B OND P RICING IN THE M ULTI -P ERIOD B INOMIAL
M ODEL
Szabolcs Sebestyén
[email protected]
Master in Finance
I NVESTMENTS
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Outline
1
The Binomial Model
Basics
The One-Period Binomial Model
Derivative Pricing in the One-Period Binomial Model
The Multi-Period Binomial Model
Replicating Strategies in the Binomial Model
Including Dividends
Pricing Forwards
Pricing Futures
2
Term Structure Models
Binomial Models for the Short Rate
The Cash Account and Pricing of Zero-Coupon Bonds
Pricing of Options on Bonds
Pricing of Bond Forwards
Pricing of Bond Futures
Pricing of Caplets and Floorlets
Pricing of Swaps and Swaptions
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The Binomial Model
Outline
1
The Binomial Model
Basics
The One-Period Binomial Model
Derivative Pricing in the One-Period Binomial Model
The Multi-Period Binomial Model
Replicating Strategies in the Binomial Model
Including Dividends
Pricing Forwards
Pricing Futures
2
Term Structure Models
Binomial Models for the Short Rate
The Cash Account and Pricing of Zero-Coupon Bonds
Pricing of Options on Bonds
Pricing of Bond Forwards
Pricing of Bond Futures
Pricing of Caplets and Floorlets
Pricing of Swaps and Swaptions
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The Binomial Model
Basics
Outline
1
The Binomial Model
Basics
The One-Period Binomial Model
Derivative Pricing in the One-Period Binomial Model
The Multi-Period Binomial Model
Replicating Strategies in the Binomial Model
Including Dividends
Pricing Forwards
Pricing Futures
2
Term Structure Models
Binomial Models for the Short Rate
The Cash Account and Pricing of Zero-Coupon Bonds
Pricing of Options on Bonds
Pricing of Bond Forwards
Pricing of Bond Futures
Pricing of Caplets and Floorlets
Pricing of Swaps and Swaptions
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The Binomial Model
Basics
Basics of the Binomial Model
Assume that the price of a stock at time zero is S0 > 0
At time 1, the stock price either goes up to S1 = uS0 , or drops to
S1 = dS0
Assume that d < u, u > 1 and 0 < d < 1
It is common to have d = 1/u
The probability of the up-move is p, and that of the down-move is
1−p
The one-period interest rate is r, and the gross interest rate is
R ≡ 1+r
Short sales are allowed
No transaction costs
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The Binomial Model
Basics
The General Structure of a Binomial Tree
t=0
t=1
t=2
t=3
u3 S0
u2 S0
u2 dS0
uS0
S0
udS0
ud2 S0
dS0
d2 S 0
d3 S0
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The Binomial Model
Basics
Example of a Binomial Tree
Stock price dynamics when S0 = 100, u = 1.07 and d = 1/u
t=0
t=1
t=2
t=3
122.50
114.49
107
107
100
100
93.46
93.46
87.34
81.63
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The Binomial Model
Basics
Some Questions
How much is an option with a strike price of K at t = 3 worth?
Does the binomial tree provide enough information to answer this
question?
Should the price depend on the utility function of the buyer and/or
the seller?
Will the price depend on the true probability, p, of an up-move in
each period? Should not the fair price be
i
1 Ph
E0 max {S3 − 100, 0} ?
3
R
Assume that at time t = 3 you lose a lot of money if the stock price
is 81.63 and gain a lot if the stock price is 122.5
Could you do anything to eliminate this risk exposure?
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The Binomial Model
Basics
The St. Petersburg Paradox (1)
Consider the following game:
toss a fair coin repeatedly until the first head appears
if the first head appears on the nth toss, you receive $2n
How much would you be willing to pay to play this game?
The expected pay-off is
EP
0 (pay-off) =
∞
∑ 2n Pr
n=1
∞
=
1
∑ 2n 2n
1st head on the nth toss =
=∞
n=1
Would you really pay an infinite amount of money to play this
game?
The fair value of a security should not necessarily be equal to its
expected pay-off
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The Binomial Model
Basics
The St. Petersburg Paradox (2)
Daniel Bernoulli resolved the paradox by introducing an
increasing and concave utility function, u (·)
In particular, when u (·) = log (·), then the expected pay-off
becomes
EP
0 [u (pay-off)] =
∞
1
∑ log (2n ) 2n
n=1
∞
= log (2)
n
<∞
2n
n=1
∑
It seems that an appropriate utility function solves our problem
But whose utility function? The buyer’s or the seller’s?
We’ll see that there is a much simpler way to price securities
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The Binomial Model
The One-Period Binomial Model
Outline
1
The Binomial Model
Basics
The One-Period Binomial Model
Derivative Pricing in the One-Period Binomial Model
The Multi-Period Binomial Model
Replicating Strategies in the Binomial Model
Including Dividends
Pricing Forwards
Pricing Futures
2
Term Structure Models
Binomial Models for the Short Rate
The Cash Account and Pricing of Zero-Coupon Bonds
Pricing of Options on Bonds
Pricing of Bond Forwards
Pricing of Bond Futures
Pricing of Caplets and Floorlets
Pricing of Swaps and Swaptions
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The Binomial Model
The One-Period Binomial Model
Type A and Type B Arbitrage
Earlier definitions of weak and strong arbitrage applied in a
deterministic world
We need more general definitions when considering randomness
Definition (Type A arbitrage)
A type A arbitrage is a security or portfolio that produces immediate positive
reward at t = 0 and has a non-negative value in every state at t = 1. Formally,
a security with initial cost V0 < 0 at t = 0, and time t = 1-value V1 ≥ 0.
Definition (Type B arbitrage)
A type B arbitrage is a security or portfolio that has a non-positive initial
cost, has positive probability of yielding a positive pay-off at t = 1 and zero
probability of yielding a negative pay-off at t = 1. Formally, a security with
initial cost V0 ≤ 0 at t = 0, and V1 ≥ 0 but V1 6= 0.
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The Binomial Model
The One-Period Binomial Model
Arbitrage in the One-Period Binomial Model
Theorem
There is no arbitrage if and only if d < R < u.
Proof.
Assume that R < d < u. Borrow S0 and buy the stock. =⇒ Type B
arbitrage.
Assume now that d < u < R. Short-sell the stock and invest the
proceeds in the cash account. =⇒ Type B arbitrage.
The opposite direction will be seen soon.
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The Binomial Model
Derivative Pricing in the One-Period Binomial Model
Outline
1
The Binomial Model
Basics
The One-Period Binomial Model
Derivative Pricing in the One-Period Binomial Model
The Multi-Period Binomial Model
Replicating Strategies in the Binomial Model
Including Dividends
Pricing Forwards
Pricing Futures
2
Term Structure Models
Binomial Models for the Short Rate
The Cash Account and Pricing of Zero-Coupon Bonds
Pricing of Options on Bonds
Pricing of Bond Forwards
Pricing of Bond Futures
Pricing of Caplets and Floorlets
Pricing of Swaps and Swaptions
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The Binomial Model
Derivative Pricing in the One-Period Binomial Model
Replicating Portfolio
Consider a derivative security with time-0 price C0 and time-1 pay-off
C1 (S1 )
t=0
p
t=1
C1 (S1 )
uS0
Cu
dS0
Cd
S0
1−p
To find C0 , first construct a replicating portfolio: buy x shares and
invest $y in the cash account at t = 1
At time t = 1 the portfolio is worth
uS0 x + Ry = Cu
dS0 x + Ry = Cd
After obtaining x and y, the fair value of the derivative will be
C0 = xS0 + y
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The Binomial Model
Derivative Pricing in the One-Period Binomial Model
Risk-Neutral Pricing
The solution to the system of equations yields
1 R−d
u−R
C0 =
Cu +
Cd =
R u−d
u−d
i
1h
1
=
qCu + (1 − q) Cd = EQ
( C1 )
R
R 0
If there is no arbitrage, then 0 < q < 1 and we call
Q the risk-neutral distribution with probabilities (q, 1 − q)
the above equation risk-neutral pricing
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The Binomial Model
Derivative Pricing in the One-Period Binomial Model
What about p?
How does the price of the derivative security depend on p?
According to the risk-neutral pricing formula, the arbitrage-free
price is independent of p
Can this be possible? Consider the following two stocks:
Stock ABC
p=0.99
110
100
1−p=0.01
90
Stock XYZ
p=0.01
110
100
1−p=0.99
90
The fair price of a call option with strike price K = 100 is the same
for both stocks!
We are asking the wrong question!
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The Binomial Model
Derivative Pricing in the One-Period Binomial Model
Example: Option Pricing
Example
Consider the following one-period binomial tree for the price of a
stock:
t=0
t=1
p
107
100
1−p
93.46
Assume that R = 1.01.
What is the time-0 price of a call option with strike price K = 102?
(Note: the option pay-off at t = 1 is max {S1 − 102, 0}.)
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The Binomial Model
Derivative Pricing in the One-Period Binomial Model
Solution
First, create a replicating portfolio: buy x shares and invest $y in cash at
t = 0.
We choose x and y so that the value of the portfolio at t = 1 equals the option
pay-off at t = 1, i.e.,
107x + 1.01y = 5
93.46x + 1.01y = 0
The solution is x = 0.3692 and y = −34.1649.
The fair/arbitrage-free value of this portfolio is
0.3692 × 100 − 34.1649 × 1 ≈ 2.76.
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The Binomial Model
The Multi-Period Binomial Model
Outline
1
The Binomial Model
Basics
The One-Period Binomial Model
Derivative Pricing in the One-Period Binomial Model
The Multi-Period Binomial Model
Replicating Strategies in the Binomial Model
Including Dividends
Pricing Forwards
Pricing Futures
2
Term Structure Models
Binomial Models for the Short Rate
The Cash Account and Pricing of Zero-Coupon Bonds
Pricing of Options on Bonds
Pricing of Bond Forwards
Pricing of Bond Futures
Pricing of Caplets and Floorlets
Pricing of Swaps and Swaptions
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The Binomial Model
The Multi-Period Binomial Model
Extension to Multi-Period Setting
The multi-period model is just a series of one-period models
sliced together
Hence, all results from the one-period model apply
We just need to multiply the one-period probabilities along
branches to obtain the multi-period probabilities
The T-period probabilities are given by
T k
q (1 − q )T −k
k
where k is the number of up-moves
Risk-neutral pricing can be done backwards as a series of
one-period models
Alternatively, it can be done in a single step:
h
i
1
C0 = T EQ
C
S
(
)
T
T
R 0
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The Binomial Model
The Multi-Period Binomial Model
Example: Multi-Period Option Pricing
Example
Consider the stock price dynamics when S0 = 100, R = 1.01, u = 1.07 and
d = 1/u:
t=0
t=1
t=2
t=3
122.50
114.49
107
107
100
100
93.46
93.46
87.34
81.63
What is the fair price of a European call option with a strike price of K = 100?
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The Binomial Model
The Multi-Period Binomial Model
Solution
The pay-off of the option at time t = 3 is max {ST − 100, 0} = {22.50, 7, 0, 0}.
We can work backwards and calculate pay-offs from a series of one-period binomial trees.
t=0
t=1
t=2
t=3
22.50
15.48
10.23
7
6.57
3.86
2.13
0
0
0
For example,
15.48 =
i
1 h
q × 22.50 + (1 − q) × 7 ,
1.01
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where q =
R−d
= 0.557.
u−d
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The Binomial Model
The Multi-Period Binomial Model
Solution (cont’d)
Alternatively, the time-0 option price can be calculated in a single step as
i
1 Qh
E
max
S
−
100,
0
=
{
}
T
R3 0
i
1 h
= 3 q3 × 22.50 + 3q2 (1 − q) × 7 + 3q (1 − q)2 × 0 + (1 − q)3 × 0 =
R
= 6.57.
C0 =
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The Binomial Model
The Multi-Period Binomial Model
The Impact of Changing the Risk-Free Rate
Consider pricing a European option in a multi-period setting
Calculate the fair price by assuming different values for the
risk-free rate
The final pay-offs remain the same
However, the option price increases when R increases
This seems counterintuitive
Don’t forget that by changing R the risk-neutral probabilities also
change!
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The Binomial Model
The Multi-Period Binomial Model
First Fundamental Theoreom of Asset Pricing
Theorem (First fundamental theoreom of asset pricing)
There exists a risk neutral distribution Q if and only if no arbitrage
opportunities exist. Then the price of any derivative security with time-T
pay-off CT will be
1
CT .
C0 = T EQ
0
R
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The Binomial Model
Replicating Strategies in the Binomial Model
Outline
1
The Binomial Model
Basics
The One-Period Binomial Model
Derivative Pricing in the One-Period Binomial Model
The Multi-Period Binomial Model
Replicating Strategies in the Binomial Model
Including Dividends
Pricing Forwards
Pricing Futures
2
Term Structure Models
Binomial Models for the Short Rate
The Cash Account and Pricing of Zero-Coupon Bonds
Pricing of Options on Bonds
Pricing of Bond Forwards
Pricing of Bond Futures
Pricing of Caplets and Floorlets
Pricing of Swaps and Swaptions
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The Binomial Model
Replicating Strategies in the Binomial Model
Trading Strategies
Let St denote the stock price at time t
Let Bt denote the value of the cash account at time t, and assume
that B0 = 1 so that Bt = Rt
Let xt denote the number of shares held between t − 1 and t for
t = 1, . . . , T
Let yt denote the number of units of cash account held between
t − 1 and t for t = 1, . . . , T
Let θt ≡ (xt , yt ) define the portfolio held
immediately after trading at time t − 1 (so it is known at time t − 1)
immediately before trading at time t
Then θt is a trading strategy, and a random process
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The Binomial Model
Replicating Strategies in the Binomial Model
Self-Financing Trading Strategies
Definition (Value process)
The value process, Vt (θ), associated with a trading strategy
θt = (xt , yt ), is defined by
(
x1 S0 + y1 B0 for t = 0
Vt =
xt St + yt Bt
for t ≥ 1.
Definition (Self-financing tranding strategy)
A self-financing trading strategy is a trading strategy θt = (xt , yt ),
where changes in Vt are due to entirely to trading gains or losses,
rather than the addition or withdrawal of cash funds. Formally, a
self-financing strategy satisfies
Vt = xt+1 St + yt+1 Bt ,
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t = 1, . . . , T − 1.
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The Binomial Model
Replicating Strategies in the Binomial Model
Self-Financing Trading Strategies: A Result
Proposition
If a trading strategy θt is self-financing, then the corresponding value process Vt
satisfies
Vt+1 − Vt = xt+1 St+1 − St + yt+1 Bt+1 − Bt ,
i.e., changes in the portfolio value can only be due to capital gains or losses and not
the injection or withdrawal of funds.
Proof.
By definition, Vt+1 = xt+1 St+1 + yt+1 Bt+1 .
Since θt is self-financing, we have Vt = xt+1 St + yt+1 Bt .
Substitution yields
Vt+1 − Vt = xt+1 St+1 + yt+1 Bt+1 − xt+1 St + yt+1 Bt =
= xt+1 St+1 − St + yt+1 Bt+1 − Bt
for all t ≥ 0.
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The Binomial Model
Replicating Strategies in the Binomial Model
Risk-Neutral Price = Price of Replicating Strategy
We priced securities in the one-period model using a replicating
portfolio, and without needing to define risk-neutral probabilities
The multi-period model allows the same strategy: construct a
self-financing trading strategy that replicates the pay-off of the
security
This is called dynamic replication
The initial cost of this replicating strategy must equal the fair
value of the security, otherwise there is arbitrage opportunity
The dynamic replication price is equal to the price obtained from
using the risk-neutral probabilities and working backwards in the
lattice
At any node, the value of the security is equal to the replicating
portfolio at that node
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The Binomial Model
Replicating Strategies in the Binomial Model
Example: Replicating Strategy for Our Option (1)
Example
Consider again the stock price dynamics, St , when S0 = 100, R = 1.01,
u = 1.07 and d = 1/u, joint with the option prices Ct
t=0
t=1
t=2
t=3
122.50
22.50
114.49
15.48
107
10.23
107
7
100
6.57
100
3.86
93.46
2.13
93.46
0
87.34
0
81.63
0
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The Binomial Model
Replicating Strategies in the Binomial Model
Example: Replicating Strategy for Our Option (2)
Example
The replicating strategy can be obtained from the system of equations:
xt+1 St+1,u + yt+1 Bt+1 = xt+1 uSt + yt+1 Bt R = Ct+1,u
xt+1 St+1,d + yt+1 Bt+1 = xt+1 dSt + yt+1 Bt R = Ct+1,d
For the node (2, 2) we have
x32 × 122.5 + y32 × 1.013 = 22.5
x32 × 107 + y32 × 1.013 = 7
The solution is x32 = 1 and y32 = −97.06. Similarly, for node (1, 1) we obtain
x21 = 0.80 and y21 = −74.83.
The self-financing condition is
xt St + yt Bt = xt+1 St + yt+1 Bt = Ct
2
0.80 × 114.49 − 74.83 × 1.01 = 1 × 114.49 − 94.06 × 1.012 = 15.48
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The Binomial Model
Replicating Strategies in the Binomial Model
Example: Replicating Strategy for Our Option (3)
Example
The lattice with the replicating strategies, [xt , yt ]:
t=0
t=1
100
6.57
[0.60,−53.25]
t=2
107
10.23
[0.80,−74.83]
93.46
2.13
[0.30,−26.11]
114.49
15.48
[1,−97.06]
100
3.86
[0.52,−46.89]
87.34
0
[0,0]
t=3
122.50
22.50
107
7
93.46
0
81.63
0
For example,
0.80 × 107 + (−74.83) × 1.01 = 10.23
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The Binomial Model
Including Dividends
Outline
1
The Binomial Model
Basics
The One-Period Binomial Model
Derivative Pricing in the One-Period Binomial Model
The Multi-Period Binomial Model
Replicating Strategies in the Binomial Model
Including Dividends
Pricing Forwards
Pricing Futures
2
Term Structure Models
Binomial Models for the Short Rate
The Cash Account and Pricing of Zero-Coupon Bonds
Pricing of Options on Bonds
Pricing of Bond Forwards
Pricing of Bond Futures
Pricing of Caplets and Floorlets
Pricing of Swaps and Swaptions
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The Binomial Model
Including Dividends
Dividends in the One-Period Model
Consider the one-period model and assume that the stock pays a
proportional dividend of cS0 at time t = 1
The no-arbitrage condition now becomes d + c < R < u + c
We can use again the replicating portfolio approach to find the fair
price, C0 , of any derivative security with pay-off C1 (S1 ) at time 1
The system of equation is now
uS0 x + cS0 x + Ry = Cu
dS0 x + cS0 x + Ry = Cd
The solution to x and y yields C0 = xS0 + y:
u+c−R
1 R−d−c
C0 =
Cu +
Cd =
R
u−d
u−d
h
i
1
1
=
qCu + 1 − q Cd = EQ
C
1
R
R 0
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The Binomial Model
Including Dividends
Extension to Several Periods
In the multi-period setting we assume a proportional dividend in
every period, so a dividend of cSt is paid time t + 1
Derivative securities can be priced backwards, as a series of
one-period models
Each one-period model has the same risk-neutral probabilities
In practice, dividends are not paid in every period, which makes
the calculation somewhat more difficult
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The Binomial Model
Including Dividends
Risk-Neutral Pricing With Dividends
Without dividends, the fair price of a T-period security is
ST
Q
S0 = E0
RT
If the security pays dividend in each period, the above expression
does not hold
Instead we have
S0 = EQ
0
T
ST
Dt
+
∑
T
R
Rt
t=1
!
where
Dt is the dividend paid at time t
ST is the ex-dividend price at time T
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The Binomial Model
Including Dividends
The Dividend-Paying Security as a Portfolio
The pricing equation is trivial if we observe that dividends and ST
can be viewed as a portfolio of securities
The tth dividend as a separate security has the price
Q Dt
Pt = E0
Rt
The owner of the underlying security holds a “portfolio” of
securities with time-0 value
T
ST
Q
P
+
E
∑ t 0 RT
t=1
But the value of the underlying security is S0
Hence we must have
!
!
T
T
ST
ST
Dt
Q
Q
S0 = E0
Pt = E0
+
+
t
RT t∑
RT t∑
=1
=1 R
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The Binomial Model
Pricing Forwards
Outline
1
The Binomial Model
Basics
The One-Period Binomial Model
Derivative Pricing in the One-Period Binomial Model
The Multi-Period Binomial Model
Replicating Strategies in the Binomial Model
Including Dividends
Pricing Forwards
Pricing Futures
2
Term Structure Models
Binomial Models for the Short Rate
The Cash Account and Pricing of Zero-Coupon Bonds
Pricing of Options on Bonds
Pricing of Bond Forwards
Pricing of Bond Futures
Pricing of Caplets and Floorlets
Pricing of Swaps and Swaptions
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The Binomial Model
Pricing Forwards
Forward Pricing
Consider a forward contract on the stock that expires after T
periods
Let G0 denote the time-0 “price” of the contract
Recall that G0 is chosen so that the contract is initially worth zero
Hence we have
0=
EQ
0
ST − G0
RT
Or, equivalently, G0 = EQ
0 ( ST )
This equation holds regardless of whether the underlying security
pays dividend
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The Binomial Model
Pricing Futures
Outline
1
The Binomial Model
Basics
The One-Period Binomial Model
Derivative Pricing in the One-Period Binomial Model
The Multi-Period Binomial Model
Replicating Strategies in the Binomial Model
Including Dividends
Pricing Forwards
Pricing Futures
2
Term Structure Models
Binomial Models for the Short Rate
The Cash Account and Pricing of Zero-Coupon Bonds
Pricing of Options on Bonds
Pricing of Bond Forwards
Pricing of Bond Futures
Pricing of Caplets and Floorlets
Pricing of Swaps and Swaptions
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The Binomial Model
Pricing Futures
What is a Futures “Price”?
Consider a futures contract on the stock that expires after T
periods
Let Ft be the time-t “price” of the futures contract for t = 0, 1, . . . , T
Then FT = ST
A common misunderstanding is that
Ft indicates how much one has to pay to buy the contract
or how much one receives when selling the contract
A futures contract always costs nothing
The “price” Ft only determines the cash flow associated with
holding the contract
± (Ft − Ft−1 ) is the pay-off at time t from a long/short position of
one contract held between t − 1 and t
Hence, a future contract can be characterised as a security that
is always worth zero
pays a “dividend” of Ft − Ft−1 at every time t
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The Binomial Model
Pricing Futures
Futures Pricing (1)
The time-T − 1 futures price, FT−1 , is the solution of
FT − FT−1
0 = EQ
T −1
R
which implies FT−1 = EQ
T −1 (FT )
In general, we have Ft = EQ
t (Ft+1 ) for t = 0, 1, . . . , T − 1 so that
Ft = EQ
t (Ft+1 ) =
h
i
Q
= EQ
E
F
=
t
+
2
t
t+1
..
.
=
Sebestyén (ISCTE-IUL)
EQ
t
EQ
t+1
h
· · · EQ
T −1
Bond Pricing
FT
i
Investments
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The Binomial Model
Pricing Futures
Futures Pricing (2)
The law of iterated expectations implies that Ft = EQ
t (FT )
Hence, the futures price is a Q-martingale
Taking t = 0 and given that FT = ST we also have that
F0 = EQ
0 ( ST )
This equation holds regardless of dividend payment, as dividends
enter only through Q
Comparing the pricing equations for forwards and futures, we
observe that F0 = G0
This is not true in general
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Term Structure Models
Outline
1
The Binomial Model
Basics
The One-Period Binomial Model
Derivative Pricing in the One-Period Binomial Model
The Multi-Period Binomial Model
Replicating Strategies in the Binomial Model
Including Dividends
Pricing Forwards
Pricing Futures
2
Term Structure Models
Binomial Models for the Short Rate
The Cash Account and Pricing of Zero-Coupon Bonds
Pricing of Options on Bonds
Pricing of Bond Forwards
Pricing of Bond Futures
Pricing of Caplets and Floorlets
Pricing of Swaps and Swaptions
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Term Structure Models
Binomial Models for the Short Rate
Outline
1
The Binomial Model
Basics
The One-Period Binomial Model
Derivative Pricing in the One-Period Binomial Model
The Multi-Period Binomial Model
Replicating Strategies in the Binomial Model
Including Dividends
Pricing Forwards
Pricing Futures
2
Term Structure Models
Binomial Models for the Short Rate
The Cash Account and Pricing of Zero-Coupon Bonds
Pricing of Options on Bonds
Pricing of Bond Forwards
Pricing of Bond Futures
Pricing of Caplets and Floorlets
Pricing of Swaps and Swaptions
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Term Structure Models
Binomial Models for the Short Rate
Introduction
Fixed-income models are inherently more complex than security
models, since we need to model the evolution of the entire term
structure of interest rates
The short rate, rt , is the key variable in fixed-income models
it is the risk-free rate that applies between t and t + 1
it is a random process, but rt is known at time t
We will specify risk-neutral probabilities for the short rate without
any reference to the true probabilities
This is in contrast to the binomial model for stocks where p was
specified and, by using replication arguments, we obtained q
We will price fixed-income derivatives via the no-arbitrage
argument
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Term Structure Models
Binomial Models for the Short Rate
Short Rate Lattice and Zero-Coupon Bonds
Our basic securities are zero-coupon bonds (ZCB); let ZTt,j denote the
price of a ZCB at time t, state j, that matures at time T
It would be nice to construct a binomial model by specifying all ZTt,j ’s at
all nodes, but it is cumbersome if we want to ensure no-arbitrage
Instead, we will specify the short rate, rt,j , at each node Nt,j , which is the
risk-free rate that applies to the next period
t=0
t=1
t=2
t=3
r3,3
r2,2
r1,1
r3,2
r0,0
r2,1
r3,1
r1,0
r2,0
r3,0
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Term Structure Models
Binomial Models for the Short Rate
Risk-Neutral Pricing in the Short Rate Model
Let Zt,j be the time-t, state-j price of a non-coupon-paying security
Let qu and qd denote the risk-neutral probabilities of an up- and
down-move at each node, with qu , qd > 0 and qu + qd = 1
The risk-neutral pricing of the above security yields
Zt,j =
1
qu Zt+1,j+1 + qd Zt+1,j
1 + rt,j
If the security pays a “coupon”, Ct+1,j , at time t + 1 and state j, its
price becomes
i
1 h
Zt,j =
qu Zt+1,j+1 + Ct+1,j+1 + qd Zt+1,j + Ct+1,j
1 + rt,j
where Zt+1,. is the ex-coupon price at time t + 1
Pricing with any of the two above equations rules out arbitrage
It is common to assume qu = qd = 1/2
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Term Structure Models
The Cash Account and Pricing of Zero-Coupon Bonds
Outline
1
The Binomial Model
Basics
The One-Period Binomial Model
Derivative Pricing in the One-Period Binomial Model
The Multi-Period Binomial Model
Replicating Strategies in the Binomial Model
Including Dividends
Pricing Forwards
Pricing Futures
2
Term Structure Models
Binomial Models for the Short Rate
The Cash Account and Pricing of Zero-Coupon Bonds
Pricing of Options on Bonds
Pricing of Bond Forwards
Pricing of Bond Futures
Pricing of Caplets and Floorlets
Pricing of Swaps and Swaptions
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Term Structure Models
The Cash Account and Pricing of Zero-Coupon Bonds
The Cash Account
The cash account is a special security that in each period earns an
interest at the short rate
It is denoted by Bt and assume that B0 = 1; it satisfies
Bt = 1 + r0,0
1 + r1 · · · 1 + rt−1
so that
Bt
1
=
Bt+1
1 + rt
The cash account is not risk free because Bt+s is unknown at time t
for any s > 1
However, it is locally risk free as Bt+1 is known at time t
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Term Structure Models
The Cash Account and Pricing of Zero-Coupon Bonds
Risk-Neutral Pricing with the Cash Account (1)
Risk-neutral pricing for a non-coupon-paying security is then
1
qu Zt+1,j+1 + qd Zt+1,j =
1 + rt,j
Bt
Zt + 1
Q
Q
= Et
= Et
Zt+1
1 + rt,j
Bt+1
Zt,j =
Or, equivalently,
Zt
= EQ
t
Bt
Zt+1
Bt + 1
Iterating the above equation results in
Zt
Q Zt + s
= Et
Bt
Bt+s
for any s > 0
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Term Structure Models
The Cash Account and Pricing of Zero-Coupon Bonds
Risk-Neutral Pricing with the Cash Account (2)
Risk-neutral pricing for a coupon-paying takes the form
i
1 h
Zt,j =
qu Zt+1,j+1 + Ct+1,j+1 + qd Zt+1,j + Ct+1,j =
1 + rt,j
Q Zt+1 + Ct+1
= Et
1 + rt,j
After rewriting it we obtain
Zt
= EQ
t
Bt
Zt+1 Ct+1
+
Bt+1
Bt+1
Iteration yields
Zt
= EQ
t
Bt
Sebestyén (ISCTE-IUL)
t+s
Zt+s
C
+ ∑ i
Bt+s i=t+1 Bi
Bond Pricing
!
Investments
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Term Structure Models
The Cash Account and Pricing of Zero-Coupon Bonds
Example of a Short-Rate Lattice
Short-rate lattice with r0,0 = 6%, u = 1.25 and d = 0.9:
t=0
t=1
t=3
t=2
t=4
t=5
18.31%
14.65%
13.18%
11.72%
10.55%
9.38%
7.5%
6%
8.44%
6.75%
9.49%
7.59%
5.4%
6.08%
4.86%
6.83%
5.47%
4.37%
4.92%
3.94%
3.54%
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Term Structure Models
The Cash Account and Pricing of Zero-Coupon Bonds
Pricing a 4-Period ZCB
t=0
t=1
t=2
t=3
t=4
100
89.51
83.08
79.27
100
92.22
87.35
77.22
84.43
100
94.27
90.64
100
95.81
100
For example,
83.08 =
Sebestyén (ISCTE-IUL)
1
(0.5 × 89.51 + 0.5 × 92.22)
1 + 0.0938
Bond Pricing
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Term Structure Models
The Cash Account and Pricing of Zero-Coupon Bonds
Compute the Term Structure
One can compute the term structure by ZCB’s of every maturity
and then backing out the spot rates for those maturities
For example, assuming per period compounding,
77.22 (1 + s4 )4 = 100
⇐⇒
s4 = 6.68%
Hence, first we compute Z10 , Z20 , Z30 , and Z40 , and then we can
calculate s1 , s2 , s3 , and s4
At t = 1 we will compute new ZCB prices and obtain a new term
structure
Therefore, the model for the short rate rt (a random variable)
defines a model for the term structure
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Term Structure Models
Pricing of Options on Bonds
Outline
1
The Binomial Model
Basics
The One-Period Binomial Model
Derivative Pricing in the One-Period Binomial Model
The Multi-Period Binomial Model
Replicating Strategies in the Binomial Model
Including Dividends
Pricing Forwards
Pricing Futures
2
Term Structure Models
Binomial Models for the Short Rate
The Cash Account and Pricing of Zero-Coupon Bonds
Pricing of Options on Bonds
Pricing of Bond Forwards
Pricing of Bond Futures
Pricing of Caplets and Floorlets
Pricing of Swaps and Swaptions
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Term Structure Models
Pricing of Options on Bonds
Pricing a European Call Option on the ZCB
Example
What is the fair price of a European call option on a 4-period ZCB with strike price
K = 84 and option expiration
n at t = 2?o
The option pay-off is max Z42,. − 94, 0 . The risk-neutral option prices are
t=0
t=1
t=2
0
1.56
2.97
3.35
4.74
6.64
For example,
1.56 =
Sebestyén (ISCTE-IUL)
1
(0.5 × 0 + 0.5 × 3.35)
1 + 0.075
Bond Pricing
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Term Structure Models
Pricing of Bond Forwards
Outline
1
The Binomial Model
Basics
The One-Period Binomial Model
Derivative Pricing in the One-Period Binomial Model
The Multi-Period Binomial Model
Replicating Strategies in the Binomial Model
Including Dividends
Pricing Forwards
Pricing Futures
2
Term Structure Models
Binomial Models for the Short Rate
The Cash Account and Pricing of Zero-Coupon Bonds
Pricing of Options on Bonds
Pricing of Bond Forwards
Pricing of Bond Futures
Pricing of Caplets and Floorlets
Pricing of Swaps and Swaptions
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Term Structure Models
Pricing of Bond Forwards
Pricing a Forward on a Coupon Bond (1)
Consider a forward contract with delivery at t = 4 of a 2-year 10%
coupon bond
Assume that the delivery takes place just after the coupon
payment
Let G0 denote the time-0 forward price, and let Z64 be the
ex-coupon bond price at t = 4
Risk-neutral pricing implies
6
Q Z4 − G0
0 = E0
B4
Rearranging terms and using the fact that G0 is known at t = 0,
6
EQ
0 Z4 /B4
G0 = Q
E0 (1/B4 )
Note that EQ
0 (1/B4 ) is the time-0 price of a ZCB maturing at t = 4
and face value equal to 1
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Term Structure Models
Pricing of Bond Forwards
Pricing a Forward on a Coupon Bond (2)
6
The term EQ
0 Z4 /B4 can be calculated backwards in the pricing
lattice
For example, for the node (5, 5) we have
10 +
1
(0.5 × 110 + 0.5 × 110) = 102.98
1 + 0.1831
Then the node (4, 4) ex coupon price is
1
(0.5 × 102.98 + 0.5 × 107.19) = 91.66
1 + 0.1465
6
Backwards induction yields EQ
0 Z4 /B4 = 79.83
Finally, the time-0 forward price is
6
EQ
79.83
0 Z4 /B4
G0 = Q
=
= 103.38
0.7722
E (1/B4 )
0
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Term Structure Models
Pricing of Bond Forwards
Pricing Lattice for the Forward Contract
t=0
t=1
t=2
t=3
t=4
t=5
t=6
110
102.98
91.66
85.08
81.53
79.99
79.83
98.44
93.27
110
110.46
103.83
90.45
89.24
110
107.19
99.85
97.67
110
112.96
108
104.99
110
114.84
111.16
110
116.24
110
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Term Structure Models
Pricing of Bond Futures
Outline
1
The Binomial Model
Basics
The One-Period Binomial Model
Derivative Pricing in the One-Period Binomial Model
The Multi-Period Binomial Model
Replicating Strategies in the Binomial Model
Including Dividends
Pricing Forwards
Pricing Futures
2
Term Structure Models
Binomial Models for the Short Rate
The Cash Account and Pricing of Zero-Coupon Bonds
Pricing of Options on Bonds
Pricing of Bond Forwards
Pricing of Bond Futures
Pricing of Caplets and Floorlets
Pricing of Swaps and Swaptions
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Term Structure Models
Pricing of Bond Futures
Pricing Futures Contracts
Let Ft denote the time-t price of a futures contract that expires
after T periods
Let St denote the time-t price of the underlying security
We know that FT = ST
We can compute the futures price at t = T − 1 by recalling that
anytime we enter a futures contract, its initial value is 0
Hence, FT−1 must satisfy
FT − FT−1
0
= EQ
T −1
BT − 1
BT
Since both BT and FT−1 are known at time T − 1, we have
FT−1 = EQ
T − 1 ( FT )
Iteration yields that Ft = EQ
t (Ft+1 ) for 0 ≤ t < T
From the law of iterated expectations it follows that
Q
F0 = EQ
0 (FT ) = E0 (ST ) because FT = ST
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Term Structure Models
Pricing of Bond Futures
Example: Pricing a Futures on a Coupon Bond (1)
Example
Consider a futures contract written on the same 2-year 10% coupon
bond with delivery at t = 4.
The values at time t = 4 are the same as the ones obtained for the
forward contract.
We then move backwards and calculate the futures prices. For
example, at node (3, 3) we have
0.5 × 91.66 + 0.5 × 98.44 = 95.05
Finally, the time-0 futures price becomes 103.22.
This is close to the forward price 103.38, but not equal.
If interest rates were deterministic, the forward and futures prices
would coincide.
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Term Structure Models
Pricing of Bond Futures
Example: Pricing a Futures on a Coupon Bond (1)
Example (cont’d)
t=0
t=1
t=2
t=3
t=4
91.66
95.05
98.09
100.81
103.22
98.44
101.14
103.52
105.64
103.83
105.91
107.75
108
109.58
111.16
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Term Structure Models
Pricing of Caplets and Floorlets
Outline
1
The Binomial Model
Basics
The One-Period Binomial Model
Derivative Pricing in the One-Period Binomial Model
The Multi-Period Binomial Model
Replicating Strategies in the Binomial Model
Including Dividends
Pricing Forwards
Pricing Futures
2
Term Structure Models
Binomial Models for the Short Rate
The Cash Account and Pricing of Zero-Coupon Bonds
Pricing of Options on Bonds
Pricing of Bond Forwards
Pricing of Bond Futures
Pricing of Caplets and Floorlets
Pricing of Swaps and Swaptions
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Term Structure Models
Pricing of Caplets and Floorlets
Definitions of Caplets and Floorlets
Caplets and floorlets are very liquidly traded fixed-income derivatives
A caplet is similar to a Europan call option on the short rate rt
It is usually settled in arrears, but sometimes in advance
The pay-off of a caplet (settled in arrears) with maturity T and strike c at
time T is
max {rT−1 − c, 0}
Hence, a caplet is a call option on the short rate prevailing at time T − 1
and settled at time T
A floorlet is the same as caplet, except for the pay-off which is
max {c − rT−1 , 0}
A cap consists of a sequence of caplets with all having the same strike
A floor consists of a sequence of floorlets with all having the same strike
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Term Structure Models
Pricing of Caplets and Floorlets
Pricing a Caplet
Consider the pricing of a caplet that expires at time t = 6 and has strike
of c = 2%
Since it is settled in arrears, it is easier to record the time-6 cash flows at
their time-5 predecessor nodes, and then discount them properly
That is, max {r5 − c, 0} at time t = 6 is worth
max{r5 −c,0}
1+r5
at t = 5
For example, the node (5, 0) price of the caplet will be
max {0.0354 − 0.02, 0}
= 0.015
1 + 0.0354
Then we can work backwards in the lattice using risk-neutral pricing
For example, the node (4, 0) price of the caplet becomes
1
(0.5 × 0.028 + 0.5 × 0.015) = 0.021
1 + 0.0394
Finally, the price of the caplet is 0.042
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Term Structure Models
Pricing of Caplets and Floorlets
Lattice of Caplet Prices
Lattice for a caplet maturing at t = 6 and with a strike of c = 2%:
t=0
t=1
t=3
t=2
t=4
t=5
0.138
0.103
0.099
0.080
0.064
0.076
0.052
0.042
0.059
0.047
0.068
0.053
0.038
0.041
0.032
0.045
0.035
0.026
0.028
0.021
0.015
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Term Structure Models
Pricing of Swaps and Swaptions
Outline
1
The Binomial Model
Basics
The One-Period Binomial Model
Derivative Pricing in the One-Period Binomial Model
The Multi-Period Binomial Model
Replicating Strategies in the Binomial Model
Including Dividends
Pricing Forwards
Pricing Futures
2
Term Structure Models
Binomial Models for the Short Rate
The Cash Account and Pricing of Zero-Coupon Bonds
Pricing of Options on Bonds
Pricing of Bond Forwards
Pricing of Bond Futures
Pricing of Caplets and Floorlets
Pricing of Swaps and Swaptions
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Term Structure Models
Pricing of Swaps and Swaptions
Pricing an Interest Rate Swap
Consider an interest rate swap with a fixed rate of 5% maturing at
t=6
The first payment occurs at t = 1 and the final payment takes
place at t = 6
Payments occur in arrears, so a payment of ± rt,j − K is made at
time t + 1 if state j occurs at time t
Hence, again it is easier to record the time-t cash flows at their
time-t − 1 predecessor nodes and discount them properly
That is, r5,4 − K at t = 6 is worth
± (r5,4 − K)
0.1318 − 0.05
=
= 0.0723
1 + r5,4
1 + 0.1318
Then we can work backwards in the lattice and use risk-neutral
pricing
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Term Structure Models
Pricing of Swaps and Swaptions
Lattice of Swap Prices
t=0
t=1
t=3
t=2
t=5
t=4
0.1125
0.1648
0.1793
0.1686
0.1403
0.0990
0.0723
0.1014
0.1021
0.0829
0.0410
0.0512
0.0400
0.0496
0.0137
0.0172
0.0122
−0.0085
−0.0008
−0.0174
−0.0141
For example,
0.1686 =
Sebestyén (ISCTE-IUL)
h
i
1
(0.0938 − 0.05) + 0.5 × 0.1793 + 0.5 × 0.1021
1 + 0.0938
Bond Pricing
Investments
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Term Structure Models
Pricing of Swaps and Swaptions
Pricing Swaptions
A swaption is an option on a swap
Consider a swaption on the swap discussed above with a strike of
0% and expiration at t = 3
The swaption value at expiration is max {S3 , 0}, where S3 denotes
the option pay-off at t = 3
Values at times 0 ≤ t < 3 are calculated in a usual backward way
Caution! The underlying cash flows of the swap are ignored at
those times!
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Term Structure Models
Pricing of Swaps and Swaptions
Lattice for Pricing Swaptions
t=0
t=1
t=2
t=3
0.1793
0.1286
0.0908
0.1021
0.0620
0.0665
0.0406
0.0400
0.0191
0
For example,
0.0908 =
Sebestyén (ISCTE-IUL)
1
(0.5 × 0.1286 + 0.5 × 0.0665)
1 + 0.075
Bond Pricing
Investments
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