Implied Trinomial Trees: Autumn 2 0 0 9

A U T U M N    2 0 0 9

IMPLIED TRINOMIAL TREES

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Numerical Methods in Finance (Implementing Market Models)

©Finbarr Murphy 2007

Lecture Objectives  Implied Trinomials  Understand State Prices  Understand the code to find state prices from market prices of

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call and put options

©Finbarr Murphy 2007

Agenda Page

Implied Trinomial Trees

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Implied Trinomial Trees  Standard or Vanilla Option Contracts traded on

exchanges such as the CBOE, contain important information within the price  Market expectations are implied in the price  E.g. Implied volatility of Dec ’07 contract = 19%

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 Implied volatility of Feb ’08 contract = 20.5%

 One can construct or imply a term structure of

volatilities from such information  Better yet a volatility surface

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Implied Trinomial Trees VolatilitySurface

0.24

Volatility

0.22

0.2

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0.18

0.16

5

5 4

maturity

4 3

2

3 2

1 1

SwapDuration

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©Finbarr Murphy 2007

Implied Trinomial Trees  We can better price non-standard or exotic options

from this implied market expectation  In particular, we can construct trees or lattices to

price exotic options that are consistent with standard option prices

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 We can “fit” or imply a tree such that its option value

equals the market price of the option  Then, putting these implied trees together, we can

price exotic options

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©Finbarr Murphy 2007

Implied Trinomial Trees  In our trinomial tree, we have  Time steps Δti — Where i = 1, … ,N — Note the i subscript

j=2

Ci+1,j+1

Δx j=1

Ci+1,j

Ci,j

Δx

 and

j=0

Si , j = Se j∆x

Δx j=-1

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Δx j=-2 i=0

Δt1

i=1

Δt2

i=2

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©Finbarr Murphy 2007

Implied Trinomial Trees  At each node, we have a state

price Qi,j  This is the current price of a

unit currency (one $) if a future state is reached  It is zero if the state is not

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reached  Like a zero bond but with the

Q2,2

j=2

j=1

j=0

Q2,1

Q1,0

Q2,0

Q2,-1

j=-1

j=-2 i=0

Q1,1

i=1

Q2,-2 i=2

added binary state

 This state price is a useful concept which we will utilise

 later This state price is a useful

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©Finbarr Murphy 2007

Implied Trinomial Trees  Now, we can generalise and say

that the price of a European Option with a strike price K and maturity NΔt is given by

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C ( K , N∆t ) =

∑ max( S N

j =− N

N, j

− K ,0 )QN , j

 It is often easier to

conceptualise by considering a simple 2-step grid as described above

Q2,2

j=2

j=1

j=0

Q2,1

Q1,0

Q2,0

Q2,-1

j=-1

j=-2 i=0

Q1,1

i=1

Q2,-2 i=2

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©Finbarr Murphy 2007

Implied Trinomial Trees  Now, what is the price of a call

option if the strike price, K, is set to SN,N-1?

C ( K = S N , N −1 , N∆t ) = ( S N , N − S N , N −1 )QN , N

Q2,2 S2,2

j=2

j=1

j=0

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j=-2 i=0

C ( K = S 2,1 , N∆t ) = ( S 2, 2 − S 2,1 )Q2, 2

Q2,1 S2,1

Q1,0

Q2,0 S2,0

K=SN,N-1 =S2,1

Q2,-1 S2,-1

j=-1

 Or, as in our specific 2-step model

Q1,1

i=1

Q2,-2 S2,-2 i=2

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©Finbarr Murphy 2007

Implied Trinomial Trees  We know the value of the call

option (observing market data)  And we know S2,2 and the strike

K=S2,1

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 So, we can calculate Q2,2

Q2,2 S2,2

j=2

j=1

j=0

Q2,1 S2,1

Q1,0

Q2,0 S2,0

K=SN,N-1 =S2,1

Q2,-1 S2,-1

j=-1

j=-2 i=0

Q1,1

i=1

Q2,-2 S2,-2 i=2

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©Finbarr Murphy 2007

Implied Trinomial Trees  Now, drop the strike one level so

that K = SN,N-2  We know that

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C ( K = S N , N − 2 , N∆t ) =

(S (S

N ,N

− S N , N − 2 ) QN , N +

N , N −1

− S N , N − 2 )QN , N −1

Q2,2 S2,2

j=2

j=1

j=0

Q2,1 S2,1

Q1,0

Q2,0 S2,0

K=SN,N-2 =S2,0

Q2,-1 S2,-1

j=-1

j=-2 i=0

Q1,1

i=1

Q2,-2 S2,-2 i=2

 Or, in our 2-step model;

C ( K = S 2, 0 , N∆t ) = ( S 2, 2 − S 2, 0 )Q2, 2 + ( S 2,1 − S 2, 0 )Q2,1 12

©Finbarr Murphy 2007

Implied Trinomial Trees  From the last equation, we know

all value except for Q2,1 which we can easily calculate

Q2,2 S2,2

j=2

j=1

 We can continue in this manner to

calculate all values of QN,j

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C ( S N , j −1 , N∆t ) = ( S N , j − S N , j −1 )QN , j + N ∑k = j +1 ( S N ,k − S N , j −1 )QN ,k  Normally, we stop at QN,0 and work

j=0

Q2,1 S2,1

Q1,0

Q2,0 S2,0

K=SN,N-2 =S2,0

Q2,-1 S2,-1

j=-1

j=-2 i=0

Q1,1

i=1

Q2,-2 S2,-2 i=2

from the bottom up using put values

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©Finbarr Murphy 2007

Implied Trinomial Trees  Time for an example: Wal-Mart Near-The-Money

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Puts and calls within a 4-month maturity

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Implied Trinomial Trees  Time for an example: We’ll start with Clewlow and

Strickland Example P136  T = 1 (year)  S = 100  R = 6% (interest rate)  N = 4 (number of steps)

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 Δx = 0.2524 ( σ3Δt)

 The code is short, easily understood but tricky to

implement  We will implement step by step

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Implied Trinomial Trees

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 Initialise variables and some pre-calculations:

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Implied Trinomial Trees  Populate the stock prices in the tree

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 Observe the output:

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Implied Trinomial Trees  We don’t have market option prices for this

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example so we’ll just calculate them:

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Implied Trinomial Trees  Start top-right, work down to the middle using our

call option prices  The tricky bit is the selection of I, j and k  Tip: First ignore the calculations and make sure you

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traverse the grid correctly

∑

N k = j +1

(S

N ,k

− S N , j −1 )QN ,k

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©Finbarr Murphy 2007

Implied Trinomial Trees  Here are the state prices after the call options

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have been used:

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Implied Trinomial Trees  Task: Calculate the state prices for the bottom

portion of the tree  Tips & Advice:  Draw the grid on a page, for each calculation, note the

grid coordinates  Replicate these coordinates with variables  E.g. 

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     

for i=N+1:-1:2 for j=N+i-1:-1:N+2 for k=j+1:N+i disp(sprintf('i = %d, j = %d, k = %d',i,j,k)); end end end

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©Finbarr Murphy 2007

Implied Trinomial Trees  Produces the following values:  i = 5, j = 8,  i = 5, j = 7,  i = 5, j = 7,  i = 5, j = 6,

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 i = 5, j = 6,  i = 5, j = 6,  i = 4, j = 7,  i = 4, j = 6,  i = 4, j = 6,  i = 3, j = 6,

k k k k k k k k k k

= = = = = = = = = =

9 8 9 7 8 9 8 7 8 7

 I.e. the coordinates required in the correct

sequence

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©Finbarr Murphy 2007

Implied Trinomial Trees  Back to our specific example.  Clearly, the available data does not easily lend

itself to a grid so we must use various techniques such as interpolation to generate our lattice

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 Look at the mid-call prices Mid-Price

Strike

1.600

42.5

0.425

45.0

 We can calculate the At-The-Money Call option

price using interpolation. E.g. Matlab code  interp1([42.5 45], [1.6 .425], 43.32)  Ans = 1.2146 23

©Finbarr Murphy 2007

Implied Trinomial Trees  Next, consider the time-steps: The analysis is on

Aug-30 2007  Maturity is Sept 19th 2007 (third Wednesday)  This is 20 days  T = 20/365

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 N=4

 Therefore  dt = 5/365

 Space steps are set at σ3Δt = 0.0405

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Implied Trinomial Trees

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 Very little difference to the C&S example:

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Implied Trinomial Trees

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 But, the resultant stock tree is of course different

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Implied Trinomial Trees  Using interpolation techniques, we estimate the

option price values

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 Giving

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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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