Trees and Martingales Multi-Step Binary Model Generalize the - - PowerPoint PPT Presentation

Trees and Martingales Multi-Step Binary Model

• Generalize the single-period model.
• Still just two assets: the stock and the risk-free bond.
• Still no transaction cost, nor limits on amounts of either

asset.

• But: market is observed at times 0 = t0 < t1 < · · · < tN = T.

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• Over each interval [ti, ti+1], the stock follows the binary

model.

• That is, given S(ti), S
• ti+1
• can take one of only two values.
• But at time ti, there are up to 2i possible states, and at time

T, up to 2N.

• Assume evenly spaced times with spacing δt = T/N, so ti =

iδt, i = 0, 1, . . . , N, and write Si = S(ti).

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S0 S1

1

S11

2

S111

3

S110

3

S10

2

S101

3

S100

3

S0

1

S01

2

S011

3

S010

3

S00

2

S001

3

S000

3

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The Cash Bond

• As before, the interest rate over [ti, ti+1] is known in advance,

at t = ti.

• But it need not be a single constant:

– It could depend on t, as r(ti); – It could even depend on the starting node, as r(ti, Si). – But it is the same along both branches from that node.

• For convenience, we continue to assume it is an overall con-

stant, r.

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Backward Induction

• How do we price a European option?
• As a European option, the payout is a function of SN, so we

know the value at every node at t = tN = T.

• Now consider any node at t = tN−1 = T − δt:

– We know the value at each of the two branches, and we know SN−1 at this node. – So we can use the one-step model to find the value at this node, which is also the cost of setting up the replicating portfolio.

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• Specifically, if SN−1 = S0 at this node, and the two values

at t = T are S0u and S0d, then: – the price of the option at this node is

• 1 − de−rδt

u − d

• C(S0u) +
• ue−rδt − 1

u − d

• C(S0d);

– the number of shares in the replicating portfolio should be φ

△

= C(S0u) − C(S0d) S0u − S0d

• Note that if (u + d)/2 = erδt, then both weights are 1

2e−rδt.

• Note also that φ, the hedge ratio, is the gradient of C(·).

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• In this way, we can calculate the value of the option at every

node at t = tN−1.

• Now consider any node at t = tN−2:

– We know the value at each of the two branches, since they end at t = tN−1, and we already calculated these values. – So we can use the one-step model to find the value at this node.

• Hence we can calculate the value of the option at every node

at t = TN−2.

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• The replicating portfolio that we set up at t = tN−2 is guar-

anteed to give the resources that we need to set up the replicating portfolio at t = tN−1.

• That is, the strategy is self-financing.
• Continue recursively to t = 0, which gives the value of the
• ption, which is also the resources needed to set up the initial

replicating portfolio.

• Note that the replicating portfolio may need to change at

any node: dynamic hedging.

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Recombining Tree

• The nodes of the tree are often chosen to reduce the number
• f distinct nodes.

– For instance, if S01

2

= S10

2

= Sud

2 , say, the tree has only 3

nodes at t2 instead of 4.

• The tree can have as few as i + 1 nodes at t = ti.

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S0 Sd

1

Sdd

2

Sddd

3

Sudd

3

Sud

2

Suud

3

Su

1

Suu

2

Suuu

3

• Also known as a binomial tree, since there are (1, 2, 1) paths

to the 3 nodes at t = t2, and so on.

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• Example 2.1.2. European call; S0 = 100, K = 100, r = 0:

  

S0 = 100 V0 = 15 φ0 = 0.50

     

80 5 0.25

     

60 0.00

  

• 40
• 80
• 

 

100 10 0.50

  

• 120

20

• 

 

120 25 0.75

     

140 40 1.00

  

• 160

60

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Path Probabilities

• The pricing exercise provides risk-neutral probabilities along

each branch. – These are conditional on the node at which the branch begins. – The product of the branch probabilities along any path is the path probability.

• The sum of the probabilities of all paths leading to a given

node is the risk-neutral probability of the node.

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• When the risk-free interest rate is constant, or time-dependent

but not state-dependent, the value of the option is the ex- pected value of the payoff, discounted to present value, with respect to the risk-neutral node probabilities.

• In general, the value is

EQ(path discount factor × path payoff)

where Q is the risk-neutral distribution over the paths. – In other words, the value of the option is the expected value of the payoff, discounted to present value, with re- spect to the risk-neutral path probabilities.

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American Option

• An American option can be exercised at maturity or at any

earlier time.

• Early exercise of an American call on a non-dividend-paying

stock is never optimal; consider two portfolios: Portfolio A: One American call plus (at t = 0) Ke−rT cash; Portfolio B: One share.

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• If the option is exercised at time t < T, then:

– Portfolio A is one share plus Ke−r(T−t) − K < 0 cash; – Portfolio B is one share, so Portfolio B is worth more.

• If the option is exercised only at t = T, if at all, then:

– Portfolio A is worth max(ST, K); – Portfolio B is worth ST, hence no more than Portfolio A.

• So early exercise does not add value.

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• American puts are more complicated:

– If the risk-free interest rate is zero, early exercise never adds value; – If the risk-free interest rate is positive, early exercise may be optimal.

• See Example 2.2.3 for the backward recursion solution for a

particular case.

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