Financial Derivatives
Section 5
The Binomial Model of Option Pricing
Michail Anthropelos
[email protected]
http://web.xrh.unipi.gr/faculty/anthropelos/
University of Piraeus
Spring 2014
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Outline
1
One Period Binomial Model
Introduction
Pricing European options
2
Two Periods Binomial Model
Pricing of European options
Pricing of American Options
The Effect of Dividend
3
Construction of a Binomial Tree
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Outline
1
One Period Binomial Model
Introduction
Pricing European options
2
Two Periods Binomial Model
Pricing of European options
Pricing of American Options
The Effect of Dividend
3
Construction of a Binomial Tree
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Option Pricing
What have we seen so far?
Payoffs and P/L functions of options.
Factors that affect the option prices.
Arbitrage-bounds of option prices (with and without dividend).
The next step...
B Is there a way to give a price to an option using only non-arbitrage arguments
as we have done with futures and swaps?
Ans: Well...let’s see.
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The Binomial Model
What is it?
The binomial model is a simplified, discrete time model -developed by Cox, Ross
& Rubinstein in 1979- which:
values the prices of European and American options,
is flexible enough to include dividends.
works as a first step to more complicated market models.
Although very simple...
The binomial model:
? incorporates the essential principles of option pricing, i.e.,
1
2
Pricing by replication
Risk-neutral valuation
? describes in a clear way the effects of all the affecting factors.
? can be used for pricing more complicated derivatives.
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The Stock and the Call
The price of the stock at time t = 0 is S(0). Suppose that at time t = T there
are two possible cases for the stock price:
where, d < 1 + r < u.
The situation of a call option written on this stock is the following:
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Replication Portfolio
The main idea
The main idea is to construct a portfolio which consists of ∆ shares of the stock
and a cash amount B invested in the free-risk interest rate such that:
Portfolio payoff = Call option payoff
Assuming NO ARBITRAGE, the price of the call should be equal to the cost of
the portfolio at time t = 0.
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Replication Portfolio cont’d
We have to choose ∆ and B such that:
∆uS(0) + B(1 + r )
= cu
∆dS(0) + B(1 + r )
= cd
If we suppose that, uS(0) > K and dS(0) < K , we have to solve:
∆uS(0) + B(1 + r )
= uS(0) − K
∆dS(0) + B(1 + r )
=
0
(two equations and two unknowns).
Perfect replication
By doing so, we perfectly replicate the call option payoff.
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The Price of European Call Option
We easily get the following solutions for ∆ and B:
∆=
cu −cd
S(0)(u−d)
B=
ucd −dcu
(u−d)(1+r )
The replication portfolio
The replicating portfolio:
Buy/short sell ∆ stocks.
Lend/borrow the amount
ucd −dcu
(u−d)(1+r ) .
The payoff of this portfolio is exactly the same as the call option payoff.
Price of call = Cost of the portfolio
Under the no arbitrage assumption, the cost of the replicating portfolio,
∆S(0) + B, should be equal to the call price:
c=
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1
1+r
h
((1+r )−d)
cu
u−d
+
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(u−(1+r ))
cd
u−d
i
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Hedging ALL the Risk on Option Positions
Seller side
The strategy of the option seller is the following:
At time t = 0: Sell the option and receive c.
Buy/short sell ∆ stocks.
Borrow/lend B in cash.
At time t = T : Use the amount ∆S(T ) + B(1 + r ) to
give the payoff max{S(T ) − K , 0}.
He starts with zero and ends up with zero (no risk exposure).
Buyer side
Similarly, the strategy of the option buyer is the following:
At time t = 0: Buy/short sell ∆ stocks.
Borrow/lend B in cash.
Buy the option by giving c.
At time t = T : Get the payoff max{S(T ) − K , 0} and use it to
return ∆S(T ) and close the interest free investment.
He starts with zero and ends up with zero(no risk exposure).
Interpretation of Option Price
By shorting an option at price c and following the replication portfolio, there
is no risk exposure.
The subjective probabilities of upward and downward stock movements do
not affect the option price.
The investor’s preference and predictions do not affect the option price.
c is the unique non-arbitrage price. If the price of an option is different than
that the non-arbitrage price, one can make an arbitrage by following the
corresponding replication strategy.
It is clear how each affecting factor influences the option price.
This method can be used for pricing any kind of derivatives written on this
stock.
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Risk-Neutral Valuation
The option price can be written as:
(1 + r ) − d
u − (1 + r )
1
cu
+ cd
.
c=
1+r
u−d
u−d
)−d
u−(1+r )
(1+r )−d
u−(1+r )
Note that (1+r
,
>
0
and
+
= 1.Hence,
u−d
u−d
u−d
u−d
c=
1
1+r [qcu
+ (1 − q)cd ] =
1
1+r Eq
[C (T )]
In other words, we may consider the option price as the discounted expectation of
the option payoff, where
the probability
of the upward movement of the stock price is given by
1+r −d
q = u−d
and
the probability of the downward movement is given by 1 − q = u−1−r
.
u−d
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Risk-Neutral Valuation cont’d
Interpretation of the probability q
If the probability of stock price upward movement is q and the probability of the
stock price downward movement is 1 − q, what is the expected return of the stock
price?
)−d
u−(1+r )
Eq [S(T )] = quS(0) + (1 − q)dS(0) = (1+r
u−d uS(0) + u−d dS(0) =
= S(0)(1 + r ).
This means that under the probabilities (q, 1 − q) the expected return of the stock
price is equal to the risk-free interest rate.
The probability measure (q, 1 − q) is called risk-neutral.
Why is that?
The reason for which the price of the option is equal to the discounted
expectation of the option payoff under the risk-neutral probability is that the
investment on option is risk-free and hence it should have the same expected
return as the risk-free investment.
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Non-Arbitrage and Risk-Neutral Pricing
Synopsis
If the perfect replication of an option payoff is possible, there is no risk
involved in the option positions, as long as the option price is the unique
non-arbitrage price.
No risk exposure means that the expected return of the option is equal to
risk-free interest rate.
By setting the probabilities of upward and downward movement equal to q
and 1 − q respectively, we assume that the world is risk-neutral in which:
I
I
individuals are indifferent to risk.
the expected return of all securities is the risk-free interest rate.
There is in fact an equivalence between the non-arbitrage pricing and the
risk-neutral valuation.
Important notice
If the perfect replication is possible, the risk-neutral valuation holds not only in
the risk-neutral world but also in the real one.
One Period Binomial Model, an example
Suppose that the Microsoft stock price is $28 today. After one year there are two
possible outcomes:
A call option written on the Microsoft stock price, with strike price $28 and
maturity after one year is described below:
One Period Binomial Model, an example
Assume also that r = 5% (annual compounding).
The parameters of the replication portfolio are
∆=
and
B=
2.8
cu − cd
=
= 0.5
S(0)(u − d)
28(1.1 − 0.9)
0.9 × 2.8
ucd − dcu
=−
= −$12.
(u − d)(1 + r )
(1.1 − 0.9)1.05
Hence, the replication strategy is the following:
Borrow $12 at the risk-free rate and
buy 0.5 stocks at the price $28.
The cost of this portfolio is 14 − 12 = $2. This means that c = $2.
)−d
The risk-neutral probabilities are q = (1+r
= 1.05−0.9
u−d
1.1−0.9 = 0.75 and 1 − q = 0.25.
Hence, the call option price can also be given by:
c=
1
1
(qcu + (1 − q)cd ) =
0.75 × 2.8 + 0 = $2.
1.05
1.05
Outline
1
One Period Binomial Model
Introduction
Pricing European options
2
Two Periods Binomial Model
Pricing of European options
Pricing of American Options
The Effect of Dividend
3
Construction of a Binomial Tree
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The Two Period Binomial Model
We take one step ahead and we impose two time periods. At each time, we
assume that the stock price increases at the rate of u or decreases at d.
Similarly, for a call option written on this stock:
Backward Induction
In order the find the value of the call option at each node of the binomial tree, we
work backwards.
First focus on two branches on the upper right:
At the node of Su , replication portfolio equations for ∆ and B are:
∆S(0)u 2 + (1 + r )B
= cuu
∆S(0)ud + (1 + r )B
= cud
ucud −dcuu
cuu −cud
and B = (u−d)(1+r
whose solutions are ∆ = uS(0)(u−d)
).
(note the similarity with the one period case).
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Backward Induction cont’d
The cost of this portfolio at time t = 1, which equals to cu is:
1
(1 + r ) − d
u − (1 + r )
∆uS(0) + B =
cuu +
cud .
1+r
u−d
u−d
The risk-neutral probabilities
As in the one period case, the value of the call option at the node of Su can be
given as the discounted (at time t = 1) expected payoff of call option, under the
)−d
risk-neutral probability q = (1+r
u−d , i.e.,
cu =
1
Eq [c(2) | S(1) = Su ] .
1+r
At the same way, we calculate that:
cd =
1
1
Eq [c(2) | S(1) = Sd ] =
1+r
1+r
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(1 + r ) − d
u−d
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cud +
u − (1 + r )
u−d
cdd .
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The Price of the Option at Time t = 0
After calculating the option value at time t = 1, we can follow exactly the same
method in order to value the option at time t = 0. We encounter the following
part of the binomial tree:
We get:
c
=
=
=
=
1
1
Eq [c(1)] =
[qcu + (1 − q)cd ] =
(1 + r )
(1 + r )
1
[q(qcuu + (1 − q)cud ) + (1 − q)(qcud + (1 − q)cdd )] =
(1 + r )2
h
i
1
q 2 cuu + 2q(1 − q)cud + (1 − q)2 cdd =
2
(1 + r )
1
Eq [c(2)].
(1 + r )2
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Two Period Binomial Model, an example
The possible cases for the Microsoft stock prices are described in the following
tree:
The situation of a call option with strike price $28 is the following:
Two Period Binomial Model, an example
The risk-neutral probability of upward movement:
q = 1.05−0.9
1.1−0.9 = 0.75
The option price at time t = 1
1
cu = 1.05
(0.75 × 5.88) = $4.2 and cd = 0 The option price at time t = 0
1
c = 1.05
(0.75 × 4.2) =
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1
2
1.052 (0.75
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× 5.88) = $3
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The Delta Hedging
For the perfect replication of the call option payoff, we should calculate the
number of the stocks that has to be purchased at each node of the binomial tree.
This is the Delta hedging strategy:
At time t = 1
5.88
∆u = 30.8×0.2
= 0.95 and ∆d = 0 And at time t = 0
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∆=
4.2
28×0.2
= 0.75
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The Case of American Options
The idea
For the price of the American option, we have to take into account the right of
the option buyer to exercise the option at each node of the binomial tree.
Simply, at each node the price of an American option is the maximum between
the price of a corresponding European option and the intrinsic option value.
In other words, for each note t, we have:
Ct = max{ct , S(t) − K } and Pt = max{pt , K − S(t)}
The early exercise
The early exercise of an American option is optimal if the intrinsic value of the
option is larger than the price of the corresponding European option.
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The Case of American Options, an example
Consider again our Microsoft example:
The value of the European put option on this stock with strike $28 is described
below:
The Case of American Options, an example
Note that the risk-neutral probability of upward movement remains the same (it
depends on the stock price and the interest rate, but not on the option payoff ):
q = 1.05−0.9
1.1−0.9 = 0.75
The option price at time t = 1
1
pu = 1.05
(0.75 × 0 + 0.25 × 0.28) = $0.067 and
1
pd = 1.05
(0.75 × 0.28 + 0.25 × 5.32) = $1.467
The option price at time t = 0
p=
1
(0.75
1.05
× 0.067 + 0.25 × 1.467) =
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1
(2
1.052
× 0.75 × 0.25 × 0.28 + 0.252 × 5.32) = $0.4
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The Case of American Options, an example
For the determination of the American put option price at time t = 1, we compare
the two following trees:
The tree of the European put option values:
And the tree of the intrinsic option values:
The Case of American Options, an example
Therefore the price of the American put option at each node of the tree is given
below:
At time t = 0, we compare the intrinsic value (=0 in this example) and the value
Eq [P(1)] = q × 0.067 + (1 − q) × 2.8 = 0.75025.
The early exercise
At the node Sd , it is optimal for the option holder to exercise the put option and
get the payoff of $2.8.
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How Dividend Enters the Tree
As we have seen, on the ex-dividend day, the stock price drops by the dividend
amount.
This decline occurs in two ways:
The dividend is given as a cash amount.
The dividend is given through a continuous dividend yield.
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How Dividend Enters the Tree cont’d
In the case there is a dividend D given at time t = 1, we have:
If the dividend is given at t = 1 through a continuous dividend yield a, the tree
becomes:
Outline
1
One Period Binomial Model
Introduction
Pricing European options
2
Two Periods Binomial Model
Pricing of European options
Pricing of American Options
The Effect of Dividend
3
Construction of a Binomial Tree
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Starting the Construction of a Tree
How to construct a tree
When we start the construction of a binomial tree, we have to specify the
following parameters:
1
2
The upward and the downward movement, i.e. u and d.
The number of the time intervals (periods) which divide the time to maturity.
I
The more time intervals, the more accuracy we get.
The parameters u and d
The parameters u and d match the stock price volatility.
It is recalled that the volatility, σ, is given by:
Var (Stock return for time period ∆t) = σ 2 ∆t.
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u, d and σ
Let µ and σ be the expected (continuous) return and the volatility of the
stock return, respectively.
E[S(∆t)] = S(0)e µ∆t .
If p and 1 − p are the subjective probabilities of the upward and the
downward movement of the stock price, we have:
pS(0)u + (1 − p)S(0)d = S(0)e µ∆t ⇒ p =
e µ∆t − d
.
u−d
Since,
Var (Stock return) = E[(Stock return)2 ] − E[Stock return]2 ,
by taking the equation Var (Stock return) = σ 2 ∆t into account, we have:
pu 2 + (1 − p)d 2 − pu + (1 − p)d 2 = σ 2 ∆t.
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u, d and σ cont’d
The next step is to replace p with
e µ∆t −d
u−d .
µ∆t
We then approximate the quantity e
exponential function:
, using the Taylor expansion of the
e µ∆t ≈ 1 + µ∆t +
(µ∆t)2
.
2
Note however that the term (∆t)2 has negligible size.
This gives (after some further calculations) the following equation:
(1 + µ∆t)(u + d) − ud − (1 + 2µ∆t) = σ 2 ∆t.
One solution of the above equation is the pair:
√
√
σ ∆t
−σ ∆t
u = e
and d = e
The above construction of a binomial tree was proposed by Cox, Ross and
Rubinstein in 1979.
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Facts about the Binomial Tree Construction
F There is a standard and constant relation between u and d:
u ∗ d = 1.
F u and d do not depend on the expected return of the stock price or the
subjective probabilities.
F What matters the most is the volatility of the stock price.
F Volatility and interest rate are constant during the life of the option.
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Binomial Tree and Option Pricing, an example
Let’s consider the following example of American put option pricing:
S(0) = 50, K = 50, r = 10% (continuous compounding),
σ = 40%, T − t = 5 months = 0.417 and ∆t = 1 month = 0.083
Therefore,
u = eσ
√
∆t
= 1.224 and d = e −σ
√
∆t
= 0.891.
And the neutral-risk probability of the upward movement
q=
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e r ∆t −d
u−d
= 0.508.
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Binomial Tree and Option Pricing, an example
The stock price and the American put option prices are given in the graph of the
binomial tree:
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