Class 8 Dynamic Replication

Model DescriptionSingle-Period treeMulti-period tree

CalibrationExtensions

Trinomial Trees

Dynamic Replication

Mario Fuenzalida Carlos Lafuente Marcelo Ortiz

Universidad Adolfo Ibanez

April 25, 2011

Fuenzalida, Lafuente, Ortiz Dynamic Replication



Trinomial Trees

Contents1 Model Description

Risky and risk-free assetRelation between u, d , and r .

2 Single-Period treeGeneral Derivative Pay-offPay-off replication ConditionReplication portfolio Composition

Risk-Neutral probabilitiesDelta

Arrow-Debreu Securities3 Multi-period tree

Convexity: Impossibility of static replicationBackward replicationPrices of the Arrow-Debreu securitiesRisk-neutral probabilities

4 CalibrationRisk-Neutral stock dynamicsDefinition of VolatilityMoment matching of normal instantaneous returnsExpression of u and d

5 Extensions6 Trinomial Trees




Trinomial Trees


Risky and risk-free asset

Relation between u, d , and r .2 Single-Period tree

General Derivative Pay-offPay-off replication ConditionReplication portfolio Composition









Trinomial Trees












Trinomial Trees



2 Single-Period tree

General Derivative Pay-offPay-off replication ConditionReplication portfolio Composition









Trinomial Trees



2 Single-Period treeGeneral Derivative Pay-off

Pay-off replication ConditionReplication portfolio Composition









Trinomial Trees



2 Single-Period treeGeneral Derivative Pay-offPay-off replication Condition

Replication portfolio CompositionRisk-Neutral probabilitiesDelta








Trinomial Trees












Trinomial Trees




Risk-Neutral probabilities

Delta








Trinomial Trees












Trinomial Trees





Arrow-Debreu Securities

3 Multi-period treeConvexity: Impossibility of static replicationBackward replicationPrices of the Arrow-Debreu securitiesRisk-neutral probabilities






Trinomial Trees












Trinomial Trees






Convexity: Impossibility of static replication

Backward replicationPrices of the Arrow-Debreu securitiesRisk-neutral probabilities






Trinomial Trees






Convexity: Impossibility of static replicationBackward replication

Prices of the Arrow-Debreu securitiesRisk-neutral probabilities






Trinomial Trees






Convexity: Impossibility of static replicationBackward replicationPrices of the Arrow-Debreu securities

Risk-neutral probabilities4 Calibration

Risk-Neutral stock dynamicsDefinition of VolatilityMoment matching of normal instantaneous returnsExpression of u and d





Trinomial Trees












Trinomial Trees







4 Calibration






Trinomial Trees







4 CalibrationRisk-Neutral stock dynamics

Definition of VolatilityMoment matching of normal instantaneous returnsExpression of u and d





Trinomial Trees







4 CalibrationRisk-Neutral stock dynamicsDefinition of Volatility

Moment matching of normal instantaneous returnsExpression of u and d





Trinomial Trees







4 CalibrationRisk-Neutral stock dynamicsDefinition of VolatilityMoment matching of normal instantaneous returns

Expression of u and d5 Extensions6 Trinomial Trees




Trinomial Trees












Trinomial Trees








5 Extensions

6 Trinomial Trees




Trinomial Trees












Trinomial Trees


Model Description

Suppose you have a risk-free asset, like a MMA (money market account)in wich you make a deposit of M USD. The rate offered by the bank atmoment t0 is rt0.

Suppose now, that you have a risky asset, like a stock , that at t0 have aprice of S0 The stock price can either move up from S0 to u ∗ S0 , whereu > 1, or down from S0 to d ∗ S0 , where d < 1, and u = 1/d .




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Obviously, (1 + r) must be lower than u, because in other way invest inrisky asset (like stock) will be less profitable than invest in risk free asset,even in the best possible return that risky asset can generate. So:

(1 + r) < u




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General Derivative Pay-offPay-off replication ConditionReplication portfolio CompositionArrow-Debreu Securities

Dynamic Replication

General derivative Pay-off

Option European Vanilla Call

No arbitrage

Option can only be exercised at the end of its life, at its maturity.

At t1 the pay-off of the option is:

Max (St1 -K,0)

We need calculate option value at t0: f0.

How can we find f0? Dynamic Replication




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Dynamic Replication



No arbitrage



Max (St1 -K,0)






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Dynamic Replication



No arbitrage



Max (St1 -K,0)






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Dynamic Replication



No arbitrage



Max (St1 -K,0)






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Dynamic Replication



No arbitrage



Max (St1 -K,0)






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Dynamic Replication



No arbitrage



Max (St1 -K,0)






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Dynamic Replication

General Derivative Pay-off

Replicate the option.

Bonds bought for the price 1 and stocks.

At t0, F0 = ψ + ∆ S0




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Dynamic Replication




At t0, F0 = ψ + ∆ S0




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Dynamic Replication




At t0, F0 = ψ + ∆ S0




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Dynamic Replication

Pay-off replication Condition

At t1 there are two possibilities:

Fu = ψ(1 + r) + ∆ S0uor

Fd = ψ(1 + r) + ∆ S0d

Solve the system of equations...

∆ =Fu − Fd

S0(u − d)

ψ =uFd − Fud

R(u − d)




Trinomial Trees


Dynamic Replication

Pay-off replication Condition

At t1 there are two possibilities:

Fu = ψ(1 + r) + ∆ S0uor

Fd = ψ(1 + r) + ∆ S0d

Solve the system of equations...

∆ =Fu − Fd

S0(u − d)

ψ =uFd − Fud

R(u − d)




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Dynamic Replication

Replication portfolio Composition

We will illustrate the concepts with a European vanilla call optionwith strike K = 100 on a non-dividend paying stock with initialvalue S0 = 100, with a risk-free interest rate r = 0, and modelparameters u = 1,2 and d = 0,8.

Figure: Single-Period treeFuenzalida, Lafuente, Ortiz Dynamic Replication



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Dynamic Replication


Replacing...

∆ =Fu − Fd

S0(u − d)

ψ =uFd − Fud

R(u − d)

∆ =20− 0

100(1, 2− 0, 8)= 1

2

ψ =1, 2 ∗ 0− 20 ∗ 0, 8

1(1, 2− 0, 8)= −40

Thus, we need to borrow 40 dollars and then buy 0.5 units of stockat t0. The cost of this will be the current value of the option:

F0 = 12 ∗ 100− 40 = 10




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Dynamic Replication


Replacing...

∆ =Fu − Fd

S0(u − d)

ψ =uFd − Fud

R(u − d)

∆ =20− 0

100(1, 2− 0, 8)= 1

2

ψ =1, 2 ∗ 0− 20 ∗ 0, 8

1(1, 2− 0, 8)= −40


F0 = 12 ∗ 100− 40 = 10




Trinomial Trees


Dynamic Replication


Replacing...

∆ =Fu − Fd

S0(u − d)

ψ =uFd − Fud

R(u − d)

∆ =20− 0

100(1, 2− 0, 8)= 1

2

ψ =1, 2 ∗ 0− 20 ∗ 0, 8

1(1, 2− 0, 8)= −40


F0 = 12 ∗ 100− 40 = 10




Trinomial Trees


Dynamic Replication


Replacing...

∆ =Fu − Fd

S0(u − d)

ψ =uFd − Fud

R(u − d)

∆ =20− 0

100(1, 2− 0, 8)= 1

2

ψ =1, 2 ∗ 0− 20 ∗ 0, 8

1(1, 2− 0, 8)= −40


F0 = 12 ∗ 100− 40 = 10




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Dynamic Replication

Replication Portfolio Condition

The variable p is the probability of a up movement, then, 1-p, is theprobability of a down movement, and the expected stock price attime T, E(ST ), is given by:

E(ST ) = pS0u + (1-p)S0dE(ST ) = pS0(u-d) + S0d

But the stock price grows on average at the risk-free rate

E(ST ) = S0erT




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Dynamic Replication


The variable p is the probability of a up movement, then, 1-p, is theprobability of a down movement, and the expected stock price attime T, E(ST ), is given by:

E(ST ) = pS0u + (1-p)S0dE(ST ) = pS0(u-d) + S0d

But the stock price grows on average at the risk-free rate

E(ST ) = S0erT




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Dynamic Replication


We now return to the example, and illustrate that risk-neutralvaluation. The stock price is currently $100 and will move either upto $120 or down to $80. The option considered is a European calloption with a strike price of $100.

We define p as the probability of an upward movement in the stockprice in risk-neutral world. We can calculate p:

100 = 100p - 80(1-p)

p = 0,5




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Dynamic Replication




100 = 100p - 80(1-p)

p = 0,5




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Dynamic Replication




100 = 100p - 80(1-p)

p = 0,5




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Dynamic Replication

Replication portfolio Condition

Delta is an important parameter in the pricing and hedging ofoptions.

The delta of a stock option is the ratio of the change in the price ofthe stock option to the change in the price of the underlying stock.

It is the number of units of the stock we should hold for each optionshorted in order to create a riskless hedge.

In the example:

20−0120−80 = 1

2

This is because when the stock price changes from $80 to $120, theoption price changes from $0 to $20.




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Dynamic Replication





In the example:

20−0120−80 = 1

2





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Dynamic Replication





In the example:

20−0120−80 = 1

2





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Dynamic Replication





In the example:

20−0120−80 = 1

2





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Dynamic Replication





In the example:

20−0120−80 = 1

2





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Dynamic Replication


Can be understood as Arrow-Debreu assets how insurance contractedfor a certain amount of money if a certain state of nature occurs.

φ1 + φ2 = 11+r = b

uS0φ1 + dS0φ2 = S0

F0 = Fuφ1 + Fdφ2

where r is the return on risk-free asset and b is the present value of abond that pays $1 in one year.

φ1 price of the Arrow-Debreu asset 1.





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Dynamic Replication



φ1 + φ2 = 11+r = b

uS0φ1 + dS0φ2 = S0

F0 = Fuφ1 + Fdφ2







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Dynamic Replication



φ1 + φ2 = 11+r = b

uS0φ1 + dS0φ2 = S0

F0 = Fuφ1 + Fdφ2







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Dynamic Replication



φ1 + φ2 = 11+r = b

uS0φ1 + dS0φ2 = S0

F0 = Fuφ1 + Fdφ2







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Dynamic Replication


Example: S0=100, uS0=120, dS0=80, 11+r =1, Fu=20, Fd=0

φ1 = ?, φ2 = ?, F0 = ?

Replacing...

120φ1 + 80φ2 = 100φ1 + φ2 = 1

Resolving:

φ1 = 0,5, φ2 = 0,5

thus, the price of option is:

20*0,5 + 0*0,5 = 10

the same previous result.




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Dynamic Replication



φ1 = ?, φ2 = ?, F0 = ?

Replacing...

120φ1 + 80φ2 = 100φ1 + φ2 = 1

Resolving:

φ1 = 0,5, φ2 = 0,5


20*0,5 + 0*0,5 = 10





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Dynamic Replication



φ1 = ?, φ2 = ?, F0 = ?

Replacing...

120φ1 + 80φ2 = 100φ1 + φ2 = 1

Resolving:

φ1 = 0,5, φ2 = 0,5


20*0,5 + 0*0,5 = 10





Trinomial Trees


Dynamic Replication



φ1 = ?, φ2 = ?, F0 = ?

Replacing...

120φ1 + 80φ2 = 100φ1 + φ2 = 1

Resolving:

φ1 = 0,5, φ2 = 0,5


20*0,5 + 0*0,5 = 10





Trinomial Trees


Dynamic Replication



φ1 = ?, φ2 = ?, F0 = ?

Replacing...

120φ1 + 80φ2 = 100φ1 + φ2 = 1

Resolving:

φ1 = 0,5, φ2 = 0,5


20*0,5 + 0*0,5 = 10





Trinomial Trees


Dynamic Replication



φ1 = ?, φ2 = ?, F0 = ?

Replacing...

120φ1 + 80φ2 = 100φ1 + φ2 = 1

Resolving:

φ1 = 0,5, φ2 = 0,5


20*0,5 + 0*0,5 = 10





Trinomial Trees


Dynamic Replication



φ1 = ?, φ2 = ?, F0 = ?

Replacing...

120φ1 + 80φ2 = 100φ1 + φ2 = 1

Resolving:

φ1 = 0,5, φ2 = 0,5


20*0,5 + 0*0,5 = 10





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Replicating portfolios may need constant adjustment to maintain theirequivalence with the targeted instrument:1) if markets in the component instruments do not exist2) the instruments themselves may exist, but they may not be liquid.3) the asset to be replicated can be highly nonlinear

Principles of dynamic replication:The strategy will combine imperfect instruments that are correlated witheach other to get a synthetic at time t0. If these price of thatinstruments (random variables) were correlated in a certain fashion, thesecorrelations can be used against each other to eliminate the need for cashinjections or withdrawals. The cost of forming the portfolio at t0 wouldthen equal the arbitrage-free value of the original asset.




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Obviously, if the instruments are time-independent, their correlations are0, so, they cant eliminate the need of cash injections or withdrawals. Ifwe want to replicate the payoff of a call at time t < T , we must lookthat their value is nonlinear.

Obviously, is impossible replicate their value using time-independentinstrument.




Trinomial Trees


Is a procedure for working from the end of a tree to its beginning, inorder to value the weight of the replications instrument.

Example:We want replicate a Bond with maturity T2 (the value of the bond at T2

is $100). Suppose the market practitioner has only two liquid markets:1)The first is the market for one-period lending/borrowing, denoted bythe symbol Bt . The Bt is the time t value of $1 invested at time t0.2) The second liquid market is for a default-free pure discount bondwhose time-t price is denoted by B(t, T3). The bond pays $100 at timeT3 and sells for the price B(t, T3) at time t.

The idea is to use the information given in trees of the lending-borrowingand the bond, to form a portfolio with (time-varying) weights θlendt andθbondt for Bt and B(t, T3).




Trinomial Trees


The portfolio should mimic the value of the medium-term bond B(t, T2)at all nodes at t = 0, 1, 2. The first condition on this portfolio is that, atT2, its value must equal $100.

This means that the θlendt and θbondt will satisfy:

θlendt Bt + θbondt B(t,T3) = B(t,T2)

, for all t.Like we want that adjustments of the weights θlendt and θbondt , observedalong the tree paths t = 0, 1, 2 will not lead to any cash injections orwithdrawals, so:




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θlendt Bupt+1 + θbondt B(t + 1,T3)up = θlendt+1B

upt+1 + θbondt+1 B(t + 1,T3)up

θlendt Bdownt+1 + θbondt B(t + 1,T3)down = θlendt+1B

downt+1 + θbondt+1 B(t + 1,T3)down

The left-hand side is the value of a portfolio chosen at time t, and valuedat a new up or down state at time t + 1.The right-hand side represents the cost of a new portfolio chosen at timet + 1, either in the up or down state.

Putting these two together, the equations imply that, regardless of whichstate occurs, the previously chosen portfolio generates just enough cashto put together a new replicating portfolio




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Imagine that are just two possible states tomorrow: Up (u) and Down(d). Denote the random variable which represents the state as S ; denotetomorrow’s random variable as S1. Thus, S1can take two values: S1 = uand S1 = d .Suppose that:1)Exist a security that pays off $1 if tomorrow’s state is u and $0 if stateis d . The price of this security is pu.2)Exist other security that pays off $1 if tomorrow’s state is d andnothing if the state is u. The price of this security is pd .

The prices pd and puare the state prices.

Theprobabilitiesof S1 = u and S1 = d affect to state prices. The morelikely a move to d is, the higher the price pd gets, since pd insures theagent against the occurrence of state d . The seller of this insurancewould demand a higher premium.




Trinomial Trees


Again, suppose that are just two possible states tomorrow: Up (u) andDown (d).Define a Price Vector, like a vector composed with the currents prices of3 instruments: risk-free deposit of B USD, an asset valued on P USD,and a their Call Option of c USD.Define now a Payoff Matrix , wich include the two possible payoff,depending of the states, for each instruments.In a free arbitrage market, there are two positive constraints ,φu ,φdallowing the next equation:BP

c

=

B(1 + r) B(1 + r)Pu Pd

cu cd

[φuφd

](1)




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Naturally, these constraints are the Arrow-Debreu securities.Dividing the first equation by B, we obtain:

1 = (1 + r)φu + (1 + r)φd

We can re-write the last equation using the next substitution:

π∗u = (1 + r)φu

π∗d = (1 + r)φd

1 = π∗u + π∗d

We can conclude that risk-neutral probabilities are nothing but theArrow-Debreu prices in disguise.Increasing the number of steps N (each step with length dT = T/N),asymptotically, the distribution becomes normal.




Trinomial Trees


Dynamic Replication

Risk-Neutral stock dynamics

The model of stock price behavior used by Black, Scholes andMerton assumes that percentage changes in the stock price in ashort period of time are normally distributed, where:

µ = Expected return on stock per year.

σ = Volatility of the stock price per year.

Standard deviation of this percentage change is σ√

∆t so that:

∆SS ∼φ(µ ∆t,σ

√∆t )

Where ∆S is the change in the stock price S in time ∆t, and φ(m, s)denotes a normal distribution with mean m and standard deviation s.




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Dynamic Replication






∆t so that:


√∆t )





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Dynamic Replication






∆t so that:


√∆t )





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Dynamic Replication






∆t so that:


√∆t )





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Dynamic Replication






∆t so that:


√∆t )





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Dynamic Replication


From Ito’s lemma

dG = (∂G∂S µS + ∂G∂t +

1

2

∂2G

∂S2σ2S2)dt + ∂G

∂S σSdz

Where G = ln(S), σ and µ constant

This means that:

ln(ST ) - ln(S0) ∼ φ[(µ− σ2

2 ), σ√T ]

From this, it follows that:

ln(ST

S0) ∼ φ[(µ− σ2

2 ), σ√T ]

andln(ST ) ∼ φ[ln(S0) + (µ− σ2

2 ), σ√T ]




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Dynamic Replication


From Ito’s lemma

dG = (∂G∂S µS + ∂G∂t +

1

2

∂2G


∂S σSdz


This means that:

ln(ST ) - ln(S0) ∼ φ[(µ− σ2

2 ), σ√T ]


ln(ST

S0) ∼ φ[(µ− σ2

2 ), σ√T ]


2 ), σ√T ]




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Dynamic Replication


From Ito’s lemma

dG = (∂G∂S µS + ∂G∂t +

1

2

∂2G


∂S σSdz


This means that:

ln(ST ) - ln(S0) ∼ φ[(µ− σ2

2 ), σ√T ]


ln(ST

S0) ∼ φ[(µ− σ2

2 ), σ√T ]


2 ), σ√T ]




Trinomial Trees


Dynamic Replication


From Ito’s lemma

dG = (∂G∂S µS + ∂G∂t +

1

2

∂2G


∂S σSdz


This means that:

ln(ST ) - ln(S0) ∼ φ[(µ− σ2

2 ), σ√T ]


ln(ST

S0) ∼ φ[(µ− σ2

2 ), σ√T ]


2 ), σ√T ]




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Dynamic Replication

Definition of Volatility

The volatility of a stock price can be defined as the standarddeviation of the return provided by the stock in 1 year when thereturn is expressed using continuous compounding.

The volatility σ of a stock is a measure of our uncertainly about thereturns provided by the stock. Stocks typically have a volatilitybetween 15% and 60%.




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Dynamic Replication

Definition of Volatility

The volatility of a stock price can be defined as the standarddeviation of the return provided by the stock in 1 year when thereturn is expressed using continuous compounding.

The volatility σ of a stock is a measure of our uncertainly about thereturns provided by the stock. Stocks typically have a volatilitybetween 15% and 60%.




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Dynamic Replication

Moment matching of normal instantaneous returns

In practice, when constructing a binomial tree to represent themovements in a stock price, we choose the parameters u and d tomatch the volatility of the stock price.

To see how this is done, we suppose that the expected return on astock is u and its volatility is σ.




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Dynamic Replication


In practice, when constructing a binomial tree to represent themovements in a stock price, we choose the parameters u and d tomatch the volatility of the stock price.

To see how this is done, we suppose that the expected return on astock is u and its volatility is σ.




Trinomial Trees


Dynamic Replication


The probability of an up movement is assumed to be p.

The expected stock price at the end of first time step is S0eu∆t . On

the tree the expected stock price at this time is:

pS0u + (1 - p)S0d

In order to match the expected return on the stock with the tree’sparameters, we must therefore have:

pS0u + (1 - p)S0d = S0 eu∆t




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Dynamic Replication





pS0u + (1 - p)S0d


pS0u + (1 - p)S0d = S0 eu∆t




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Dynamic Replication





pS0u + (1 - p)S0d


pS0u + (1 - p)S0d = S0 eu∆t




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Dynamic Replication


The volatility of a stock price is defined so that σ√

∆t, that is thestandard deviation of the return on the stock price in a short periodof time of lenght ∆t.

Equivalently, the variance of the return is σ2∆t.

Variance of X is: E(X 2) - [E (X )]2

Then:

pu2 + (1-p)d2 - [pu + (1− p)d ]2

In order to match the stock price volatility with the tree’sparameters, we must therefore have:

pu2 + (1-p)d2 - [pu + (1− p)d ]2 = σ2∆t




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Dynamic Replication






Then:

pu2 + (1-p)d2 - [pu + (1− p)d ]2


pu2 + (1-p)d2 - [pu + (1− p)d ]2 = σ2∆t




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Dynamic Replication






Then:

pu2 + (1-p)d2 - [pu + (1− p)d ]2


pu2 + (1-p)d2 - [pu + (1− p)d ]2 = σ2∆t




Trinomial Trees


Dynamic Replication






Then:

pu2 + (1-p)d2 - [pu + (1− p)d ]2


pu2 + (1-p)d2 - [pu + (1− p)d ]2 = σ2∆t




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Dynamic Replication






Then:

pu2 + (1-p)d2 - [pu + (1− p)d ]2


pu2 + (1-p)d2 - [pu + (1− p)d ]2 = σ2∆t




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Dynamic Replication

Expression of u and d

Substituting:

eu∆t(u+d) - ud - e2u∆t = σ2∆t

Using the series expansion: ex = 1 + x + x2

2 + x3

3 + ...

u = eσ√

∆t

d = e−σ√

∆t

These are the evalues of u and d proposed by Cox, Ross andRubinstein (1979) for matching u and d.




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Dynamic Replication


Substituting:



2 + x3

3 + ...

u = eσ√

∆t

d = e−σ√

∆t





Trinomial Trees


Dynamic Replication


Substituting:



2 + x3

3 + ...

u = eσ√

∆t

d = e−σ√

∆t





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Dynamic Replication

Early exercise:

An early exercise of an American option will not realized at anymoment of the life of the contract if: K > S(t), but if S(t) > Kmay not be exercised if the investor thinks that the price of thestock will be higher.

Pricing American options is more complex than the pricing ofEuropean options since it is required an optimization, related to theoptimal time for the investor for exercising.

EEP is the early exercise premium which expresses the value of theAmerican option as the corresponding European option value plusthe gain from early exercise which is the present value of thedividend benefits in the exercise region net of the interest losses onthe payments incurred upon exercise.




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Dynamic Replication

Early exercise:







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Early exercise:







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Early exercise:







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Dynamic Replication

Early exercise: : Calls on a non-dividend-paying stock

It is never optimal to exercise an American call option over a NDPSif the investor plans to keep the stock: T = 1 month;SK =40;S(t)=50

If the investor wants to hold the stock after the contract he will payUSD 40 one month later, and invest that value for the entire monthmaking interest for that period, the stock pays no dividends.

Also there is some probability that the stock price in one moremonth may be lower than USD 40.

If he thinks the stock is overvalued is better to sell the option orshort the stock and keep the option, so the price obtained will begrater than the intrinsic value of S(t)− K = 10




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The value of and option must satisfies that: c >= S0 − Ke−rt

Because in an American call option the investor has all the period todecide when to exercise the contract, its value is grater than of anequivalence European call option: C >= c ;C >= S0 − Ke−rt

Since r > 0, then: C > S0 − K , and if it were optimal to exercisebefore the contract gets the maturity it is required that:C = S0 − K .

The value of a call is always above its intrinsic value,max(S0 − K , 0), and while r, T and the volatility increases, the callprice also increase.




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Holding the option until maturity gives a protection to the investorin case the stock prices gets lower than the strike price (insurance).

Time value of money, is better to pay latter than sooner.




Trinomial Trees

Dynamic Replication



Time value of money, is better to pay latter than sooner.




Trinomial Trees

Dynamic Replication



Time value of money, is better to pay latter than sooner.Fuenzalida, Lafuente, Ortiz Dynamic Replication



Trinomial Trees

Dynamic Replication

Early exercise: : Calls on a dividend-paying stock

It is optimal to exercise an American call option over a DPS only ata time immediately before the stock goes ex-dividend.

Assume that remains n ex-dividend dates at time t1, t2, tn and theirpresent values are D1, D2, Dn. The early exercise will be consideredfor the tn, the investor will receive S(tn)− K if he decided toexercise the option.

The lower bound for an American call on a DPS is:C >= S0 − D − Ke−rT

For the specific case shown: C >= S(tn)− Dn − Ke−r(T−tn).

Then: S(tn)− Dn − Ke−r(T−tn) > S(tn)− K , makingDn <= K [1− e−r(T−tn)].

In that case is not optimal to early exercise the option at time tn.




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Trinomial Trees

Dynamic Replication


On the opposite way if: Dn >= K [1− e−r(T−tn)].

It is always optimal to exercise the option if the stock value at timen is sufficiently high. The inequality will be satisfied if n is close tothe expiry date and the dividend is large.

The relation between the dividend and the price of the stock iscalled dividend yield.




Trinomial Trees

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Trinomial Trees

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Trinomial Trees

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Trinomial Trees

Dynamic Replication

Dividend Yield

The relation between the dividend and the price of the stock iscalled dividend yield. This ratioshows how much a company paysout in dividends each year relative to its stock price. Isa way tomeasure how much cash flow is getting for each dollar invested in anequity position.

Calculation is as follows:(

AnnualdividendsperstockStockprice

)




Trinomial Trees

Dynamic Replication

Dividend Yield




)




Trinomial Trees

Dynamic Replication

Dividend Yield




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Trinomial Trees

Dynamic Replication

Discrete dividends

In the Black-Scholes model, any dividends on stocks are paidcontinuously, but in reality dividends are always paid discretely.

It is not entirely clear how such discrete dividends are to be handled;simple perturbations of the Black-Scholes model often fall intocontradictions.

The market convention is to recognize the stock price as the netpresent value of all future dividends, and to model the (discrete)dividend process directly.

Arbitrage pricing theory (APT) states the price of a stock as thepresent value of the future dividend payments.

The problem is to write down the joint distribution of the tworandom variables, dividends and stocks price, because sometimesleads to an inconsistency.




Trinomial Trees

Dynamic Replication

Discrete dividends









Trinomial Trees

Dynamic Replication

Discrete dividends









Trinomial Trees

Dynamic Replication

Discrete dividends









Trinomial Trees

Dynamic Replication

Discrete dividends









Trinomial Trees

Dynamic Replication

Discrete dividends









Trinomial Trees

Dynamic Replication

Discrete dividends

Modeling the stock price with a discrete set of dividends paymentsat dates t1 < t2 , is should be useful to modeling the dividendpayments, is then a consequence of the model assumed for thedividends the stock price process.




Trinomial Trees

Dynamic Replication

Discrete dividends

Modeling the stock price with a discrete set of dividends paymentsat dates t1 < t2 , is should be useful to modeling the dividendpayments, is then a consequence of the model assumed for thedividends the stock price process.




Trinomial Trees

Dynamic Replication

Replacing the stock with a future

One way to do this is using a Total Return Swap, which is anagreement to exchange the total return of an asset for LIBOR plus aspread. The total returns involves all the cash flows related to theasset, for the case of a stock will pays dividends and capital gains,the receiver pays interest at LIBOR plus a spread, both based on aprincipal amount.

The LIBOR is set for next period at the last dividend payment date.

When the contract gets it maturity, there is a payment forcompensating the change in the stock value.




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Trinomial Trees

Dynamic Replication


Other way is to use a Dividend Swap, in which investor can buy orsell the dividends paid by a stock or an index. They can be used forhedging or managing cash flows from stocks portfolios.

Dividend Swap, compared to the Total Return Swap, exchange fixedby floating amounts, were the floating is related to the dividendsmade.

Their payouts are determined by the dividends reach by the stock,the fixed rate and the principal amount agreed




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Trinomial Trees

Dynamic Replication

Construction - General aspects

Improves upon the binomial model by allowing a stock price to moveup, down or stay the same with certain probabilities, as shown in thediagram below:

They are considered an effective method of numerical calculation ofoption prices within B-S model.




Trinomial Trees

Dynamic Replication







Trinomial Trees

Dynamic Replication







Trinomial Trees

Dynamic Replication

Construction - Model definition

The price of a stock in the next period under the trinomial treecould have the following values:

S(t + ∆t) = S(t)u with probability pu

S(t + ∆t) = S(t) with probability 1− pu − pd

S(t + ∆t) = S(t)d with probability pd




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Dynamic Replication


Matching the first two moments of the models distribution to the noarbitrage condition:

E [S(ti+1|S(ti ] = er∆tS(ti )

VAR[S(ti+1|S(ti ] = ∆t ∗ S(ti )2σ2 + o(∆t)

Assuming constant volatility and that the stock price follows ageometric Brownian motion. The first condition states anequilibrium assumption where the average return of the stock shouldbe equal to the risk free return. Also it is required that the upwardjump most be identical to the downward jump.




Trinomial Trees

Dynamic Replication


Matching the first two moments of the models distribution to the noarbitrage condition:

E [S(ti+1|S(ti ] = er∆tS(ti )

VAR[S(ti+1|S(ti ] = ∆t ∗ S(ti )2σ2 + o(∆t)

Assuming constant volatility and that the stock price follows ageometric Brownian motion. The first condition states anequilibrium assumption where the average return of the stock shouldbe equal to the risk free return. Also it is required that the upwardjump most be identical to the downward jump.




Trinomial Trees

Dynamic Replication


Having the values of the jump sizes u and d , and the transitionprobabilities pu and pd , it is possible to reach the value of the stockfor any sequence of price movements.

Other thing to specify is the number of jumps Nu, Nd and Nm, afterthat it is possible to reach the stock price at node j for time i as:

Si,j = uNudNdS(t0), where Nu + Nd + Nm = n




Trinomial Trees

Dynamic Replication


Having the values of the jump sizes u and d , and the transitionprobabilities pu and pd , it is possible to reach the value of the stockfor any sequence of price movements.

Other thing to specify is the number of jumps Nu, Nd and Nm, afterthat it is possible to reach the stock price at node j for time i as:

Si,j = uNudNdS(t0), where Nu + Nd + Nm = n




Trinomial Trees

Dynamic Replication

Construction - Setting the parameters

The sizes of the jumps are: u = eσ√

2∆t , d = e−σ√

2∆t

The transition probabilities are given by:

pu =

(e(r∆t)/2 − e−σ

√(∆t)/2

eσ√

(∆t)/2 − e−σ√

(∆t)/2

)2

pd =

(eσ√

(∆t)/2 − e(r∆t)/2

eσ√

(∆t)/2 − e−σ√

(∆t)/2

)2

pm = 1− pu − pd




Trinomial Trees

Dynamic Replication

Construction - Setting the parameters

The sizes of the jumps are: u = eσ√

2∆t , d = e−σ√

2∆t

The transition probabilities are given by:

pu =

(e(r∆t)/2 − e−σ

√(∆t)/2

eσ√

(∆t)/2 − e−σ√

(∆t)/2

)2

pd =

(eσ√

(∆t)/2 − e(r∆t)/2

eσ√

(∆t)/2 − e−σ√

(∆t)/2

)2

pm = 1− pu − pd




Trinomial Trees

Dynamic Replication

Construction - Pricing options

Transition probabilities of various stock price movements, pu, pd andpm.

The sizes of moves up, down and middle, u, d and m=1.

The payoff of the option at maturity.

Apply the backward algorithm derived from the risk-neutralityprinciple where ”i” and ”j” are the time and space positionsrespectively, so it can be calculated the price of the option at time i,Ci as the option price of an up move pu ∗ Ci+1, plus the option pricemiddle move by pm ∗ Ci+1 plus the option prices down move bypd ∗ Ci+1 discounted by one time step e−r∆t , so it is possible to getthe option price at any node of the tree. It is only needed to valuethe option at maturity and then do it backwards.

fi,j = e−r∆t [pufi+1,j+1 + pmfi+1,j + pd fi+1,j−1]




Trinomial Trees

Dynamic Replication

Construction - Pricing options

Transition probabilities of various stock price movements, pu, pd andpm.

The sizes of moves up, down and middle, u, d and m=1.

The payoff of the option at maturity.

Apply the backward algorithm derived from the risk-neutralityprinciple where ”i” and ”j” are the time and space positionsrespectively, so it can be calculated the price of the option at time i,Ci as the option price of an up move pu ∗ Ci+1, plus the option pricemiddle move by pm ∗ Ci+1 plus the option prices down move bypd ∗ Ci+1 discounted by one time step e−r∆t , so it is possible to getthe option price at any node of the tree. It is only needed to valuethe option at maturity and then do it backwards.

fi,j = e−r∆t [pufi+1,j+1 + pmfi+1,j + pd fi+1,j−1]




Trinomial Trees

Dynamic Replication

Comparison of the convergence speed between a binomial and a trinomialtree

The main difference between binomial and trinomial trees is just thenumber of transitions probabilities allowed for pricing a stock.

When a small number of tree steps is used the trinomial model tendsto give more accurate results than the binomial model. As thenumber of steps increases the results from the binomial andtrinomial models (for vanilla options) rapidly converge.

When a trading desk is asked for pricing a security, time (and price)are the relevant variable for making money.




Trinomial Trees

Dynamic Replication

Comparison of the convergence speed between a binomial and a trinomialtree

The main difference between binomial and trinomial trees is just thenumber of transitions probabilities allowed for pricing a stock.

When a small number of tree steps is used the trinomial model tendsto give more accurate results than the binomial model. As thenumber of steps increases the results from the binomial andtrinomial models (for vanilla options) rapidly converge.

When a trading desk is asked for pricing a security, time (and price)are the relevant variable for making money.




Trinomial Trees

Dynamic Replication

Matlab Algorithm




Trinomial Trees

Dynamic Replication

Replication cost




Trinomial Trees

Dynamic Replication

Replication cost


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Class 8 Dynamic Replication