• Cd = Max[0, dS0 – E]
Binomial (Cont…)
• Our objective is to find that value of the call
option one period earlier, that is right now.
• We will denote this unknown value by C0.
A Risk-less Portfolio
• In order to price the option, we will
consider the following investment strategy.
• Let us buy  shares of stock and write one
call option.
• The current value of this portfolio is:
• S0 – C0.
• The negative sign in front of the option
value indicates a short position.
Risk-less Portfolio (Cont…)
• In the up state, the portfolio will have a
value of: uS0 – Cu
• In the down state, the portfolio will have a
value of: dS0 – Cd
• Suppose we were to select  in such a way
that the value of this portfolio is the same in
both the up as well as the down state?
Risk-less Portfolio (Cont…)
• Then this portfolio may be said to be riskless since there are only two possible states
of nature in the next period.
• So if: uS0 – Cu = dS0 – Cd
Risk-less Portfolio (Cont…)
•  is known as the hedge ratio.
• Since the portfolio is risk-less it must earn
the risk-less rate of return.
• Let us define r as 1 + risk-less rate.
• If so:
• uS0 – Cu = dS0 – Cd = (S0 – C0)r
Risk-less Portfolio (Cont…)
• Substituting for , we get:
The Option Premium
• Let us denote (r-d)/(u-d) by p.
• Therefore, (u-r)/(u-d) = 1-p.
• We can then write C0 as:
The Option Premium (Cont…)
• This is the one period binomial option
pricing formula.
• p is known as the Risk Neutral probability.
Numerical Illustration
• Let the current stock price be 100 and the
exercise price of a call option be 100.
• Let there be a possibility of a 20% up move
in the next period and a 20% down move in
the next period.
• Let the risk-less rate be 5% per period.
• Therefore r = 1.05.
Illustration (Cont…)
•
•
•
•
•
p = (1.05 - .8)/(1.2-.8) = .625
1-p = .375
Cu = Max[0, 120 – 100] = 20
Cd = Max[0, 80 – 100] = 0
C0 = .625x20 + .375x0
------------------------------------ =
1.05
11.9048
The Two-Period Situation
• Now let us extend the model to a case
where there are two periods to expiration.
• That is, the option expires at T, whereas we
are standing at T-2.
• We will denote the current stock price by S0
• The stock price tree can be depicted as
follows.
The Stock Price Tree
uuS0
uS0
S0
udS0
dS0
T-2
T-1
ddS0
T
Two Periods (Cont…)
• Once again, we know the payoff from the
option at expiration.
• Let us go back one period, that is to time
• T-1.
• At this point in time the problem is
essentially a one-period problem, to which
we have a solution already.
Two Periods (Cont…)
• Let us denote the option premia
corresponding to values of the stock at time
T, as Cuu, Cud, and Cdd.
• If so, then:
Two Periods (Cont…)
• Knowing Cu and Cd, we can work
backwards to get C0, using an iterative
process.
• This procedure can be extended to any
number of periods.
Numerical Illustration
• Let the current stock price be 100.
• Consider a call option with two periods to
expiration and an exercise price of 100.
• Assume that given a stock price, the price
next period can be 20% more or 20% less.
• Let the risk-less rate of interest be 5%.
• The stock price tree will look as follows.
The Stock Price Tree
144
120
100
96
80
T-2
64
T-1
T
Illustration (Cont…)
•
•
•
•
p = 0.625 and 1-p = .375.
Cuu = Max[0, 144- 100] = 44
Cud = Max[0, 96-100] = 0
Cdd = Max[0,64-100] = 0
Illustration (Cont…)
Impact of Time to Expiration
• As you can see, the value of a two period
call is greater than that of a one period call.
• Obviously, because European call options
always have a non-negative time value.
Pricing European Puts
• We will illustrate the procedure for the one
period case.
• The procedure is similar to the one used for
call options.
• It can easily be extended to the multi-period
case.
European Puts (Cont…)
• Assume that we have a stock with a price of
S0, which can either go up to uS0 or go
down to dS0.
• Consider a one-period put option with an
exercise price of E.
• Pu = Max[0, E – uS0]
• Pd = Max[0, E – dS0]
European Puts (Cont…)
• Using similar arguments, we can show that:
and
European Puts (Cont…)
• p and 1-p, have the same definitions as
before.
Numerical Illustration
•
•
•
•
•
We will use the same data as before.
That is: S0 = E = 100
u = 1.20; d = 0.80; r = 1.05
Pu = Max[0, 100 – 120] = 0
Pd = Max[0,100 – 80] = 20
Illustration (Cont…)
Extension to the Multiperiod
Case
• In the case of a call option with N periods
left to expiration:
Options on Dividend Paying
Stocks
• Whenever a stock pays dividends,
theoretically, the share price should decline
by the amount of the dividend.
• This feature can be inbuilt into the binomial
option pricing model.
• It is critical to know as to when exactly the
dividend will be paid.
Dividends (Cont…)
• Let the stock price be 100, and let there be a
possibility of a 20% increase or a 20%
decline every period.
• Let the risk-less rate of return be 5% per
period.
• Consider a European call option with three
periods to expiration.
Dividends (Cont…)
• The interesting feature is that the stock will
pay a dividend of Rs 16 with one period
remaining to expiration.
• That is, if the option expires at T, then at
T-1, the stock will trade ex-dividend.
• What this means is that the dividend is paid
an instant before the stock trades at T-1.
Dividends (Cont…)
• In other words, the observable price at T-1,
is post-dividend.
• The stock price tree may be modeled as
follows.
The Stock Price Tree
153.6
144
128
102.4
96
120
96
100
80
57.6
80
64
64
64
38.4
T-3
T-2
T-1
T
Dividends (Cont…)
• Notice that because of the dividend, you get
additional branches at time T.
Dividends (Cont…)
Dividends (Cont…)
• Using these values, we can work backwards
to T-2.
Dividends (Cont…)
• Working backwards once again:
American Options
• Now let us extend the model to the pricing
of American calls.
• We will apply the valuation method to the
case of the stock paying dividends.
• The stock price tree will be as depicted
earlier.
American Options (Cont…)
• The major difference, is that, at each node
of the tree, we have to consider as to
whether the option will be exercised (killed)
or kept alive.
• In other words, we must compare the payoff
from exercising the option, to the price
given by the model if the option is kept
alive.
American Options (Cont…)
• Let us go back to time T-1.
• Cuu,T-1 = 32.7619
• If the option is exercised the payoff will be 128 –
100 = 28, which is less.
• So the option will not be exercised early.
• Cud,T-1 = Cdd,T-1 = 0
• Since the option is out of the money in either case,
there is no question of early exercise.
American Options (Cont…)
• Now let us go to T-2.
• Cu,T-2 = 19.5011
• If exercised early, the payoff = 120 –100 =
20.
• So the option will be exercised early.
• Cd,T-2 = 0.
• Since the option is out of the money, there is
no question of early exercise.
American Options (Cont…)
• If we find that at a particular node, the
option will be exercised early, then we
should take the payoff from early exercise
while working backwards to the previous
node, and not the value given by the model.
American Options (Cont…)
• Therefore at T-3, the option value will be
calculated as:
American Options (Cont…)
• Once again you have to check for early
exercise.
• At T-3, the payoff if exercised is
100 –100 = 0, and so there is no question of
early exercise.
American Options (Cont…)
• Thus the option value is 11.9048.
• The option is more valuable than the
corresponding European call, which was
priced at 11.6078.
• This is because the early exercise option is
clearly valuable in this case.
European versus American Puts
• We will consider the same data used for the
earlier example.
• That is, the current stock price is equal to
the exercise price is equal to 100.
• Every period the stock price can increase by
20% or decline by 20%.
• The riskless rate is 5%.
• The stock pays no dividends.
Puts (Cont…)
• Consider puts with 3 periods to expiration.
• The stock price tree can be expressed as
follows.
Stock Price Tree
172.8
144
120
115.2
96
100
80
76.8
64
51.2
Valuing a European Put
European Put (Cont…)
Valuing an American Put
• In this case, at each node, we have to
compare the model value with the intrinsic
value of the option.
• The greater of the two values should be
used for subsequent calculations.
• At uuS0, the model value is zero and so is
the intrinsic value.
American Puts (Cont…)
• At udS0 the model price is 8.2857 and the
intrinsic value is 4.
• So we will take the model value.
• Thus Pu,T-1 is identical for both European as
well as American puts.
• At ddS0 the model price is 31.2381.
• The intrinsic value is however 36.
American Puts (Cont…)
• Therefore:
• Pd,T-1 = .625 x 8.2857 + .375 x 36
-------------------------------1.05
= 17.7891
American Puts (Cont…)
• At dS0, the intrinsic value is 20, which is
greater than 17.7891.
• Therefore:
P0 = .625 x 2.9592 + .375 x 20
------------------------------- = 8.9043
1.05
American Puts (Cont…)
• Even at this last stage, the model value must
be compared with the intrinsic value.
• In this case the intrinsic value is zero.
• So the option will be valued at 8.9043.
• Not surprisingly the American put is valued
at a higher premium than the European put.
Pseudo Probabilities
• We have seen that:
• p = (r – d)
--------(u – d)
• In order for p and (1-p) to be positive, it
must be the case that u > r > d.
• We will demonstrate that if this is not the
case, then one can make arbitrage profits.
Proof
• Assume u > d > r
• Consider the following strategy.
• At T-1, borrow S0 for one period at the
riskless rate and buy a share of stock.
• At T, you will get either uS0 or dS0.
• After repaying the loan with interest you
will get either (u-r)S0 or (d-r)S0, both of
which are positive by assumption.
Proof (Cont…)
• This is clearly a case of arbitrage.
• Therefore we require that r should be
greater than d.
• Now let us consider the other possibility,
that is:
• r>u>d
Proof (Cont…)
• Now consider the following strategy.
• Short sell the stock at T-1 and deposit the
money at the riskless rate.
• At T, you will have to pay either uS0 or dS0
to reacquire the stock.
• The profit is therefore (r-u)S0 or (r-d)S0,
both of which are positive by assumption.
Proof (Cont…)
• Once again this is clearly a case of
arbitrage.
• So in order to preclude both forms of
arbitrage, we require that:
• u>r>d
• Which implies that both p and (1-p) will be
positive.
Pseudo probabilities &
Risk neutrality
• p and (1-p) are referred to as the pseudo
probabilities of reaching the upstate and the
downstate respectively.
• Let us assume that the actual probability of
reaching the upstate is q and that of
reaching the downstate is 1-q.
• As you can see, these values do not appear
anywhere in the pricing equation.
Risk neutrality (Cont…)
• Thus two different people may disagree on
the probabilities of the stock reaching the
upstate and the downstate, and yet they will
agree on the price of the option if this model
is applicable.
• Also notice that risk preferences of
individual investors do not appear in the
pricing equation.
Risk neutrality (Cont…)
• For, we have made no reference to any utility
function.
• Consequently it is irrelevant if the investor is risk
averse, risk seeking or risk neutral.
• So what is p?
• If risk aversion is irrelevant from the standpoint of
pricing then everyone including a risk neutral
investor will agree on the option value that is
calculated.
Risk neutrality (Cont…)
• How will a risk neutral investor value an
option?
• He will determine its expected payoff and
will then discount it at the riskless rate.
• Let us assume that q is the actual
probability of an up move and (1-q) that of
a down move and that we have a world full
of risk neutral investors.
Risk neutrality (Cont…)
• The expected rate of return on the stock is
therefore equal to the riskless rate in this
scenario.
• Thus:
quS0 + (1-q)dS0 – S0
rS0 – S0
--------------------------- = ----------S0
S0
Risk neutrality (Cont…)
• This implies that:
q = (r - d)
--------(u - d)
Which is the same as our p.
• So p is nothing but the probability of an up
move in a world full of identical risk neutral
investors.
Risk neutrality (Cont…)
• As explained earlier, a risk neutral investor
would simply calculate the expected payoff
from the option and discount it at the
riskless rate.
• Thus for him:
C0 = qCu + (1-q)Cd
------------------- =
r
pCu + (1-p)Cd
-----------------r
Risk neutrality (Cont…)
• So the option price according to the
binomial model is the value of the option if
every investor were to have an identical risk
neutral profile.
• Therefore we can pretend as if every
investor is risk neutral and value the option
accordingly.
Risk neutrality (Cont…)
• We are not saying that everyone is risk
neutral.
• What we are saying is that for the purpose
of valuation we can proceed under the
assumption that everyone is risk neutral.
• How will a risk averse person value this
option?
Risk neutrality (Cont…)
• He will use the actual probabilities to
calculate the expected payoff from the
option: qCu + (1-q)Cd
• He will then discount this expected payoff
using his required rate of return.
• The discount rate will be some rate k, which
will not equal the riskless rate.
Risk neutrality (Cont…)
• This is because, being risk averse, he will
demand a risk premium over and above the
riskless rate.
• Thus according to him the value will be:
qCu + (1-q)Cd
----------------k
Risk neutrality (Cont…)
• The problem with this approach is that we do not
know q, nor do we know k.
• However the equivalence with the risk neutral
approach ensures that this value of the option is
the same as that given by the formula:
pCu + (1-p)Cd
-----------------r
Risk neutrality (Cont…)
• In this case, we can easily value this option
since we know u, d, and r, and consequently
p.
• Thus the risk neutral approach simplifies
option valuation.
Exploiting Mispriced Options
• Let us take the one period call.
• The correct value of the option is 11.9048.
• The question is are there arbitrage
opportunities if the call is mispriced, and if
so, how will we exploit them.
Overpriced Calls
• Assume that the call price is 12.50.
• The hedge ratio is:
 = Cu – Cd
20 – 0
----------- = --------------- = 0.50
S0(u-d)
100(1.2-0.8)
Overpriced Calls (Cont…)
• Take the case of an arbitrageur who buys
500 shares and sells 1000 call options.
• His initial investment is:
500 x 100 – 12.5 x 1000 = 37,500
• In the up state, the portfolio will be worth:
500 x 120 – 20 x 1000 = 40000
Overpriced Calls (Cont…)
• In the down state, the portfolio will be worth:
500 x 80 – 0 = 40000
So the portfolio as expected is riskless.
The rate of return is
40,000 – 37,500
------------------- = .0667  6.67%
37,500
Overpriced Calls (Cont…)
• Since the cost of borrowing is only 5%, this
is clearly an arbitrage opportunity.
Underpriced Calls
• What if the call price were to be only 11.00?
• How will an arbitrageur make profits?
• He will short sell 500 shares and buy 1000
calls.
• The net inflow is: 50000 – 11000 = 39000
Underpriced Calls
• At expiration in the up state, the cash flow
is:
• -500 x 120 + 20 x 10000 = -40000
• In the down state the cash flow is:
• -500 x 80 + 0 = -4000
• So the arbitrageur receives 39,000 upfront
and pays 40000 back after one period.
Underpriced Calls
• The cost of borrowing is:
40000 – 39000
------------------ = .0256  2.56%
39000
The initial inflow can be lent out at 5%.
So clearly there is an arbitrage opportunity.
Risk & Risk Neutrality
• Risk neutrality is a critical concept.
• However it is an easily misunderstood
concept.
• Worse, it is an often misapplied concept.
Risk (Cont…)
• What is risk preference?
– Assume that you are offered a game wherein a
coin is tossed.
– If you get a head you get $1.
– If you get a tail you get nothing.
• Will you play this game?
• Obviously it would depend on whether you
have to pay to play, and if so, how much.
Risk (Cont…)
• Let us first assume that you have to pay
nothing.
– Obviously everyone would like to play.
– This is a manifestation of non-satiation which is
the first axiom of utility theory.
– That is, given the opportunity to increase your
wealth without taking any risk, everyone would
go ahead.
Risk (Cont…)
• Now assume that you have to pay $0.50 to
play.
– That is, the cost of playing is equal to the
expected payoff.
– We will assume that the riskless rate is zero.
– Therefore the rate of return from this game is
equal to the riskless rate.
– This is therefore a fair game.
Risk (Cont…)
• The question is will you play?
– Remember that on an average you will only get back
what you put in.
– A risk averse investor will obviously not play.
– For, an expected return equal to the riskless rate will not
compensate him for the risk taken.
– Psychologically, he will be unwilling to face the
disappointment and frustration of losing, if the expected
return is nothing but the riskless rate.
Risk (Cont…)
• A risk averter would therefore be willing to pay
only an amount that is less than the expected
payoff from the game.
– In other words he would demand a positive risk
premium.
• A risk neutral investor would however gladly pay
$0.50.
– He is not bothered by the spectre of a loss.
– He sees it only as a possible aberration in the path to
future gains.
Risk (Cont…)
• A risk lover would go one step further.
– He would actually pay more than $0.50 to play
the game since he gets a thrill from taking on
risk.
Risk and Finance Theory
• In modern Finance we assume that all
investors are risk averse.
– So the price of an asset would be set in such a
way that the investor can expect a return equal
to the riskfree rate plus a risk premium.
– This does not mean that the actual return will
be greater than the riskfree rate.
– It only means that ex-ante one would expect a
positive risk premium.
Risk and Finance Theory
(Cont…)
– Ex-post the rate of return may or may not
exceed the riskless rate.
– However, in the long run, the actual return must
be greater than the riskless rate.
– Else people will simply stop investing.
Cash and Carry- A Revisit
• Consider an investor who buys an asset and
holds it till a future date.
• The spot price of the asset is a given.
• Assume that he takes a short position in a
forward contract to deliver the asset on the
future date.
• If arbitrage is to be ruled out:
F = S(1+r)
Cash and Carry (Cont…)
• Notice that we have not mentioned anything
about the nature of the investor – he may be
risk averse, risk neutral or risk loving.
• Thus, once we have a spot price S, we will
determine F as equal to S(1+r), without
knowing the conditions under which F was
obtained.
Cash and Carry (Cont…)
• If the risk preferences of the investors are
not of consequence for determining the
price of the derivative security we might as
well invoke risk neutrality in order to value
the derivative.
Invariance of Relative Prices or
Absolute Prices
• Let us suppose that on Mars, everyone is
risk neutral.
• However, Mars has a financial market
identical to that of earth and with the same
securities.
• Let us assume that Reliance trades on both
the planets with the same probability
distribution for the future stock price.
Invariance (Cont…)
• On Mars, the expected rate of return on
Reliance will be rf, the riskless rate.
• On Earth where investors are risk averse, it
will be more than rf.
• So the stock price on Earth SE, will be less
than that on Mars, SM.
Invariance (Cont…)
• The price of a forward contract on Earth
will be:
FE = SE x (1+rf)
The price of a forward contract on Mars will
be:
FM = SM x (1+rf)
Obviously FM > FE.
Invariance (Cont…)
• Does this therefore mean that risk preferences are
important for the price of the derivative?
• And that therefore, we are not justified in invoking
risk neutrality?
• The answer is that the ability to invoke risk
neutrality does not imply that risk preferences are
not of consequence for evaluating the absolute
value of a derivative asset.
Invariance (Cont…)
• It only means that risk preferences do not
matter for calculating the price of the
derivative relative to that of the underlying
asset.
• In other words once we know S we can
proceed to find F using the standard
arbitrage arguments.
Invariance (Cont…)
• How S was determined is irrelevant.
• Obviously however, risk preferences would
have had a role in the determination of S.
• Thus although FE  FM
FE
FM
---- = ----- = (1+rf)
SE
SM
Invariance (Cont…)
• Risk neutrality is a convenient tool for solving
complex derivatives pricing problems.
• The use of risk neutrality does not mean that all
investors are risk neutral.
• We are merely invoking risk neutrality to simplify
the analysis.
• We are not assuming that everyone is risk neutral.
Invariance (Cont…)
• Thus while absolute values of derivative
assets are not independent of risk
preferences, the relative price, that is, the
price relative to the price of the underlying
asset, is independent of risk preferences.
Risk Neutral Probabilities
• Assume that S = 100 and X = 100.
• Consider a one period call option in a
binomial world where the quantum of an up
move is 25% and that of a down move is
20%.
• The riskfree rate is 7%.
• Assume that the probability of an up move
is 0.65 while that of a down move is 0.35.
Probabilities (Cont…)
• The expected terminal stock price is:
0.65 x 125 + 0.35 x 80 = 109.25
The expected rate of return is therefore 9.25%
The risk premium is therefore 2.25%.
As per the binomial model:
C0 = pCu + (1-p)Cd
-----------------r
Probabilities (Cont…)
• p = 1.07 – 0.80
--------------- = 0.60
1.25 – 0.80
1-p = 0.40
Therefore:
C0 = .6 x 25 + .4 x 0
------------------- = 14.02
1.07
Probabilities (Cont…)
• We have already identified p as the risk
neutral probability of an up move.
• Consequently by invoking risk neutrality
and using p as the corresponding probability
we can determine the value of the call
option as 14.02.
• It is not just the option whose price is
invariant with respect to these probabilities.
Probabilities (Cont…)
• We can also apply the risk neutral probability to
value the stock:
• S = 0.6 x 125 + 0.4 x 80
------------------------- = 100
1.07
Thus despite the fact that the current stock price
embodies the risk preferences of investors who
may not be risk neutral, we can derive the same
price by invoking risk neutrality.
Probabilities (Cont…)
• Thus both the stock price as well as the
option price are invariant to the
transformation of the probability.
• This transformation of the probability is
called a change of measure.
• A change of measure refers to a shift in the
probabilities that preserves the essential
properties of the probability distribution.
Probabilities (Cont…)
• By essential properties, we mean the
properties required to properly price the
underlying asset and its derivatives.
• It does not mean that the probability of a
particular event or a group of events
occurring is preserved.