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Pricing Equity Derivatives Using Trees

Computing Instrument Prices

The portfolio pricing functions crrprice, eqpprice, and ittprice calculate the price of any set of supported instruments based on a binary equity price tree, an implied trinomial price tree, or a standard trinomial tree. These functions are capable of pricing the following instrument types:

  • Vanilla stock options

    • American and European puts and calls

  • Exotic options

    • Asian

    • Barrier

    • Compound

    • Lookback

    • Stock options (Bermuda put and call schedules)

The syntax for calling the function crrprice is:

[Price, PriceTree] = crrprice(CRRTree, InstSet, Options)

The syntax for eqpprice is:

[Price, PriceTree] = eqpprice(EQPTree, InstSet, Options)

The syntax for ittprice is:

Price = ittprice(ITTTree, ITTInstSet, Options)

The syntax for sttprice is:

[Price, PriceTree] = sttprice(STTTree, InstSet, Name, Value)

These functions require two input arguments: the equity price tree and the set of instruments, InstSet, and allow a third optional argument.

Required Arguments

CRRTree is a CRR equity price tree created using crrtree. EQPTree is an equal probability equity price tree created using eqptree. ITTTree is an ITT equity price tree created using itttree. STTTree is a standard trinomial equity price tree created using stttree. See Building Equity Binary Trees and Building Implied Trinomial Trees to learn how to create these structures.

InstSet is a structure that represents the set of instruments to be priced independently using the model.

Optional Argument

You can enter a third optional argument, Options, used when pricing barrier options. For more specific information, see Derivatives Pricing Options.

These pricing functions internally classify the instruments and call the appropriate individual instrument pricing function for each of the instrument types. The CRR pricing functions are asianbycrr, barrierbycrr, compoundbycrr, lookbackbycrr, and optstockbycrr. A similar set of functions exists for EQP, ITT, and STT pricing. You can also use these functions directly to calculate the price of sets of instruments of the same type. See the reference pages for these individual functions for further information.

Computing Prices Using CRR

Consider the following example, which uses the portfolio and stock price data in the MAT-file deriv.mat included in the toolbox. Load the data into the MATLAB® workspace.

load deriv.mat

Use the MATLAB whos command to display a list of the variables loaded from the MAT-file.

Name              Size            Bytes  Class     Attributes

  BDTInstSet        1x1             27344  struct              
  BDTTree           1x1              7322  struct              
  BKInstSet         1x1             27334  struct              
  BKTree            1x1              8532  struct              
  CRRInstSet        1x1             21066  struct              
  CRRTree           1x1              7086  struct              
  EQPInstSet        1x1             21066  struct              
  EQPTree           1x1              7086  struct              
  HJMInstSet        1x1             27336  struct              
  HJMTree           1x1              8334  struct              
  HWInstSet         1x1             27334  struct              
  HWTree            1x1              8532  struct              
  ITTInstSet        1x1             21070  struct              
  ITTTree           1x1             12660  struct              
  STTInstSet        1x1             21070  struct              
  STTTree           1x1              7782  struct              
  ZeroInstSet       1x1             17458  struct              
  ZeroRateSpec      1x1              2152  struct     

CRRTree and CRRInstSet are the required input arguments to call the function crrprice.

Use instdisp to examine the set of instruments contained in the variable CRRInstSet.

instdisp(CRRInstSet)
Index Type     OptSpec Strike Settle         ExerciseDates  AmericanOpt Name  Quantity
1     OptStock call    105    01-Jan-2003    01-Jan-2005    1           Call1 10      
2     OptStock put     105    01-Jan-2003    01-Jan-2006    0           Put1   5      
 
Index Type    OptSpec Strike Settle         ExerciseDates  AmericanOpt BarrierSpec Barrier Rebate Name     Quantity
3     Barrier call    105    01-Jan-2003    01-Jan-2006    1           ui          102     0      Barrier1 1       
 
Index Type     UOptSpec UStrike USettle        UExerciseDates UAmericanOpt COptSpec CStrike CSettle        CExerciseDates CAmericanOpt Name      Quantity
4     Compound call     130     01-Jan-2003    01-Jan-2006    1            put      5       01-Jan-2003    01-Jan-2005    1            Compound1 3       
 
Index Type     OptSpec Strike Settle         ExerciseDates  AmericanOpt Name      Quantity
5     Lookback call    115    01-Jan-2003    01-Jan-2006    0           Lookback1 7       
6     Lookback call    115    01-Jan-2003    01-Jan-2007    0           Lookback2 9       
 
Index Type  OptSpec Strike Settle         ExerciseDates  AmericanOpt AvgType    AvgPrice AvgDate Name   Quantity
7     Asian put     110    01-Jan-2003    01-Jan-2006    0           arithmetic NaN      NaN     Asian1 4       
8     Asian put     110    01-Jan-2003    01-Jan-2007    0           arithmetic NaN      NaN     Asian2 6       

Note

Because of space considerations, the compound option above (Index 4) has been condensed to fit the page. The instdisp command displays all compound option fields on your computer screen.

The instrument set contains eight instruments:

  • Two vanilla options (Call1, Put1)

  • One barrier option (Barrier1)

  • One compound option (Compound1)

  • Two lookback options (Lookback1, Lookback2)

  • Two Asian options (Asian1, Asian2)

Each instrument has a corresponding index that identifies the instrument prices in the price vector returned by crrprice.

Now use crrprice to calculate the price of each instrument in the instrument set.

Price = crrprice(CRRTree, CRRInstSet)
Price =

    8.2863
    2.5016
   12.1272
    3.3241
    7.6015
   11.7772
    4.1797
    3.4219

Computing Prices Using EQP

Load the data into the MATLAB workspace.

load deriv.mat

Use the MATLAB whos command to display a list of the variables loaded from the MAT-file.

Name              Size            Bytes  Class     Attributes

  BDTInstSet        1x1             27344  struct              
  BDTTree           1x1              7322  struct              
  BKInstSet         1x1             27334  struct              
  BKTree            1x1              8532  struct              
  CRRInstSet        1x1             21066  struct              
  CRRTree           1x1              7086  struct              
  EQPInstSet        1x1             21066  struct              
  EQPTree           1x1              7086  struct              
  HJMInstSet        1x1             27336  struct              
  HJMTree           1x1              8334  struct              
  HWInstSet         1x1             27334  struct              
  HWTree            1x1              8532  struct              
  ITTInstSet        1x1             21070  struct              
  ITTTree           1x1             12660  struct              
  STTInstSet        1x1             21070  struct              
  STTTree           1x1              7782  struct              
  ZeroInstSet       1x1             17458  struct              
  ZeroRateSpec      1x1              2152  struct     

EQPTree and EQPInstSet are the input arguments required to call the function eqpprice.

Use the command instdisp to examine the set of instruments contained in the variable EQPInstSet.

instdisp(EQPInstSet)
Index Type     OptSpec Strike Settle         ExerciseDates  AmericanOpt Name  Quantity
1     OptStock call    105    01-Jan-2003    01-Jan-2005    1           Call1 10      
2     OptStock put     105    01-Jan-2003    01-Jan-2006    0           Put1   5      
 
Index Type    OptSpec Strike Settle         ExerciseDates  AmericanOpt BarrierSpec Barrier Rebate Name     Quantity
3     Barrier call    105    01-Jan-2003    01-Jan-2006    1           ui          102     0      Barrier1 1       
 
Index Type     UOptSpec UStrike USettle        UExerciseDates UAmericanOpt COptSpec CStrike CSettle        CExerciseDates CAmericanOpt Name      Quantity
4     Compound call     130     01-Jan-2003    01-Jan-2006    1            put      5       01-Jan-2003    01-Jan-2005    1            Compound1 3       
 
Index Type     OptSpec Strike Settle         ExerciseDates  AmericanOpt Name      Quantity
5     Lookback call    115    01-Jan-2003    01-Jan-2006    0           Lookback1 7       
6     Lookback call    115    01-Jan-2003    01-Jan-2007    0           Lookback2 9       
 
Index Type  OptSpec Strike Settle         ExerciseDates  AmericanOpt AvgType    AvgPrice AvgDate Name   Quantity
7     Asian put     110    01-Jan-2003    01-Jan-2006    0           arithmetic NaN      NaN     Asian1 4       
8     Asian put     110    01-Jan-2003    01-Jan-2007    0           arithmetic NaN      NaN     Asian2 6       
 
>> instdisp(EQPInstSet)
Index Type     OptSpec Strike Settle         ExerciseDates  AmericanOpt Name  Quantity
1     OptStock call    105    01-Jan-2003    01-Jan-2005    1           Call1 10      
2     OptStock put     105    01-Jan-2003    01-Jan-2006    0           Put1   5      
 
Index Type    OptSpec Strike Settle         ExerciseDates  AmericanOpt BarrierSpec Barrier Rebate Name     Quantity
3     Barrier call    105    01-Jan-2003    01-Jan-2006    1           ui          102     0      Barrier1 1       
 
Index Type     UOptSpec UStrike USettle        UExerciseDates UAmericanOpt COptSpec CStrike CSettle        CExerciseDates CAmericanOpt Name      Quantity
4     Compound call     130     01-Jan-2003    01-Jan-2006    1            put      5       01-Jan-2003    01-Jan-2005    1            Compound1 3       
 
Index Type     OptSpec Strike Settle         ExerciseDates  AmericanOpt Name      Quantity
5     Lookback call    115    01-Jan-2003    01-Jan-2006    0           Lookback1 7       
6     Lookback call    115    01-Jan-2003    01-Jan-2007    0           Lookback2 9       
 
Index Type  OptSpec Strike Settle         ExerciseDates  AmericanOpt AvgType    AvgPrice AvgDate Name   Quantity
7     Asian put     110    01-Jan-2003    01-Jan-2006    0           arithmetic NaN      NaN     Asian1 4       
8     Asian put     110    01-Jan-2003    01-Jan-2007    0           arithmetic NaN      NaN     Asian2 6       
 

Note

Because of space considerations, the compound option above (Index 4) has been condensed to fit the page. The instdisp command displays all compound option fields on your computer screen.

The instrument set contains eight instruments:

  • Two vanilla options (Call1, Put1)

  • One barrier option (Barrier1)

  • One compound option (Compound1)

  • Two lookback options (Lookback1, Lookback2)

  • Two Asian options (Asian1, Asian2)

Each instrument has a corresponding index that identifies the instrument prices in the price vector returned by eqpprice.

Now use eqpprice to calculate the price of each instrument in the instrument set.

Price = eqpprice(EQPTree, EQPInstSet)
Price =

    8.4791
    2.6375
   12.2632
    3.5091
    8.7941
   12.9577
    4.7444
    3.9178

Computing Prices Using ITT

Consider the following example, which uses the portfolio and stock price data in the MAT-file deriv.mat included in the toolbox. Load the data into the MATLAB workspace.

load deriv.mat

Use the MATLAB whos command to display a list of the variables loaded from the MAT-file.

Name              Size            Bytes  Class     Attributes

  BDTInstSet        1x1             27344  struct              
  BDTTree           1x1              7322  struct              
  BKInstSet         1x1             27334  struct              
  BKTree            1x1              8532  struct              
  CRRInstSet        1x1             21066  struct              
  CRRTree           1x1              7086  struct              
  EQPInstSet        1x1             21066  struct              
  EQPTree           1x1              7086  struct              
  HJMInstSet        1x1             27336  struct              
  HJMTree           1x1              8334  struct              
  HWInstSet         1x1             27334  struct              
  HWTree            1x1              8532  struct              
  ITTInstSet        1x1             21070  struct              
  ITTTree           1x1             12660  struct              
  STTInstSet        1x1             21070  struct              
  STTTree           1x1              7782  struct              
  ZeroInstSet       1x1             17458  struct              
  ZeroRateSpec      1x1              2152  struct     

ITTTree and ITTInstSet are the input arguments required to call the function ittprice. Use the command instdisp to examine the set of instruments contained in the variable ITTInstSet.

instdisp(ITTInstSet)
Index Type     OptSpec Strike Settle         ExerciseDates  AmericanOpt Name  Quantity
1     OptStock call    95     01-Jan-2006    31-Dec-2008    1           Call1 10      
2     OptStock put     80     01-Jan-2006    01-Jan-2010    0           Put1   4      
 
Index Type    OptSpec Strike Settle         ExerciseDates  AmericanOpt BarrierSpec Barrier Rebate Name     Quantity
3     Barrier call    85     01-Jan-2006    31-Dec-2008    1           ui          115     0      Barrier1 1       
 
Index Type     UOptSpec UStrike USettle        UExerciseDates UAmericanOpt COptSpec CStrike CSettle        CExerciseDates CAmericanOpt Name      Quantity
4     Compound call     99      01-Jan-2006    01-Jan-2010    1            put      5       01-Jan-2006    01-Jan-2010    1            Compound1 3       
 
Index Type     OptSpec Strike Settle         ExerciseDates  AmericanOpt Name      Quantity
5     Lookback call    85     01-Jan-2006    01-Jan-2008    0           Lookback1 7       
6     Lookback call    85     01-Jan-2006    01-Jan-2010    0           Lookback2 9       
 
Index Type  OptSpec Strike Settle         ExerciseDates  AmericanOpt AvgType    AvgPrice AvgDate Name   Quantity
7     Asian call    55     01-Jan-2006    01-Jan-2008    0           arithmetic NaN      NaN     Asian1 5       
8     Asian call    55     01-Jan-2006    01-Jan-2010    0           arithmetic NaN      NaN     Asian2 7       

The instrument set contains eight instruments:

  • Two vanilla options (Call1, Put1)

  • One barrier option (Barrier1)

  • One compound option (Compound1)

  • Two lookback options (Lookback1, Lookback2)

  • Two Asian options (Asian1, Asian2)

Each instrument has a corresponding index that identifies the instrument prices in the price vector returned by ittprice.

Now use ittprice to calculate the price of each instrument in the instrument set.

Price = ittprice(ITTTree, ITTInstSet)
Price =

    1.6506
   10.6832
    2.4074
    3.2294
    0.5426
    6.1845
    3.2052
    6.6074

Computing Prices Using STT

Consider the following example, which uses the portfolio and stock price data in the MAT-file deriv.mat included in the toolbox. Load the data into the MATLAB workspace.

load deriv.mat

Use the MATLAB whos command to display a list of the variables loaded from the MAT-file.

 Name              Size            Bytes  Class     Attributes

  BDTInstSet        1x1             27344  struct              
  BDTTree           1x1              7322  struct              
  BKInstSet         1x1             27334  struct              
  BKTree            1x1              8532  struct              
  CRRInstSet        1x1             21066  struct              
  CRRTree           1x1              7086  struct              
  EQPInstSet        1x1             21066  struct              
  EQPTree           1x1              7086  struct              
  HJMInstSet        1x1             27336  struct              
  HJMTree           1x1              8334  struct              
  HWInstSet         1x1             27334  struct              
  HWTree            1x1              8532  struct              
  ITTInstSet        1x1             21070  struct              
  ITTTree           1x1             12660  struct              
  STTInstSet        1x1             21070  struct              
  STTTree           1x1              7782  struct              
  ZeroInstSet       1x1             17458  struct              
  ZeroRateSpec      1x1              2152  struct     

STTTree and STTInstSet are the input arguments required to call the function sttprice. Use the command instdisp to examine the set of instruments contained in the variable STTInstSet.

instdisp(STTInstSet)
Index Type     OptSpec Strike Settle         ExerciseDates  AmericanOpt Name  Quantity
1     OptStock call    100    01-Jan-2009    01-Jan-2011    1           Call1 10      
2     OptStock put      80    01-Jan-2009    01-Jan-2012    0           Put1   5      
 
Index Type    OptSpec Strike Settle         ExerciseDates  AmericanOpt BarrierSpec Barrier Rebate Name     Quantity
3     Barrier call    105    01-Jan-2009    01-Jan-2012    1           ui          115     0      Barrier1 1       
 
Index Type     UOptSpec UStrike USettle        UExerciseDates UAmericanOpt COptSpec CStrike CSettle        CExerciseDates CAmericanOpt Name      Quantity
4     Compound call     95      01-Jan-2009    01-Jan-2012    1            put      5       01-Jan-2009    01-Jan-2011    1            Compound1 3       
 
Index Type     OptSpec Strike Settle         ExerciseDates  AmericanOpt Name      Quantity
5     Lookback call    90     01-Jan-2009    01-Jan-2012    0           Lookback1 7       
6     Lookback call    95     01-Jan-2009    01-Jan-2013    0           Lookback2 9       
 
Index Type  OptSpec Strike Settle         ExerciseDates  AmericanOpt AvgType    AvgPrice AvgDate Name   Quantity
7     Asian call    100    01-Jan-2009    01-Jan-2012    0           arithmetic NaN      NaN     Asian1 4       
8     Asian call    100    01-Jan-2009    01-Jan-2013    0           arithmetic NaN      NaN     Asian2 6       
 

The instrument set contains eight instruments:

  • Two vanilla options (Call1, Put1)

  • One barrier option (Barrier1)

  • One compound option (Compound1)

  • Two lookback options (Lookback1, Lookback2)

  • Two Asian options (Asian1, Asian2)

Each instrument has a corresponding index that identifies the instrument prices in the price vector returned by sttprice.

Now use sttprice to calculate the price of each instrument in the instrument set.

Price = sttprice(STTTree, STTInstSet)
Price =

    4.5025
    3.0603
    3.7977
    1.7090
   11.7296
   12.9120
    1.6905
    2.6203

Examining Output from the Pricing Functions

The prices in the output vector Price correspond to the prices at observation time zero (tObs = 0), which is defined as the valuation date of the equity tree. The instrument indexing within Price is the same as the indexing within InstSet.

In the CRR example, the prices in the Price vector correspond to the instruments in this order.

InstNames = instget(CRRInstSet, 'FieldName','Name')
InstNames =

Call1
Put1
Barrier1
Compound1 
Lookback1  
Lookback2  
Asian1     
Asian2     

So, in the Price vector, the fourth element, 3.3241, represents the price of the fourth instrument (Compound1), and the sixth element, 11.7772, represents the price of the sixth instrument (Lookback2).

In the ITT example, the prices in the Price vector correspond to the instruments in this order.

InstNames = instget(ITTInstSet, 'FieldName','Name')
InstNames =

Call1
Put1
Barrier1
Compound1 
Lookback1  
Lookback2  
Asian1     
Asian2     

So, in the Price vector, the first element, 1.650, represents the price of the first instrument (Call1), and the eighth elements, 6.607, represents the price of the eighth instrument (Asian2).

Price Tree Output for CRR

If you call a pricing function with two output arguments, for example:

[Price, PriceTree] = crrprice(CRRTree, CRRInstSet)

you generate a price tree structure along with the price information.

This price tree structure PriceTree holds all pricing information.

PriceTree =
FinObj: 'BinPriceTree'
PTree: {[8x1 double] [8x2 double] [8x3 double] [8x4 double] [8x5 double]}
tObs: [0 1 2 3 4]
dObs: [731582 731947 732313 732678 733043]

The first field of this structure, FinObj, indicates that this structure represents a price tree. The second field, PTree, is the tree holding the prices of the instruments in each node of the tree. Finally, the third and fourth fields, tObs and dObs, represent the observation time and date of each level of PTree, with tObs using units in terms of compounding periods.

Using the command-line interface, you can directly examine PriceTree.PTree, the field within the PriceTree structure that contains the price tree with the price vectors at every state. The first node represents tObs = 0, corresponding to the valuation date.

PriceTree.PTree{1}
ans =
8.2863
2.5016
12.1272
3.3241
7.6015
11.7772
4.1797
3.4219

With this interface, you can observe the prices for all instruments in the portfolio at a specific time.

The function eqpprice also returns a price tree that you can examine in the same way.

Price Tree Output for ITT

If you call a pricing function with two output arguments, for example:

[Price, PriceTree] = ittprice(ITTTree, ITTInstSet)

you generate a price tree structure along with the price information.

This price tree structure PriceTree holds all pricing information.

PriceTree = 

    FinObj: 'TrinPriceTree'
     PTree: {[8x1 double]  [8x3 double]  [8x5 double]  [8x7 double]  [8x9 double]}
      tObs: [0 1 2 3 4]
      dObs: [732678 733043 733408 733773 734139]

The first field of this structure, FinObj, indicates that this structure represents a trinomial price tree. The second field, PTree is the tree holding the prices of the instruments in each node of the tree. Finally, the third and fourth fields, tObs and dObs, represent the observation time and date of each level of PTree, with tObs using units in terms of compounding periods.

Using the command-line interface, you can directly examine PriceTree.PTree, the field within the PriceTree structure that contains the price tree with the price vectors at every state. The first node represents tObs = 0, corresponding to the valuation date.

PriceTree.PTree{1}
ans =

    1.6506
   10.6832
    2.4074
    3.2294
    0.5426
    6.1845
    3.2052
    6.6074

With this interface, you can observe the prices for all instruments in the portfolio at a specific time.

Prices for Lookback and Asian Options for Equity Trees

Lookback options and Asian options are path-dependent, and, as such, there are no unique prices for any node except the root node. So, the corresponding values for lookback and Asian options in the price tree are set to NaN, the only exception being the root node. This becomes apparent if you examine the prices in the second node (tobs = 1) of the CRR price tree:

PriceTree.PTree{2}
ans =

   11.9176         0
    0.9508    7.1914
   16.4600    2.6672
    2.5896    5.0000
       NaN       NaN
       NaN       NaN
       NaN       NaN
       NaN       NaN

Examining the prices in the second node (tobs = 1) of the ITT price tree displays:

PriceTree.PTree{2}  
 ans =

    3.9022         0         0
    6.3736   13.3743   22.1915
    5.6914         0         0
    2.7663    3.8594    5.0000
       NaN       NaN       NaN
       NaN       NaN       NaN
       NaN       NaN       NaN
       NaN       NaN       NaN

Graphical Representation of Equity Derivative Trees

You can use the function treeviewer to display a graphical representation of a tree, allowing you to examine interactively the prices and rates on the nodes of the tree until maturity. The graphical representations of CRR, EQP, and LR trees are equivalent to Black-Derman-Toy (BDT) trees, given that they are all binary recombining trees. The graphical representations of ITT and STT trees are equivalent to Hull-White (HW) trees, given that they are all trinomial recombining trees. See Graphical Representation of Trees for an overview on the use of treeviewer with CRR trees, EQP trees, LR trees, ITT trees, and STT trees and their corresponding option price trees. Follow the instructions for BDT trees.

See Also

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