Option Markets MGMTMFE 406
Binomial Option Pricing (weeks 3 and 4)
Christian Dorion
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During these two lectures (6 hours) we will learn:
I Lecture 3
I option price intuition
I how to build binomial trees
I how to price European & American options using binomial trees
I how to price options with “exotic” payoffs using binomial trees
I how to price path-dependent options using binomial trees
I Lecture 4
I how to price convertible bonds using binomial trees
I how to apply derivatives theory to the operation and valuation of real
investment projects (real options)
I how to use simulation to price American options
2 / 99
Outline
I Pricing Options: Intuitive Comparative Statics 5
II Discrete-Time Option Pricing: The Binomial Model 7
III Call Options 10
A One-Period Binomial Tree 11
The Binomial Solution 19
A Two-Period Binomial Tree 26
Many Binomial Periods 31
Self-Financing Strategy 33
IV Put Options 36
V Path-Dependent Options 41
VI Uncertainty in the Binomial Model 46
VII Risk-Neutral Pricing 52
VIII American Options 65
IX Discrete Dividends 73
X Convertible Bonds 78
XI Real Options 84
XII Least Squares Monte Carlo (LSM) 91
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Outline
I Pricing Options: Intuitive Comparative Statics 5
II Discrete-Time Option Pricing: The Binomial Model 7
III Call Options 10
A One-Period Binomial Tree 11
The Binomial Solution 19
A Two-Period Binomial Tree 26
Many Binomial Periods 31
Self-Financing Strategy 33
IV Put Options 36
V Path-Dependent Options 41
VI Uncertainty in the Binomial Model 46
VII Risk-Neutral Pricing 52
VIII American Options 65
IX Discrete Dividends 73
X Convertible Bonds 78
XI Real Options 84
XII Least Squares Monte Carlo (LSM) 91
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Intuitive Comparative Statics
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Outline
I Pricing Options: Intuitive Comparative Statics 5
II Discrete-Time Option Pricing: The Binomial Model 7
III Call Options 10
A One-Period Binomial Tree 11
The Binomial Solution 19
A Two-Period Binomial Tree 26
Many Binomial Periods 31
Self-Financing Strategy 33
IV Put Options 36
V Path-Dependent Options 41
VI Uncertainty in the Binomial Model 46
VII Risk-Neutral Pricing 52
VIII American Options 65
IX Discrete Dividends 73
X Convertible Bonds 78
XI Real Options 84
XII Least Squares Monte Carlo (LSM) 91
7 / 99
Discrete-Time Option Pricing: The Binomial Model
I Until now, we have looked only at some basic principles of option
pricing
I We examined payoff and profit diagrams, and upper/lower bounds on
option prices
I We saw that with put-call parity we could price a put or a call based on
the prices of the combinations on instruments that make up the
synthetic version of the put or call.
I What we need to be able to do is price a put or a call without the
other instrument.
I In this section, we introduce a simple but powerful means of pricing
an option.
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Discrete-Time Option Pricing: The Binomial Model
(cont’d)
I The approach we take here is called the binomial tree.
I The word “binomial” refers to the fact that there are only two
outcomes at a given point in time: the underlying price can move to
only one of two possible new prices.
I It may appear that this framework oversimplifies things, but the
model can be extended in several ways to encompass a vast array of
possible prices, and trees are used to price derivatives in practice.
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Outline
I Pricing Options: Intuitive Comparative Statics 5
II Discrete-Time Option Pricing: The Binomial Model 7
III Call Options 10
A One-Period Binomial Tree 11
The Binomial Solution 19
A Two-Period Binomial Tree 26
Many Binomial Periods 31
Self-Financing Strategy 33
IV Put Options 36
V Path-Dependent Options 41
VI Uncertainty in the Binomial Model 46
VII Risk-Neutral Pricing 52
VIII American Options 65
IX Discrete Dividends 73
X Convertible Bonds 78
XI Real Options 84
XII Least Squares Monte Carlo (LSM) 91
10 / 99
The Binomial Option Pricing Model
I The binomial option pricing model assumes that, over a period of
time, the price of the underlying asset can only move up or down by a
specified amount—that is, the asset price follows a binomial
distribution:
S
u×S
d×S
prob
abili
ty u
p
probability down
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A Simple Example
I XYZ does not pay dividends and its current price is $41. In one year
the price can be either $59.954 or $32.903, i.e., u = 1.4623 and
d = 0.8025.
I Consider a European call option on the stock of XYZ, with a $40
strike an 1 year to expiration. The continuously compounded risk-free
interest rate is 8%.
I We wish to determine the option price:
S = 41
C =?
uS = 59.954
Cu = max [0,59.954−40] = 19.954
dS = 32.903
Cd = max [0,32.903−40] = 0
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A Simple Example (cont’d)
I Let us try to find a portfolio that mimics the option (replicating
portfolio).
I We have two instruments: shares of stock and a position in bonds
(i.e., borrowing or lending).
I To be specific, we wish to find a portfolio consisting of ∆ shares of
stock and a dollar amount B in borrowing or lending, such that the
portfolio imitates the option whether the stock rises or falls.
I The value of this replicating portfolio at maturity is:{
59.954∆ + e0.08B = 19.954
32.903∆ + e0.08B = 0
(1)
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A Simple Example (cont’d)
I The unique solution of this system of 2 equations with 2 unknowns is
∆ = 0.7376, B =−22.4047, (2)
i.e., buy 0.7376 shares of XYZ and borrow $22.4047 at the risk-free
rate.
I In computing the payoff for the replicating portfolio, we assume that
we sell the shares at the market price and that we repay the borrowed
amount, plus interest.
I Thus, we obtain that the option and the replicating portfolio have the
same payoff: $19.954 if the stock price goes up and $0 if the stock
price goes down.
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A Simple Example (cont’d)
I By the law of one price, positions that have the same payoff should
have the same cost.
I The price of the option must be
C = 0.7376×$41︸ ︷︷ ︸
risky
−$22.4047︸ ︷︷ ︸
risk−free
= $7.839 (3)
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Arbitraging a Mispriced Option
I Suppose that the market price for the option is $8 instead of $7.839
(the option is overpriced).
I We can sell the option and buy a synthetic option at the same time
(buy low and sell high). The initial cash flow is
I and there is no risk at expiration:
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Arbitraging a Mispriced Option (cont’d)
I Suppose that the market price for the option is $7.5 instead of $7.839
(the option is underpriced).
I We can buy the option and sell a synthetic option at the same time
(buy low and sell high). The initial cash flow is
$7.839−$7.5 = $0.339 (4)
and there is no risk at expiration:
Stock Price in 1 Year
$32.903 $59.954
Purchased call $0 $19.954
∆ units of short-sold shares -$24.271 -$44.225
“Lend” 22.4047
(Buy T-bill at t=0, sell at T) $24.271 $24.271
Total payoff $0 $0
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A Remarkable Result
I So far we have not specified the probabilities of the stock going up
and down.
I In fact, probabilities were not used anywhere in the option price
calculations.
I This is a remarkable result: Since the strategy of holding ∆ shares
and B bonds replicates the option whichever way the stock
moves, the probability of an up or down movement in the stock
is irrelevant for pricing the option.
I However, as we will see, the probabilities are linked to the underlying
asset prices through state pricing.
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The Binomial Solution
I Suppose that the stock has a continuous dividend yield of δ, which is
reinvested in the stock. Thus, if you buy one share at time 0 and the
length of a period is h, at time h you will have eδh shares.
I The up and down movements of the stock price reflect the
ex-dividend price. We can write the stock price as uS0 when the
stock goes up and dS0 when the stock goes down. We can represent
the tree for the stock and the option as follows, where C st denotes the
value of the call at date t in state s.
S0
C0
uS0
Cu1
dS0
Cd1
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The Binomial Solution (cont’d)
I If the length of a period is h, the interest factor per period is erh.
I A successful replicating portfolio will satisfy{
∆×S0×u× eδh +B× erh = Cu1
∆×S0×d× eδh +B× erh = Cd1
(5)
I This is a system of two equations in two unknowns ∆ and B. Solving
for ∆ and B gives ∆ = e−δh
Cu1−Cd1
S0(u−d)
B = e−rh uC
d
1−dCu1
u−d
(6)
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The Binomial Solution (cont’d)
I Given the expressions (6) for ∆ and B, we can derive a simple
formula for the value of the option. The cost of creating the option is
the net cash required to buy the shares and bonds. Thus, the cost of
the option is
C0 = ∆S0 +B
= e−rh
(
Cu1
e(r−δ)h−d
u−d +C
d
1
u− e(r−δ)h
u−d
)
= e−rh
(
Cu1 p∗+Cd1 (1−p∗)
)
with p∗ = e
(r−δ)h−d
u−d
(7)
I Note that if we are interested only in the option price, it is not
necessary to solve for ∆ and B; that is just an intermediate step. If
we want to know only the option price, we can use equation (7)
directly.
I Interestingly, p∗ and 1−p∗ look like probabilities
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The Binomial Solution (cont’d)
I p∗ and 1−p∗ are referred to as risk-neutral probabilities. We
NEVER assume that agents are risk-neutral! On the contrary, we
perfectly hedge the call to find its value. We’ll return to the notion of
risk-neutral pricing below.
I Note that if we are interested only in the option price, it is not
necessary to solve for ∆ and B; that is just an intermediate step. If
we want to know only the option price, we can use equation (7)
directly.
I Note that because ∆ is the number of shares in the replicating
portfolio, it can also be interpreted as the sensitivity of the option to
a change in the stock price. If the stock prices changes by $1, then
the option price, ∆S +B, changes by ∆.
I The assumed stock price movements, u and d , should not give rise to
arbitrage opportunities. In particular, we require that
d < e(r−δ)h < u (8)
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Problem 10.23: Suppose that u < e(r−δ)h. Show that
there is an arbitrage opportunity. Now suppose that
d > e(r−δ)h. Show again that there is an arbitrage
opportunity.
I If u < e(r−δ)h, we short a tailed position of the stock and invest the
proceeds at the interest rate. There is an arbitrage opportunity:
t = 0 state = d state = u
Short stock +e−δhS −d×S −u×S
Lend money −e−δhS +e(r−δ)hS +e(r−δ)hS
Total 0 >0 >0
I If d > e(r−δ)h, we buy a tailed position of the stock and borrow at
the interest rate. There is an arbitrage opportunity:
t = 0 state = d state = u
Buy stock −e−δhS +d×S +u×S
Borrow money e−δhS −e(r−δ)hS −e(r−δ)hS
Total 0 >0 >0
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The Binomial Solution, Special Case: δ = 0 and h = 1
I The solution for ∆ and B reduces to∆ =
Cu1−Cd1
S0(u−d)
B = e−r C
d
1 u−Cu1 d
u−d
(9)
I The option price further simplifies to
C0 = ∆S0 +B = e−r
(
Cu1
er −d
u−d +C
d
1
u− er
u−d
)
(10)
24 / 99
Problem 10.1.a: Let S = $100, K = $105, r = 8%
(continuously compounded), T = 0.5, and δ = 0. Let
u = 1.3, d = 0.8, and the number of binomial periods
n = 1. What are the premium, ∆, and B for a European
call?
25 / 99
A Two-Period Binomial Tree
I We can extend the previous example to price a 2-year option,
assuming all inputs are the same as before.
S0 = 41
C0 =?
Su1 = 59.954
Cu1 =?
Sd1 = 32.903
Cd1 =?
Suu2 = 87.669
Cuu2 = 47.669
Sud2 = 48.114
Cud2 = 8.114
Sdd2 = 26.405
Cdd2 = 0
I Note that an up move followed by a down move (Sud2 ) generates the
same stock price as a down move followed by an up move (Sdu2 ). This
is called a recombining tree.
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A Two-Period Binomial Tree (cont’d)
I To price the option when we have two binomial periods, we need to
work backward through the tree.
I Suppose that in period 1 the stock price is Su1 = $59.954. We can use
equation (10) to derive the option price:
Cu1 = e−r
(
Cuu2
er −d
u−d +C
ud
2
u− er
u−d
)
= $23.029 (11)
I Using equations (9), we can also solve for the composition of the
replicating portfolio:
∆ = 1, B =−36.925 (12)
i.e., buy 1 share of XYZ and borrow $36.925 at the risk-free rate,
which costs 1×$59.954−$36.925 = $23.029.
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A Two-Period Binomial Tree (cont’d)
I Suppose that in period 1 the stock price is Sd1 = $32.903. We can use
equation (10) to derive the option price:
Cd1 = e−r
(
Cud2
er −d
u−d +C
dd
2
u− er
u−d
)
= $3.187 (13)
I Using equations (9), we can also solve for the composition of the
replicating portfolio:
∆ = 0.374, B =−9.111 (14)
i.e., buy 0.374 shares of XYZ and borrow $9.111 at the risk-free rate,
which costs 0.374×$32.903−$9.111 = $3.187.
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A Two-Period Binomial Tree (cont’d)
I Move backward now at period 0. The stock price is S0 = 41. We can
use equation (10) to derive the option price:
C0 = e−r
(
Cu1
er −d
u−d +C
d
1
u− er
u−d
)
= $10.737 (15)
I Using equations (9), we can also solve for the composition of the
replicating portfolio:
∆ = 0.734, B =−19.337 (16)
i.e., buy 0.734 shares of XYZ and borrow $19.337 at the risk-free
rate, which costs 0.734×$41−$19.337 = $10.737.
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A Two-Period Binomial Tree (cont’d)
I The two-period binomial tree with the option price at each node as
well as the details of the replicating portfolio is:
S0 = $41
C0 = $10.737
∆ = 0.734
B =−$19.337
Su1 = $59.954
Cu1 = $23.029
∆ = 1
B =−$36.925
Sd1 = $32.903
Cd1 = $3.187
∆ = 0.374
B =−$9.111
Suu2 = $87.669
Cuu2 = $47.669
Sud2 = $48.114
Cud2 = $8.114
Sdd2 = $26.405
Cdd2 = $0
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Many Binomial Periods: Three-Period Example
I Once we understand the two-period option it is straightforward to
value an option using more than two binomial periods.
I Sometimes we may need different notation for
I The number of periods (the number of up or down increments or sets
of arrows) N,
I the time to expiration in years T, and
I the length in years of one increment h.
I For notation, we will try to stick to the convention by which T =
time to expiration in years, n = number of time increments and
h=T/n is the length of each period.
I While I will attempt to use this notation consistently in the notes, you
should always read carefully to understand what T, N and h represent
in a given environment – there is no guarantee of consistent notation
in the real world.
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Many Binomial Periods: Three-Period Example
I Once we understand the two-period option it is straightforward to
value an option using more than two binomial periods.
I The important principle is to work backward through the tree:
S0 = $41
C0 = $12.799
∆ = 0.798
B =−$19.899
Su1 = $59.954
Cu1 = $26.258
∆ = 0.981
B =−$32.580
Sd1 = $32.903
Cd1 = $4.685
∆ = 0.549
B =−$13.390
Suu2 = $87.669
Cuu2 = $50.745
∆ = 1
B =−$36.925
Sud2 = $48.114
Cud2 = $11.925
∆ = 0.956
B =−$34.086
Sdd2 = $26.405
Cdd2 = $0
∆ = 0
B = $0
Suuu3 = $128.198
Cuuu3 = $88.198
Suud3 = $70.356
Cuud3 = $30.356
Sudd3 = $38.612
Cudd3 = $0
Sddd3 = $21.191
Cddd3 = $0
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Self-Financing Strategy
I In the two-period binomial tree example, suppose that the stock
moves from S0 = $41 to Su1 = $59.954.
I The replicating portfolio should be modified as follows:
1. Buy 1−0.734 = 0.266 shares of XYZ (increase the XYZ position from
0.734 shares to 1 share), which costs 0.266×$59.954 = $15.977.
2. Increase borrowing from $19.337× e0.08 = $20.947 to $36.925, which
yields $15.977.
I The amount necessary to buy shares is equal to the amount obtained
from increased borrowing.
I Modifying the portfolio does not require additional cash. Thus, the
replicating portfolio is self-financing.
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Self-Financing Strategy (cont’d)
I If the stock moves from S0 = $41 to Sd1 = $32.903, the replicating
portfolio should be modified as follows:
1. Sell 0.734−0.374 = 0.360 shares of XYZ (decrease the XYZ position
from 0.734 shares to 0.374 shares), which yields
0.360×$32.903 = $11.836.
2. Decrease borrowing from $19.337× e0.08 = $20.947 to $9.111, which
costs $11.836.
I The amount obtained from selling shares is equal to the amount
necessary to decrease borrowing.
I Modifying the portfolio does not require additional cash. Once again,
the replicating portfolio is self-financing.
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Consider a derivative X :
X0 = ∆0S0 +B0
Xu = ∆u(eδhuS0) +Bu
= ∆0(eδhuS) + erhB0
Xd = ∆d(eδhdS0) +Bd
= ∆0(eδhdS) + erhB0
Xuu
Xud
Xdd
where we had obtained ∆0 and B0 solving
∆0(eδhuS0) + erhB0 = Xu
∆0(eδhdS0) + erhB0 = Xd
which we repeat at time h to solve for ∆u and Bu in the up node, and for
∆d and Bd in the down node.
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Outline
I Pricing Options: Intuitive Comparative Statics 5
II Discrete-Time Option Pricing: The Binomial Model 7
III Call Options 10
A One-Period Binomial Tree 11
The Binomial Solution 19
A Two-Period Binomial Tree 26
Many Binomial Periods 31
Self-Financing Strategy 33
IV Put Options 36
V Path-Dependent Options 41
VI Uncertainty in the Binomial Model 46
VII Risk-Neutral Pricing 52
VIII American Options 65
IX Discrete Dividends 73
X Convertible Bonds 78
XI Real Options 84
XII Least Squares Monte Carlo (LSM) 91
36 / 99
Put Options
I We compute put option prices using the same stock price tree and in
the same way as call option prices.
I The only difference with an European put option occurs at expiration:
Instead of computing the price as max [0,S−K ], we use
max [0,K −S].
I Here is a two-period binomial tree for an European put option with a
$40 strike:
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Put Options (cont’d)
S0 = $41
P0 = $3.823
∆ =−0.266
B = $14.749
Su1 = $59.954
Pu1 = $0
∆ = 0
B = $0
Sd1 = $32.903
Pd1 = $7.209
∆ =−0.626
B = $27.814
Suu2 = $87.669
Puu2 = $0
Sud2 = $48.114
Pud2 = $0
Sdd2 = $26.405
Pdd2 = $13.595
I The proof that the replicating portfolio is self-financing is left as an
exercise.
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Put Options (Using the Parity Relationship)
I For non-dividend paying stocks, the basic parity relationship for
European options with the same strike price and time to expiration is
Ct −Pt = St −PV (strike price) (17)
I We can use this relationship to find the put price at all nodes:
S0 = 41
C0 = 10.737
P0 = 3.823
Su1 = 59.954
Cu1 = 23.029
Pu1 = 0
Sd1 = 32.903
Cd1 = 3.187
Pd1 = 7.209
Suu2 = 87.669
Cuu2 = 47.669
Puu2 = 0
Sud2 = 48.114
Cud2 = 8.114
Pud2 = 0
Sdd2 = 26.405
Cdd2 = 0
Pdd2 = 13.595
39 / 99
Problem 10.1.b: Let S = $100, K = $105, r = 8%
(continuously compounded), T = 0.5, and δ = 0. Let
u = 1.3, d = 0.8, and the number of binomial periods
n = 1. What are the premium, ∆, and B for a European
put?
40 / 99
Outline
I Pricing Options: Intuitive Comparative Statics 5
II Discrete-Time Option Pricing: The Binomial Model 7
III Call Options 10
A One-Period Binomial Tree 11
The Binomial Solution 19
A Two-Period Binomial Tree 26
Many Binomial Periods 31
Self-Financing Strategy 33
IV Put Options 36
V Path-Dependent Options 41
VI Uncertainty in the Binomial Model 46
VII Risk-Neutral Pricing 52
VIII American Options 65
IX Discrete Dividends 73
X Convertible Bonds 78
XI Real Options 84
XII Least Squares Monte Carlo (LSM) 91
41 / 99
Non-Recombining Binomial Trees
I We show how to modify our binomial tree to deal with path
dependent options
I We illustrate the ideas with Asian options (also called Average
Rate Options)
I Using a non-recombining binomial tree will mean that at time t = n
there will be 2n states of the world as opposed to n+1 states with
recombining trees
I This causes problems as the number of states is growing exponentially
42 / 99
Asian Option Example
I Let’s go back to the previous example with two periods: S0 = $41,
u = 1.4623, d = 0.8025, K = 40, T = 2, h = 1, and r = 0.08.
I Consider an Asian call option which pays-off the following
non-negative amount at maturity:
GT = max
[
1
3
2∑
k=0
Sk −40,0
]
(18)
I The two-period binomial tree with the option price at each node as
well as the replicating portfolio is:
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Asian Option Example (cont’d)
44 / 99
Asian Option Example (cont’d)
I As an exercise, consider how to price the option whose payoff depends
on the geometric average:
GT = max
( 2∏
k=0
Sk
)1/3
−40,0
 (19)
I See Chapter 14.2 in McDonald.
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Outline
I Pricing Options: Intuitive Comparative Statics 5
II Discrete-Time Option Pricing: The Binomial Model 7
III Call Options 10
A One-Period Binomial Tree 11
The Binomial Solution 19
A Two-Period Binomial Tree 26
Many Binomial Periods 31
Self-Financing Strategy 33
IV Put Options 36
V Path-Dependent Options 41
VI Uncertainty in the Binomial Model 46
VII Risk-Neutral Pricing 52
VIII American Options 65
IX Discrete Dividends 73
X Convertible Bonds 78
XI Real Options 84
XII Least Squares Monte Carlo (LSM) 91
46 / 99
Uncertainty in the Binomial Model
I A natural measure of uncertainty about the stock return is the
annualized standard deviation of the continuously compounded
stock return, which we will denote by σ.
I If we split the year into n periods of length h (so that h = 1/n), the
standard deviation over the period of length h, σh, is (assuming
returns are uncorrelated over time)
σh = σ
√
h (20)
I In other words, the standard deviation of the stock return is
proportional to the square root of time.
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Uncertainty in the Binomial Model (cont’d)
I We incorporate uncertainty into the binomial tree by modeling the
up and down moves of the stock price. Without uncertainty, the
stock price next period must equal:
St+h = Ste(r−δ)h (21)
I To interpret this, without uncertainty, the rate of return on the stock
must be the risk-free rate. Thus, the stock price must rise at the
risk-free rate less the dividend yield, r − δ.
I We now model the stock price evolution as
uSt = Ste(r−δ)he+σ
√
h
dSt = Ste(r−δ)he−σ
√
h
(22)
48 / 99
Uncertainty in the Binomial Model (cont’d)
I We can rewrite this as
u = e(r−δ)h+σ
√
h
d = e(r−δ)h−σ
√
h
(23)
I Return has two parts, one of which is certain [(r − δ)h], and the other
of which is uncertain and generates the up and down stock price
moves
(
σ
√
h
)
.
I Note that if we set volatility equal to zero, we are back to (21) and we
have St+h = uSt = dSt = Ste(r−δ)h. Zero volatility does not mean
that prices are fixed; it means that prices are known in advance.
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Uncertainty in the Binomial Model (cont’d)
I In our example we assumed that u = 1.4623 and d = 0.8025. These
correspond to an annual stock price volatility of 30%:
u = e(0.08−0)×1+0.3×
√
1 = 1.4623
d = e(0.08−0)×1−0.3×
√
1 = 0.8025
(24)
I We will use the equations in (23) to construct binomial trees. This
approach (called the forward tree approach) is very convenient
because it never violates the no arbitrage restriction
d < e(r−δ)h < u
50 / 99
Problem 10.19: For a stock index, S = $100, σ = 30%,
r = 5%, δ = 3%, and T = 3. Let n = 3.
I What is the price of a European call option with a strike of $95?
I What is the price of a European put option with a strike of $95?
51 / 99
Outline
I Pricing Options: Intuitive Comparative Statics 5
II Discrete-Time Option Pricing: The Binomial Model 7
III Call Options 10
A One-Period Binomial Tree 11
The Binomial Solution 19
A Two-Period Binomial Tree 26
Many Binomial Periods 31
Self-Financing Strategy 33
IV Put Options 36
V Path-Dependent Options 41
VI Uncertainty in the Binomial Model 46
VII Risk-Neutral Pricing 52
VIII American Options 65
IX Discrete Dividends 73
X Convertible Bonds 78
XI Real Options 84
XII Least Squares Monte Carlo (LSM) 91
52 / 99
Risk-Neutral Pricing
I There is a probabilistic interpretation of the binomial solution for the
price of an option (equation 7, restated below):
C0 = e−rh
(
Cu1
e(r−δ)h−d
u−d +C
d
1
u− e(r−δ)h
u−d
)
(25)
I The terms e(r−δ)h−du−d and
u−e(r−δ)h
u−d sum to 1 and are both positive (this
follows from inequality 8).
I Thus, we can interpret these terms as probabilities. Equation (7) can
be written as
C0 = e−rh
[
p∗Cu1 + (1−p∗)Cd1
]
(26)
I This expression has the appearance of an expected value discounted
at the risk-free rate. Thus, we will call p∗ ≡ e(r−δ)h−du−d the risk-neutral
probability of an increase in the stock price.
53 / 99
Risk-Neutral Pricing (cont’d)
I Pricing options using risk-neutral probabilities can be done in one
step, no matter how large the number of periods. In our example,
p∗ = 0.4256. For the one-period European call option we have:
Call Price in 1 Year Probability
$19.954 0.4256
$0 0.5744
I The call price is
C0 = e−0.08 (0.4256×$19.954+0.5744×$0) = $7.839 (27)
54 / 99
Risk-Neutral Pricing (cont’d)
I The probability of reaching any given node is the probability of one
path reaching that node times the number of paths reaching that
node. For example, the probability of reaching the node
Sud2 = $48.114 is 2p∗ (1−p∗).
I It can be easily verified that the sum of probabilities in the table
below is 1.
I For the two-period European call option we have:
Call Price in 2 Years Probability
$47.669 p∗2 = 0.1811
$8.114 2p∗ (1−p∗) = 0.4889
$0 (1−p∗)2 = 0.3300
I The call price is
C0 = e−0.08×2 (0.1811×$47.669+0.4889×$8.114+0.3300×$0)
= $10.737
(28)
55 / 99
Risk-Neutral Pricing (cont’d)
I It is left as an exercise to find the price of the two-period European
put using risk-neutral probabilities.
56 / 99
Risk-Neutral Pricing (cont’d)
I For the three-period European call option we have:
Call Price in 3 Years Probability
$88.198 p∗3 = 0.0771
$30.356 3p∗2 (1−p∗) = 0.3121
$0 3p∗ (1−p∗)2 = 0.4213
$0 (1−p∗)3 = 0.1896
I The call price is
C0 = e−0.08×3
3∑
k=0
3!
k! (3−k)!p
∗k (1−p∗)3−k max
[
S0ukd3−k −K ,0
]
= $12.799
(29)
I It can be easily verified that the sum of probabilities in the table
above is 1.
57 / 99
Risk-Neutral Pricing (cont’d)
I For an arbitrary number of periods n, the price of an European call
option is given by
C0 = e−rT
n∑
k=0
n!
k! (n−k)!p
∗k (1−p∗)n−k max
[
S0ukdn−k −K ,0
]
(30)
I We will use this formula later on when talking about Black-Scholes:
when the number of steps becomes great enough the price of the
option appears to approach a limiting value. This value is given by
the Black-Scholes formula.
58 / 99
Problem 11.12: Let S = $100, σ = 0.3, r = 0.08, T = 1,
and δ = 0. Use equation (30) to compute the risk-neutral
probability of reaching a terminal node and the price at
that node for n = 3. Plot the risk-neutral distribution of
year-1 stock prices.
59 / 99
Problem 11.12: Let S = $100, σ = 0.3, r = 0.08, T = 1,
and δ = 0. Use equation (30) to compute the risk-neutral
probability of reaching a terminal node and the price at
that node for n = 3. Plot the risk-neutral distribution of
year-1 stock prices.
For n = 3, u and d are calculated as
follows:
u = e(0.08−0)×1/3+0.3×
√
1/3 = 1.2212
d = e(0.08−0)×1/3−0.3×
√
1/3 = 0.8637
It follows that p∗ = 0.4568, and
n−k Stock price Probability
0 182.14 0.0953
1 128.81 0.3400
2 91.10 0.4044
3 64.43 0.1603
64.43 91.10 128.81 182.14
0.1
0.2
0.3
0.4
59 / 99
Understanding Risk-Neutral Pricing
I A risk-neutral investor is indifferent between a sure thing and a risky
bet with an expected payoff equal to the value of the sure thing.
I A risk-averse investor prefers a sure thing to a risky bet with an
expected payoff equal to the value of the sure thing.
I Formula (26) suggests that we are discounting at the risk-free rate,
even though the risk of the option is at least as great as the risk of
the stock.
I Thus, the option pricing formula, equation (26), can be said to price
options as if investors are risk-neutral.
I Is this option pricing consistent with standard discounted cash
flow calculations?
60 / 99
Understanding Risk-Neutral Pricing (cont’d)
I Assume, as in our example, that a stock does not pay dividends and
the length of a period is 1 year (δ = 0 and h = 1).
Risk-Neutral Pricing
I The risk-free rate is the discount
rate for any asset including the
stock:
S0 = e−r [p∗uS0 + (1−p∗)dS0]
Solving for p∗ gives us
p∗ = e
r −d
u−d
Standard DCF calculation
I Suppose that the continuously
compounded expected return on
the stock is α. If p is the true
probability of the stock going up, p
must be consistent with u, d , and
α.
S0 = e−α [puS0 + (1−p)dS0]
Solving for p gives us
p = e
α−d
u−d
61 / 99
Understanding Risk-Neutral Pricing (cont’d)
Risk-Neutral Pricing
I We discount the option payoff at
the risk-free rate, r .
Standard DCF calculation
I At what rate do we discount the
option payoff? Since an option is
equivalent to holding a portfolio
consisting of ∆ shares of stock and
B bonds, the expected return on
this portfolio is
eγ = S0∆S0∆ +B
eα+ BS0∆ +B
er
where γ is the discount rate for the
option.
I The option price is
C0 = e−r
[
p∗Cu1 + (1−p∗)Cd1
] I The option price is
C0 = e−γ
[
pCu1 + (1−p)Cd1
]
62 / 99
Understanding Risk-Neutral Pricing (cont’d)
I Are these two prices the same? Yes (proof left as an exercise).
I Note that it does not matter whether we have the “correct” value of
α to start with.
I Any consistent pair of α and γ will give the same option price.
I Risk-neutral pricing is valuable because setting α = r results in the
simplest pricing procedure.
63 / 99
Understanding Risk-Neutral Pricing (cont’d)
I We need to emphasize that at no point are we assuming that
investors are risk-neutral.
I On the contrary, we could even say that (e.g) the market-maker is so
risk averse that she choses to perfectly hedge the underlying risk
embedded in the option she just sold (or bought).
I Rather, risk-neutral pricing is an interpretation of the formulas
above.
64 / 99
Outline
I Pricing Options: Intuitive Comparative Statics 5
II Discrete-Time Option Pricing: The Binomial Model 7
III Call Options 10
A One-Period Binomial Tree 11
The Binomial Solution 19
A Two-Period Binomial Tree 26
Many Binomial Periods 31
Self-Financing Strategy 33
IV Put Options 36
V Path-Dependent Options 41
VI Uncertainty in the Binomial Model 46
VII Risk-Neutral Pricing 52
VIII American Options 65
IX Discrete Dividends 73
X Convertible Bonds 78
XI Real Options 84
XII Least Squares Monte Carlo (LSM) 91
65 / 99
American Options
I An European option can only be exercised at the expiration date,
whereas an American option can be exercised at any time.
I Because of this added flexibility, an American option must always be
at least as valuable as an otherwise identical European option:
CAmer (S,K ,T )≥ CEur (S,K ,T )
PAmer (S,K ,T )≥ PEur (S,K ,T )
I Combining these statements, together with the maximum and
minimum option prices (from the first Section), gives us
S ≥ CAmer (S,K ,T )≥ CEur (S,K ,T )≥max [0,S−PV (Div)−PV (K )]
K ≥ PAmer (S,K ,T )≥ PEur (S,K ,T )≥max [0,PV (K )−S +PV (Div)]
66 / 99
American Call
I An American-style call option on a nondividend-paying stock should
never be exercised prior to expiration (proof in class).
I For an American call on a dividend-paying stock it might be beneficial
to exercise the option prior to expiration (by exercising the call, the
owner will be entitled to dividend payments that she would not have
otherwise received).
I Consider the previous example, with one exception: δ = 0.065, i.e.,
XYZ stock has a continuous dividend yield of 6.5% per year.
67 / 99
American Call (cont’d)
S0 = $41
C0 = $8.635
C0,NO = $8.635
C0,EX = $1
Su1 = $56.181
Cu1 = $18.255
Cu1,NO = $18.255
Cu1,EX = $16.181
Sd1 = $30.833
Cd1 = $2.761
Cd1,NO = $2.761
Cd1,EX = $0
Suu2 = $76.982
Cuu2 = $36.982
Cuu2,NO = $35.213
Cuu2,EX = $36.982
Sud2 = $42.249
Cud2 = $7.029
Cud2,NO = $7.029
Cud2,EX = $2.249
Sdd2 = $23.187
Cdd2 = $0
Cdd2,NO = $0
Cdd2,EX = $0
Suuu3 = $105.485
Cuuu3 = $65.485
Suud3 = $57.892
Cuud3 = $17.892
Sudd3 = $31.772
Cudd3 = $0
Sddd3 = $17.437
Cddd3 = $0C·,NO = value of call if not exercised
C·,EX = value of call if exercised
I Because of dividends, early exercise
is optimal at the node where the
stock price is $76.982.
68 / 99
American Put
I When the underlying stock pays no dividend, a call will not be
early-exercised, but a put might be.
I Suppose a company is bankrupt and the stock price falls to zero.
Then a put that would not be exercised until expiration will be worth
PV (K ), which is smaller than K for a positive interest rate.
I Therefore, early exercise would be optimal in order to receive the
strike price earlier.
I By put-call parity:
Pamer ≥ Peuro =−S +Ke−rT +C
The put can be seen as a short position on the stock, with a call
providing insurance. The tension between the value of this insurance
and the time-value of money is at the heart of the decision to exercise
early or not.
69 / 99
American Put (cont’d)
S0 = $41
P0 = $6.546
P0,NO = $6.546
P0,EX = $0
Su1 = $56.181
Pu1 = $2.314
Pu1,NO = $2.314
Pu1,EX = $0
Sd1 = $30.833
Pd1 = $10.630
Pd1,NO = $10.630
Pd1,EX = $9.167
Suu2 = $76.982
Puu2 = $0
Puu2,NO = $0
Puu2,EX = $0
Sud2 = $42.249
Pud2 = $4.363
Pud2,NO = $4.363
Pud2,EX = $0
Sdd2 = $23.187
Pdd2 = $16.813
Pdd2,NO = $15.197
Pdd2,EX = $16.813
Suuu3 = $105.485
Puuu3 = $0
Suud3 = $57.892
Puud3 = $0
Sudd3 = $31.772
Pudd3 = $8.228
Sddd3 = $17.437
Pddd3 = $22.563P·,NO = value of put if not exercised
PC·,EX = value of put if exercised
I Early exercise is optimal at the
node where the stock price is
$23.187.
70 / 99
Problem 10.20: For a stock index, S = $100, σ = 30%,
r = 5%, δ = 3%, and T = 3. Let n = 3.
I What is the price of an American call option with a strike of $95?
I What is the price of an American put option with a strike of $95?
71 / 99
Understanding Early Exercise
I In deciding whether to early-exercise an option, the option holder
compares the value of exercising immediately with the value of
continuing to hold the option.
I Consider the cost and benefits of early exercise for a call option and a
put option. By exercising, the option holder
Call Option
I Receives the stock and therefore
receives future dividends
I Bears the interest cost of paying
the strike price prior to expiration
I Loses the insurance implicit in the
call (the option holder is protected
against the possibility that the
stock price will be less than the
strike price at expiration).
Put Option
I Dividends are lost by giving up the
stock
I Receives the strike price sooner
rather than later
I Loses the insurance implicit in the
put (the option holder is protected
against the possibility that the
stock price will be more than the
strike price at expiration)
72 / 99
Outline
I Pricing Options: Intuitive Comparative Statics 5
II Discrete-Time Option Pricing: The Binomial Model 7
III Call Options 10
A One-Period Binomial Tree 11
The Binomial Solution 19
A Two-Period Binomial Tree 26
Many Binomial Periods 31
Self-Financing Strategy 33
IV Put Options 36
V Path-Dependent Options 41
VI Uncertainty in the Binomial Model 46
VII Risk-Neutral Pricing 52
VIII American Options 65
IX Discrete Dividends 73
X Convertible Bonds 78
XI Real Options 84
XII Least Squares Monte Carlo (LSM) 91
73 / 99
Stocks Paying Discrete Dividends
I It is reasonable to assume that a stock index pays dividends
continuously
I Individual stocks pay dividends in discrete lumps, quarterly or
annually. In addition, over short horizons it is frequently possible to
predict the amount of the dividend
I How should we price an option when the stock will pay a known dollar
dividend (or a known dividend yield) during the life of the option?
I The binomial tree can be adjusted to accommodate this case
74 / 99
Constant Dividend Yield
I A very simple way to incorporate dividends is to assume a constant
dividend yield at the payment date
I Assume that
St+1 =
(
1− δ1{t+1∈D}
)
Stξt+1 (31)
where δ > 0 is the constant dividend yield, D⊆ {1, ...,T} is the set of
dividend dates, ξt+1 ∈ {u,d}, and the variable 1{t+1∈D} takes the
value of 1 if t +1 is a dividend date and 0 otherwise.
I Everything works out the same as before, except for u and d , which
now are defined u = 1/d = eσ
√
h and p∗ = erh−du−d
I An important feature of this model is that the tree for the ex-dividend
stock price St is recombining
75 / 99
Example
Two periods, u = 1/d = 2, S0 = 4, erh = 1.25, δ = 0.25, D= {1}
Exercise: Find the price of an American call with strike K = 2
76 / 99
Discrete Dividends: Selected Readings
I Chapter 11.4 in McDonald
I Schroder, 1988, “Adapting the Binomial Model to Value Options on
Assets with Fixed-Cash Payouts,” Financial Analyst Journal, 44(6),
54-62
77 / 99
Outline
I Pricing Options: Intuitive Comparative Statics 5
II Discrete-Time Option Pricing: The Binomial Model 7
III Call Options 10
A One-Period Binomial Tree 11
The Binomial Solution 19
A Two-Period Binomial Tree 26
Many Binomial Periods 31
Self-Financing Strategy 33
IV Put Options 36
V Path-Dependent Options 41
VI Uncertainty in the Binomial Model 46
VII Risk-Neutral Pricing 52
VIII American Options 65
IX Discrete Dividends 73
X Convertible Bonds 78
XI Real Options 84
XII Least Squares Monte Carlo (LSM) 91
78 / 99
Convertible Bonds: Basics
I Bonds issued by a company where the holder has the option to
exchange the bonds for a certain number (conversion ratio) of
shares of the company’s stock at certain times in the future
I Convertible bonds are typically callable: the issuer has the right to
buy them back at certain times at predetermined prices. There is a
“tug-of-war” going between the issuer and bondholders:
I Bondholders decide whether to hold or convert (maximizing value)
I The issuer decides whether to call (minimizing value)
I See also McDonald (3rd Edition), pages 479-485.
79 / 99
Convertible Bonds: Credit Risk
I Credit risk plays an important role in the valuation of convertible
bonds: if credit risk is ignored, bond prices are overvalued.
I The stock price process can be represented by varying the usual
binomial tree so that at each node there is:
1. A probability of up movement:
pu =
e(r−δ)h−de−λh
u−d (32)
2. A probability of down movement:
pd =
ue−λh− e(r−δ)h
u−d (33)
3. A probability of default:
p0 = 1− e−λh (34)
where the variable λ is the default intensity.
I One can easily verify that pu +pd +p0 = 1.
80 / 99
Convertible Bonds: Example from Hull (8th Edition),
pages 608-611:
I Consider a 9-month (0.75 years) zero-coupon bond issued by
company XYZ with a face value of $100
I Suppose that it can be exchanged for two shares of company XYZ’s
stock at any time during the 9 months
I Assume also that it is callable for $113 at any time
I The initial stock price is $50, its volatility is 30% per annum, there
are no dividends, and the risk-free rate is 5%
I The default intensity λ is 1% per year. Suppose that in the event of a
default the bond is worth $40 (i.e., the “recovery rate” is 40%)
I Evaluate the convertible bond using a 3-period binomial tree
81 / 99
Solution (to be discussed in class)
82 / 99
Convertible Bonds: Extensions
I When interest is paid on the debt, it must be taken into account
I Parameters λ, σ, and r can be functions of time
I The default intensity could be a function of the stock price as well
83 / 99
Outline
I Pricing Options: Intuitive Comparative Statics 5
II Discrete-Time Option Pricing: The Binomial Model 7
III Call Options 10
A One-Period Binomial Tree 11
The Binomial Solution 19
A Two-Period Binomial Tree 26
Many Binomial Periods 31
Self-Financing Strategy 33
IV Put Options 36
V Path-Dependent Options 41
VI Uncertainty in the Binomial Model 46
VII Risk-Neutral Pricing 52
VIII American Options 65
IX Discrete Dividends 73
X Convertible Bonds 78
XI Real Options 84
XII Least Squares Monte Carlo (LSM) 91
84 / 99
Real Option = the right, but not the obligation, to make a
particular business decision
I The application of derivatives theory to the operation and valuation
of real investment projects
I A call option is the right to pay a strike price to receive the present
value of a stream of future cash flows
I An investment project is the right to pay an investment cost to
receive the present value of a future cash flow stream:
Investment Project Call Option
Investment Cost = Strike Price
Present Value of Project = Price of Underlying Asset
85 / 99
Investment Under Uncertainty
I A project requires an initial investment of $100.
⇒ K = 100 (35)
I The project is expected to generate a perpetual cash flow stream,
with a first cash flow $18 in one year, expected to grow at 3%
annually. Assume a discount rate of 15%.
⇒

Perpetual growing annuity⇒ PV = $180.15−0.03 = $150
Static NPV = $150−$100 = $50
Cont. compounded dividend yield δ = ln
(
1+ $18$150
)
= 0.1133
(36)
I The cont. compounded risk-free rate is r = 6.766%. The cash flows
of the project are normally distributed with a volatility of σ = 50%.
⇒
{
u = eσ
√
h = 1.6487
d = e−σ
√
h = 0.6065
(37)
86 / 99
Investment Under Uncertainty (cont.)
I Suppose the project can be delayed: we can decide whether to accept
the project at time 0, 1, or 2.
I Should the project be accepted? If yes, when?
I Trade-off between three factors:
1. Foregone initial cash flow: $18
2. Interest savings: 100(e0.06766−1) = $7
⇒ Loss of $11 if delayed (38)
3. Value of preserved insurance: Is it more than the loss due to
delay?
87 / 99
Binomial tree for project cash flows & project value
$18
PV = $150
$29.68
PV = $247.31
$10.92
PV = $90.98
$48.93
PV = $407.74
$18
PV = $150
$6.62
PV = $55.19
I The risk-neutral probability of the project value increasing in any
period, p∗, is given by
p∗ = e
r−δ−d
u−d = 0.3347 (39)
88 / 99
Value of the investment option
I Notice that the initial value of the project option is $55.80, which is
greater than the static NPV of $50.
I If we invest immediately, the project is worth $50. The ability to wait
increases that value by $5.80.
89 / 99
Real Options in Practice
I The decision about whether and when to invest in a project ∼ call
option
I The ability to shut down, restart, and permanently abandon a project
∼ project + put option
I Strategic options: the ability to invest in projects that may give rise
to future options ∼ compound option
I Flexibility options: the ability to switch between inputs, outputs, or
technologies ∼ rainbow option
90 / 99
Outline
I Pricing Options: Intuitive Comparative Statics 5
II Discrete-Time Option Pricing: The Binomial Model 7
III Call Options 10
A One-Period Binomial Tree 11
The Binomial Solution 19
A Two-Period Binomial Tree 26
Many Binomial Periods 31
Self-Financing Strategy 33
IV Put Options 36
V Path-Dependent Options 41
VI Uncertainty in the Binomial Model 46
VII Risk-Neutral Pricing 52
VIII American Options 65
IX Discrete Dividends 73
X Convertible Bonds 78
XI Real Options 84
XII Least Squares Monte Carlo (LSM) 91
91 / 99
Longstaff and Schwartz (RFS, 2001)
I Consider an American option with final expiration date T
I At any possible exercise time, the holder should compare the payoff
from immediate exercise to the expected value from keeping the
option alive
I The optimal decision is to exercise if the value of immediate exercise
is positive and larger than the expected value of continuation
I In a simulation approach the problem is that we cannot simply use
next period values to determine the expected pathwise value from
continuation (this corresponds to assuming perfect foresight and
would lead to biased price estimates)
I Longstaff and Schwartz suggest estimating the conditional
expectation of the payoff from continuation using the cross-sectional
information in the simulation
92 / 99
Longstaff and Schwartz (RFS, 2001)
I Consider an option that can be exercised at a discrete set of dates
t1, t2, ..., tN
I Such “Bermudan options” are widely traded
I The price of a true American option (which is continuously
exercisable) is obtained by letting the number of exercise dates
increase towards infinity
93 / 99
LSM simple example
I Consider an American put option on a non-dividend paying stock
I The option may be exercised at time 1,2, and 3, where time 3 is the
final expiration date
I Strike is 1.10
I Risk-free rate is 0.06
94 / 99
LSM
Stock Price Paths
Path t0 t1 t2 t3
1 1.00 1.09 1.08 1.34
2 1.00 1.16 1.26 1.54
3 1.00 1.22 1.07 1.03
4 1.00 .93 .97 .92
5 1.00 1.11 1.56 1.52
6 1.00 .76 .77 .90
7 1.00 .92 .84 1.01
8 1.00 .88 1.22 1.34
I Simulate 8 sample paths under the risk-neutral probability measure
I Exercise decision at time 3 is trivial
Cash Flow Matrix
Path t1 t2 t3
1 – ...– –
2 – – –
3 – – .07
4 – – .18
5 – – –
6 – – .20
7 – – .09
8 – – –
95 / 99
LSM
Stock Price Paths
Path t0 t1 t2 t3
1 1.00 1.09 1.08 1.34
2 1.00 1.16 1.26 1.54
3 1.00 1.22 1.07 1.03
4 1.00 .93 .97 .92
5 1.00 1.11 1.56 1.52
6 1.00 .76 .77 .90
7 1.00 .92 .84 1.01
8 1.00 .88 1.22 1.34
I Simulate 8 sample paths under the risk-neutral probability measure
I Exercise decision at time 3 is trivial
Cash Flow Matrix
Path t1 t2 t3
1 – ...– –
2 – – –
3 – – .07
4 – – .18
5 – – –
6 – – .20
7 – – .09
8 – – –
95 / 99
LSM
I X is the stock price at time 2 for the paths that are in-the-money
I Y is the corresponding discounted cash flow if the option is not exercised
I The expected continuation is approximated by the fitted value of a cross-sectional least squares
I Exercise the option if payoff is larger than the discounted expected value of keeping the option alive
Stock Price Paths
Path t0 t1 t2 t3
1 1.00 1.09 1.08 1.34
2 1.00 1.16 1.26 1.54
3 1.00 1.22 1.07 1.03
4 1.00 .93 .97 .92
5 1.00 1.11 1.56 1.52
6 1.00 .76 .77 .90
7 1.00 .92 .84 1.01
8 1.00 .88 1.22 1.34
Cash Flow Matrix
Path t1 t2 t3
1 – ...– –
2 – – –
3 – – .07
4 – – .18
5 – – –
6 – – .20
7 – – .09
8 – – –
Regression & Optimal Exercise
Path Y X EX NO
1 .00× 0.94176 1.08 .02 .0369
2 – – – –
3 .07× 0.94176 1.07 .03 .0461
4 .18× 0.94176 .97 .13 .1176
5 – – – –
6 .20× 0.94176 .77 .33 .1520
7 .09× 0.94176 .84 .26 .1565
8 – – – –
E[Y ] =−1.070+ 2.983X − 1.813X2
Cash Flow Matrix
Path t1 t2 t3
1 – – –
2 – – –
3 – – .07
4 – .13 –
5 – – –
6 – .33 –
7 – .26 –
8 – – –
95 / 99
LSM
I X is the stock price at time 2 for the paths that are in-the-money
I Y is the corresponding discounted cash flow if the option is not exercised
I The expected continuation is approximated by the fitted value of a cross-sectional least squares
I Exercise the option if payoff is larger than the discounted expected value of keeping the option alive
Stock Price Paths
Path t0 t1 t2 t3
1 1.00 1.09 1.08 1.34
2 1.00 1.16 1.26 1.54
3 1.00 1.22 1.07 1.03
4 1.00 .93 .97 .92
5 1.00 1.11 1.56 1.52
6 1.00 .76 .77 .90
7 1.00 .92 .84 1.01
8 1.00 .88 1.22 1.34
Cash Flow Matrix
Path t1 t2 t3
1 – ...– –
2 – – –
3 – – .07
4 – – .18
5 – – –
6 – – .20
7 – – .09
8 – – –
Regression & Optimal Exercise
Path Y X EX NO
1 .00× 0.94176 1.08 .02 .0369
2 – – – –
3 .07× 0.94176 1.07 .03 .0461
4 .18× 0.94176 .97 .13 .1176
5 – – – –
6 .20× 0.94176 .77 .33 .1520
7 .09× 0.94176 .84 .26 .1565
8 – – – –
E[Y ] =−1.070+ 2.983X − 1.813X2
Cash Flow Matrix
Path t1 t2 t3
1 – – –
2 – – –
3 – – .07
4 – .13 –
5 – – –
6 – .33 –
7 – .26 –
8 – – –
95 / 99
LSM
I X is the stock price at time 2 for the paths that are in-the-money
I Y is the corresponding discounted cash flow if the option is not exercised
I The expected continuation is approximated by the fitted value of a cross-sectional least squares
I Exercise the option if payoff is larger than the discounted expected value of keeping the option alive
Stock Price Paths
Path t0 t1 t2 t3
1 1.00 1.09 1.08 1.34
2 1.00 1.16 1.26 1.54
3 1.00 1.22 1.07 1.03
4 1.00 .93 .97 .92
5 1.00 1.11 1.56 1.52
6 1.00 .76 .77 .90
7 1.00 .92 .84 1.01
8 1.00 .88 1.22 1.34
Cash Flow Matrix
Path t1 t2 t3
1 – ...– –
2 – – –
3 – – .07
4 – – .18
5 – – –
6 – – .20
7 – – .09
8 – – –
Regression & Optimal Exercise
Path Y X EX NO
1 .00× 0.94176 1.08 .02 .0369
2 – – – –
3 .07× 0.94176 1.07 .03 .0461
4 .18× 0.94176 .97 .13 .1176
5 – – – –
6 .20× 0.94176 .77 .33 .1520
7 .09× 0.94176 .84 .26 .1565
8 – – – –
E[Y ] =−1.070+ 2.983X − 1.813X2
Cash Flow Matrix
Path t1 t2 t3
1 – – –
2 – – –
3 – – .07
4 – .13 –
5 – – –
6 – .33 –
7 – .26 –
8 – – –
95 / 99
LSM
I X is the stock price at time 2 for the paths that are in-the-money
I Y is the corresponding discounted cash flow if the option is not exercised
I The expected continuation is approximated by the fitted value of a cross-sectional least squares
I Exercise the option if payoff is larger than the discounted expected value of keeping the option alive
Stock Price Paths
Path t0 t1 t2 t3
1 1.00 1.09 1.08 1.34
2 1.00 1.16 1.26 1.54
3 1.00 1.22 1.07 1.03
4 1.00 .93 .97 .92
5 1.00 1.11 1.56 1.52
6 1.00 .76 .77 .90
7 1.00 .92 .84 1.01
8 1.00 .88 1.22 1.34
Cash Flow Matrix
Path t1 t2 t3
1 – ...– –
2 – – –
3 – – .07
4 – – .18
5 – – –
6 – – .20
7 – – .09
8 – – –
Regression & Optimal Exercise
Path Y X EX NO
1 .00× 0.94176 1.08 .02 .0369
2 – – – –
3 .07× 0.94176 1.07 .03 .0461
4 .18× 0.94176 .97 .13 .1176
5 – – – –
6 .20× 0.94176 .77 .33 .1520
7 .09× 0.94176 .84 .26 .1565
8 – – – –
E[Y ] =−1.070+ 2.983X − 1.813X2
Cash Flow Matrix
Path t1 t2 t3
1 – – –
2 – – –
3 – – .07
4 – .13 –
5 – – –
6 – .33 –
7 – .26 –
8 – – –
95 / 99
LSM
I X is the stock price at time 2 for the paths that are in-the-money
I Y is the corresponding discounted cash flow if the option is not exercised
I The expected continuation is approximated by the fitted value of a cross-sectional least squares
I Exercise the option if payoff is larger than the discounted expected value of keeping the option alive
Stock Price Paths
Path t0 t1 t2 t3
1 1.00 1.09 1.08 1.34
2 1.00 1.16 1.26 1.54
3 1.00 1.22 1.07 1.03
4 1.00 .93 .97 .92
5 1.00 1.11 1.56 1.52
6 1.00 .76 .77 .90
7 1.00 .92 .84 1.01
8 1.00 .88 1.22 1.34
Cash Flow Matrix
Path t1 t2 t3
1 – ...– –
2 – – –
3 – – .07
4 – – .18
5 – – –
6 – – .20
7 – – .09
8 – – –
Regression & Optimal Exercise
Path Y X EX NO
1 .00× 0.94176 1.08 .02 .0369
2 – – – –
3 .07× 0.94176 1.07 .03 .0461
4 .18× 0.94176 .97 .13 .1176
5 – – – –
6 .20× 0.94176 .77 .33 .1520
7 .09× 0.94176 .84 .26 .1565
8 – – – –
E[Y ] =−1.070+ 2.983X − 1.813X2
Cash Flow Matrix
Path t1 t2 t3
1 – – –
2 – – –
3 – – .07
4 – .13 –
5 – – –
6 – .33 –
7 – .26 –
8 – – –
95 / 99
LSM
I Repeat the procedure at time 1
I Discount cash-flows back to time 0 and average over all 8 paths⇒ American put = .1144 (European = .0564)
Stock Price Paths
Path t0 t1 t2 t3
1 1.00 1.09 1.08 1.34
2 1.00 1.16 1.26 1.54
3 1.00 1.22 1.07 1.03
4 1.00 .93 .97 .92
5 1.00 1.11 1.56 1.52
6 1.00 .76 .77 .90
7 1.00 .92 .84 1.01
8 1.00 .88 1.22 1.34
Cash Flow Matrix
Path t1 t2 t3
1 – ...– –
2 – – –
3 – – .07
4 – – .18
5 – – –
6 – – .20
7 – – .09
8 – – –
Regression & Optimal Exercise
Path Y X EX NO
1 .00× 0.94176 1.08 .02 .0369
2 – – – –
3 .07× 0.94176 1.07 .03 .0461
4 .18× 0.94176 .97 .13 .1176
5 – – – –
6 .20× 0.94176 .77 .33 .1520
7 .09× 0.94176 .84 .26 .1565
8 – – – –
Cash Flow Matrix
Path t1 t2 t3
1 – – –
2 – – –
3 – – .07
4 – .13 –
5 – – –
6 – .33 –
7 – .26 –
8 – – –
E[Y ] =−1.070+ 2.983X − 1.813X2
Regression & Optimal Exercise
Path Y X EX NO
1 .00× 0.94176 1.09 .01 .0139
2 – – – –
3 – – – –
4 .13× 0.94176 .93 .17 .1092
5 – – – –
6 .33× 0.94176 .76 .34 .2866
7 .26× 0.94176 .92 .18 .1175
8 .00× 0.94176 .88 .22 .1533
E[Y ] = 2.038− 3.335X + 1.356X2
Cash Flow Matrix
Path t1 t2 t3
1 – – –
2 – – –
3 – – .07
4 .17 – –
5 – – –
6 .34 – –
7 .18 – –
8 .22 – –
95 / 99
LSM
I Repeat the procedure at time 1
I Discount cash-flows back to time 0 and average over all 8 paths⇒ American put = .1144 (European = .0564)
Stock Price Paths
Path t0 t1 t2 t3
1 1.00 1.09 1.08 1.34
2 1.00 1.16 1.26 1.54
3 1.00 1.22 1.07 1.03
4 1.00 .93 .97 .92
5 1.00 1.11 1.56 1.52
6 1.00 .76 .77 .90
7 1.00 .92 .84 1.01
8 1.00 .88 1.22 1.34
Cash Flow Matrix
Path t1 t2 t3
1 – ...– –
2 – – –
3 – – .07
4 – – .18
5 – – –
6 – – .20
7 – – .09
8 – – –
Regression & Optimal Exercise
Path Y X EX NO
1 .00× 0.94176 1.08 .02 .0369
2 – – – –
3 .07× 0.94176 1.07 .03 .0461
4 .18× 0.94176 .97 .13 .1176
5 – – – –
6 .20× 0.94176 .77 .33 .1520
7 .09× 0.94176 .84 .26 .1565
8 – – – –
Cash Flow Matrix
Path t1 t2 t3
1 – – –
2 – – –
3 – – .07
4 – .13 –
5 – – –
6 – .33 –
7 – .26 –
8 – – –
E[Y ] =−1.070+ 2.983X − 1.813X2
Regression & Optimal Exercise
Path Y X EX NO
1 .00× 0.94176 1.09 .01 .0139
2 – – – –
3 – – – –
4 .13× 0.94176 .93 .17 .1092
5 – – – –
6 .33× 0.94176 .76 .34 .2866
7 .26× 0.94176 .92 .18 .1175
8 .00× 0.94176 .88 .22 .1533
E[Y ] = 2.038− 3.335X + 1.356X2
Cash Flow Matrix
Path t1 t2 t3
1 – – –
2 – – –
3 – – .07
4 .17 – –
5 – – –
6 .34 – –
7 .18 – –
8 .22 – –
95 / 99
LSM
I Repeat the procedure at time 1
I Discount cash-flows back to time 0 and average over all 8 paths⇒ American put = .1144 (European = .0564)
Stock Price Paths
Path t0 t1 t2 t3
1 1.00 1.09 1.08 1.34
2 1.00 1.16 1.26 1.54
3 1.00 1.22 1.07 1.03
4 1.00 .93 .97 .92
5 1.00 1.11 1.56 1.52
6 1.00 .76 .77 .90
7 1.00 .92 .84 1.01
8 1.00 .88 1.22 1.34
Cash Flow Matrix
Path t1 t2 t3
1 – ...– –
2 – – –
3 – – .07
4 – – .18
5 – – –
6 – – .20
7 – – .09
8 – – –
Regression & Optimal Exercise
Path Y X EX NO
1 .00× 0.94176 1.08 .02 .0369
2 – – – –
3 .07× 0.94176 1.07 .03 .0461
4 .18× 0.94176 .97 .13 .1176
5 – – – –
6 .20× 0.94176 .77 .33 .1520
7 .09× 0.94176 .84 .26 .1565
8 – – – –
Cash Flow Matrix
Path t1 t2 t3
1 – – –
2 – – –
3 – – .07
4 – .13 –
5 – – –
6 – .33 –
7 – .26 –
8 – – –
E[Y ] =−1.070+ 2.983X − 1.813X2
Regression & Optimal Exercise
Path Y X EX NO
1 .00× 0.94176 1.09 .01 .0139
2 – – – –
3 – – – –
4 .13× 0.94176 .93 .17 .1092
5 – – – –
6 .33× 0.94176 .76 .34 .2866
7 .26× 0.94176 .92 .18 .1175
8 .00× 0.94176 .88 .22 .1533
E[Y ] = 2.038− 3.335X + 1.356X2
Cash Flow Matrix
Path t1 t2 t3
1 – – –
2 – – –
3 – – .07
4 .17 – –
5 – – –
6 .34 – –
7 .18 – –
8 .22 – –
95 / 99
LSM
I Repeat the procedure at time 1
I Discount cash-flows back to time 0 and average over all 8 paths⇒ American put = .1144 (European = .0564)
Stock Price Paths
Path t0 t1 t2 t3
1 1.00 1.09 1.08 1.34
2 1.00 1.16 1.26 1.54
3 1.00 1.22 1.07 1.03
4 1.00 .93 .97 .92
5 1.00 1.11 1.56 1.52
6 1.00 .76 .77 .90
7 1.00 .92 .84 1.01
8 1.00 .88 1.22 1.34
Cash Flow Matrix
Path t1 t2 t3
1 – ...– –
2 – – –
3 – – .07
4 – – .18
5 – – –
6 – – .20
7 – – .09
8 – – –
Regression & Optimal Exercise
Path Y X EX NO
1 .00× 0.94176 1.08 .02 .0369
2 – – – –
3 .07× 0.94176 1.07 .03 .0461
4 .18× 0.94176 .97 .13 .1176
5 – – – –
6 .20× 0.94176 .77 .33 .1520
7 .09× 0.94176 .84 .26 .1565
8 – – – –
Cash Flow Matrix
Path t1 t2 t3
1 – – –
2 – – –
3 – – .07
4 – .13 –
5 – – –
6 – .33 –
7 – .26 –
8 – – –
E[Y ] =−1.070+ 2.983X − 1.813X2
Regression & Optimal Exercise
Path Y X EX NO
1 .00× 0.94176 1.09 .01 .0139
2 – – – –
3 – – – –
4 .13× 0.94176 .93 .17 .1092
5 – – – –
6 .33× 0.94176 .76 .34 .2866
7 .26× 0.94176 .92 .18 .1175
8 .00× 0.94176 .88 .22 .1533
E[Y ] = 2.038− 3.335X + 1.356X2
Cash Flow Matrix
Path t1 t2 t3
1 – – –
2 – – –
3 – – .07
4 .17 – –
5 – – –
6 .34 – –
7 .18 – –
8 .22 – –
95 / 99
LSM
I Repeat the procedure at time 1
I Discount cash-flows back to time 0 and average over all 8 paths⇒ American put = .1144 (European = .0564)
Stock Price Paths
Path t0 t1 t2 t3
1 1.00 1.09 1.08 1.34
2 1.00 1.16 1.26 1.54
3 1.00 1.22 1.07 1.03
4 1.00 .93 .97 .92
5 1.00 1.11 1.56 1.52
6 1.00 .76 .77 .90
7 1.00 .92 .84 1.01
8 1.00 .88 1.22 1.34
Cash Flow Matrix
Path t1 t2 t3
1 – ...– –
2 – – –
3 – – .07
4 – – .18
5 – – –
6 – – .20
7 – – .09
8 – – –
Regression & Optimal Exercise
Path Y X EX NO
1 .00× 0.94176 1.08 .02 .0369
2 – – – –
3 .07× 0.94176 1.07 .03 .0461
4 .18× 0.94176 .97 .13 .1176
5 – – – –
6 .20× 0.94176 .77 .33 .1520
7 .09× 0.94176 .84 .26 .1565
8 – – – –
Cash Flow Matrix
Path t1 t2 t3
1 – – –
2 – – –
3 – – .07
4 – .13 –
5 – – –
6 – .33 –
7 – .26 –
8 – – –
E[Y ] =−1.070+ 2.983X − 1.813X2
Regression & Optimal Exercise
Path Y X EX NO
1 .00× 0.94176 1.09 .01 .0139
2 – – – –
3 – – – –
4 .13× 0.94176 .93 .17 .1092
5 – – – –
6 .33× 0.94176 .76 .34 .2866
7 .26× 0.94176 .92 .18 .1175
8 .00× 0.94176 .88 .22 .1533
E[Y ] = 2.038− 3.335X + 1.356X2
Cash Flow Matrix
Path t1 t2 t3
1 – – –
2 – – –
3 – – .07
4 .17 – –
5 – – –
6 .34 – –
7 .18 – –
8 .22 – –
95 / 99
LSM
I Repeat the procedure at time 1
I Discount cash-flows back to time 0 and average over all 8 paths⇒ American put = .1144 (European = .0564)
Stock Price Paths
Path t0 t1 t2 t3
1 1.00 1.09 1.08 1.34
2 1.00 1.16 1.26 1.54
3 1.00 1.22 1.07 1.03
4 1.00 .93 .97 .92
5 1.00 1.11 1.56 1.52
6 1.00 .76 .77 .90
7 1.00 .92 .84 1.01
8 1.00 .88 1.22 1.34
Cash Flow Matrix
Path t1 t2 t3
1 – ...– –
2 – – –
3 – – .07
4 – – .18
5 – – –
6 – – .20
7 – – .09
8 – – –
Regression & Optimal Exercise
Path Y X EX NO
1 .00× 0.94176 1.08 .02 .0369
2 – – – –
3 .07× 0.94176 1.07 .03 .0461
4 .18× 0.94176 .97 .13 .1176
5 – – – –
6 .20× 0.94176 .77 .33 .1520
7 .09× 0.94176 .84 .26 .1565
8 – – – –
Cash Flow Matrix
Path t1 t2 t3
1 – – –
2 – – –
3 – – .07
4 – .13 –
5 – – –
6 – .33 –
7 – .26 –
8 – – –
E[Y ] =−1.070+ 2.983X − 1.813X2
Regression & Optimal Exercise
Path Y X EX NO
1 .00× 0.94176 1.09 .01 .0139
2 – – – –
3 – – – –
4 .13× 0.94176 .93 .17 .1092
5 – – – –
6 .33× 0.94176 .76 .34 .2866
7 .26× 0.94176 .92 .18 .1175
8 .00× 0.94176 .88 .22 .1533
E[Y ] = 2.038− 3.335X + 1.356X2
Cash Flow Matrix
Path t1 t2 t3
1 – – –
2 – – –
3 – – .07
4 .17 – –
5 – – –
6 .34 – –
7 .18 – –
8 .22 – –
95 / 99
Longstaff and Schwartz (RFS, 2001)
I An important point to understand is that the early exercise decision is
based on the fitted regression in which the coefficients are common
on each sample path; the decision is not based on the knowledge of
the future prices along each sample path
I Clement, Lamberton, and Protter (2002) prove that the LSM
algorithm converges (for a given number of basis functions) as
M→∞
I The algorithm can be applied to many realistic cases (jump diffusion
processes, American-Bermuda-Asian options,...)
I See Chapter 19.6 in McDonald or Chapter 26.8 in Hull (8th edition)
96 / 99
Final Thoughts: Is the Binomial Model Realistic?
I Volatility is constant
I There is ample evidence that volatility changes over time
I Large stock price movements do not occur
I It appears that on occasion stocks move by a large amount (jumps)
I Returns are independent over time
I There is strong evidence that stock returns are correlated across time,
with positive correlations at the short to medium term (momentum)
and negative correlation at long horizons (reversal).
I Continuous dividend yield δ
I Stocks pay dividends in discrete lumps, quarterly or annually
I In addition, over short horizons it is frequently possible to predict the
amount of the dividend
I The binomial tree can be adjusted to accommodate this case (see
Chapter 11.4 in McDonald)
97 / 99
I Problem Set 2 will be available by morning on Wed. January 26
98 / 99
Midterm:
Monday February 7, in class
I Length: 2 hours
I Open book, open notes
I Computers, calculators: allowed
I Midterm covers the first 4 lectures
99 / 99