Implied Transition Probabilities: Autumn 2 0 0 9

A U T U M N    2 0 0 9

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Numerical Methods in Finance (Implementing Market Models)

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Lecture Objectives

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 Implied Transition Probabilities

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Agenda Page

Implied Transition Probabilities

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2 2

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Implied Transition Probabilities  The implied transition probabilities can now be

computed from the state prices  We start with some basic premises j+1

Pu ,i , j + Pm ,i , j + Pd ,i , j = 1

Pu,i,j

Pm,i,j

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j

Pd,i,j

j-1 i

i+1

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Implied Transition Probabilities  We also assume that the current price of a stock

must be equal to 1. The sum of the expected values times the probabilities 2. Discounted at the risk free rate

Si , j = e

− r∆t

Si+1,j+1

 Pu ,i , j Si +1, j +1 + ...  P S  + P S m , i , j i + 1 , j d , i , j i +1, j −1  S 

Pu,i,j

Pm,i,j

Si+1,j

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i,j

Pd,i,j

Si+1,j-1

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Implied Transition Probabilities  Finally, the forward evolution of the state price

must be consistent

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 Pu ,i , j Qi , j + ...   − r∆t  Qi +1, j +1 = e  Pm ,i , j +1Qi , j +1 + ...  P  Q  d ,i , j + 2 i , j + 2 

Qi+1,j+2

Qi,j+2 Pd,i,j+2 Pm,i,j+1

Qi,j+1

Qi+1,j+1

Pu,i,j

 This can be rearranged as

Pu ,i , j =

Qi+1,j

Qi,j

e r∆t Qi +1, j +1 − Pm ,i , j +1Qi , j +1 − Pd ,i , j + 2Qi , j + 2 Qi , j

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Implied Transition Probabilities  The other probabilities are:

e Si , j − Si +1, j −1 − Pu ,i , j ( Si +1, j +1 − Si +1, j −1 ) r∆t

Pm ,i , j =

Si +1, j − Si +1, j −1

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Pd ,i , j = 1 − pm ,i , j − pu ,i , j  These probability equations suggest that if we know

future probabilities (above node i,j), we can calculate current values 7

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Implied Transition Probabilities  So using our “simple” 2-step

structure, the probability Pu,1,1 is calculated as

Pu ,1,1 =

r∆t

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say:

Q1,1

Δx

Pu ,i , j =

e Qi +1,i +1

Q1,1

j=1

e Q2 , 2

r∆t

Pu,1,1

Δx

Δx

 Or, in a general sense we can

Q2,2

j=2

Pu,0,0

Pm,1,1

Q2,1

Pd,1,1 Q2,0

j=0

j=-1 Δx j=-2 i=0

Δt1

i=1

Δt2

i=2

Qi ,i 8

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Implied Transition Probabilities  Using the previous equations,

we can calculate Pm,1,1 and Pd,1,1  So, in the same way top-right-

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down manner as we calculated the state prices, we can calculate the transition probabilities  We also move down to the

Q2,2

j=2 Pu,1,1

Δx Q1,1

j=1 Pu,0,0

Δx

Pm,1,1

Q2,1

Pd,1,1 Q2,0

j=0 Δx j=-1 Δx j=-2 i=0

Δt1

i=1

Δt2

i=2

centre and up to the centre

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Implied Transition Probabilities  To ensure that the transition probabilities remain

positive and the explicit finite difference methods stability condition

∆x ≥ σ 3∆t  Is satisfied, we need to calculate the local

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volatility at each node  We can do this using the formula:

var[ ∆x ] = σ

2 local

[ ]

∆t = E ∆x − ( E [ ∆x ] ) 2

2

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Implied Transition Probabilities  Time to look at some code again. We’ll continue

with the example from C&S page 143  Recall:  S = 100  T=1  R = 6%

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 N=4  Δx = 0.2524

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Implied Transition Probabilities  Here is the code to work down through the

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transition probabilities

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Implied Transition Probabilities

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 Here are the full implied probabilities

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Implied Transition Probabilities  The code to implement the implied state prices

and implied transition probabilities is reasonably complex  But the advantages of such an implied tree are

many

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 To illustrate, we will briefly look at an exotic

option and apply our implied tree

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Implied Transition Probabilities  We consider the family of barrier options  Down and in call or put  Down and out call or put  Up and in call or put  Up and out call or put

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 These options are activated (=“in”) if the asset

price is below (=“down”) or above (=“up”) a predetermined level during specified dates  These options are de-activated (=“out”) if the

asset price is below (=“down”) or above (=“up”) a predetermined level during specified dates 15

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Implied Transition Probabilities  Some analytical solutions are available for these

options but these solutions are limited  We could use Monte Carlo methods or  We could apply the implied tree results to the

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pricing of the option.

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Implied Transition Probabilities  Consider an American Up-An-Out Put option  This gives the holder the right to sell an asset at

any time unless the asset price has breached a barrier during a specified time period

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 The following graph shows three possible paths

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Implied Transition Probabilities  Path 1 23 finishes sonot expires finishesout-of-the-money in-the-money andand didbroke breach the up-

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worthless and-out the barrier barrier so pays H so max(K-S expires worthless T,0)

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Implied Transition Probabilities  The mathematical representation for this

particular option is

max( 0, K − ST ) |min( St 1 ,..., Stm )< H

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 Using the same stock price processes as per the

examples in C&S, we can see what the option values at maturity will be

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Implied Transition Probabilities

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 Look again at the stock price trinomial tree

 What are the terminal option values if K = 100

and H = 110?

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Implied Transition Probabilities  The final nodes are easy to calculate

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max( 0, K − ST )

 What is the value of the option at C3,-2?

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Implied Transition Probabilities  The first check is to see if the value of the stock

at 3,-2 is greater than the barrier, H If (S3,-2 > H) C3,-2 = 0 else

max( 0, K − ST )

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continue …

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Implied Transition Probabilities  We have calculated the implied transition

probabilities from a standard option price, so we can use these here  For node 3,-2, these were

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Pu,3,-2 Pm,3,-2 Pd,3,-2

 Therefore, the option at 3,-2 is given by the

discounted expectation

C3, −2 = e

− r∆t

(p

u , 3, −2

C4 , −1 + pm ,3, −2C4 , −3 + pd ,3, −2C4 , −3 ) 23

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Implied Transition Probabilities  This given an option value of C3,-2 = 38.1486

 Now apply early exercise conditions

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 C3,-2 = max(C3,-2,K - S3,-2); 

= max(38.1486, 100-60.3626)



= 39.6374

 So early exercise is optimal at this point

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Implied Transition Probabilities  To summarise:  We have seen how implied trinomial trees can be

constructed from standard option market data  Using various maturity options along with

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interpolation, we can construct an implied state and probability transition tree  Using this data, we can calculate the price of

more exotic options where analytical solutions are unavailable

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Recommended Texts  Required/Recommended  Clewlow, L. and Strickland, C. (1996) Implementing derivative

models, 1st ed., John Wiley and Sons Ltd. — Chapter 5

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 Additional/Useful

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