Pricing Options Structure

Introduction

The MATLAB^® Options structure provides additional input to most pricing functions. The Options structure

• Tells pricing functions how to use the interest-rate tree to calculate instrument prices.

• Determines what additional information the Command Window displays along with instrument prices.

• Tells pricing functions which method to use in pricing barrier options.

The pricing options structure is primarily used in the pricing of interest-rate-based financial derivatives. However, the BarrierMethod field in the structure allows you to use it in pricing equity barrier options as well.

You provide pricing options in an optional Options argument passed to a pricing function. (See, for example, bondbyhjm, bdtprice, barrierbycrr, barrierbyeqp, or barrierbyitt.)

Default Structure

If you do not specify the Options argument in the call to a pricing function, the function uses a default structure. To observe the default structure, use derivset without any arguments.

Options = derivset
 
Options = 
 
    Diagnostics: 'off'
       Warnings: 'on'
      ConstRate: 'on'
  BarrierMethod: 'unenhanced'

The Options structure has four fields: Diagnostics, Warnings, ConstRate, and BarrierMethod.

Diagnostics Field

Diagnostics indicates whether additional information is displayed if the tree is modified. The default value for this option is 'off'. If Diagnostics is set to 'on' and ConstRate is set to 'off', the pricing functions display information such as the number of nodes in the last level of the tree generated for pricing purposes.

Warnings Field

Warnings indicates whether to display warning messages when the input tree is not adequate for accurately pricing the instruments. The default value for this option is 'on'. If both ConstRate and Warnings are 'on', a warning is displayed if any of the instruments in the input portfolio have a cash flow date between tree dates. If ConstRate is 'off', and Warnings is 'on', a warning is displayed if the tree is modified to match the cash flow dates on the instruments in the portfolio.

ConstRate Field

ConstRate indicates whether the interest rates should be assumed constant between tree dates. By default this option is 'on', which is not an arbitrage-free assumption. So the pricing functions return an approximate price for instruments featuring cash flows between tree dates. Instruments featuring cash flows only on tree nodes are not affected by this option and return exact (arbitrage-free) prices. When ConstRate is 'off', the pricing function finds the cash flow dates for all instruments in the portfolio. If these cash flows do not align exactly with the tree dates, a new tree is generated and used for pricing. This new tree features the same volatility and initial rate specifications of the input tree but contains tree nodes for each date in which at least one instrument in the portfolio has a cash flow. Keep in mind that the number of nodes in a tree grows exponentially with the number of tree dates. So, setting ConstRate 'off' dramatically increases the memory and processor demands on the computer.

BarrierMethod Field

When using binomial trees to price barrier options, this may require many tree steps to achieve an accurate result when tree nodes do not align with the barrier level. With the BarrierMethod field, the toolbox provides an enhancement method that improves the accuracy of the results without having to use large trees.

The BarrierMethod field can be set to 'unenhanced' (default) or 'interp'. If you specify 'unenhanced', no correction calculation is used. Otherwise, if you specify 'interp', the toolbox provides an enhanced valuation by interpolating between nodes on barrier boundaries.

You specify the barrier method in the last input argument, Options, of the functions barrierbycrr, barrierbyeqp, barrierbyitt, crrprice, eqpprice, ittprice, crrsens, eqpsens, or ittsens. Options is a structure that you create with the function derivset. Using derivset, you specify whether to use the enhanced or the unenhanced method.

For more information about this algorithm, see Derman, E., I. Kani, D. Ergener and I. Bardhan, “Enhanced Numerical Methods for Options with Barriers,” Financial Analysts Journal, (Nov. - Dec. 1995), pp. 65–74.

Customizing the Structure

Customize the Options structure by passing property name/property value pairs to the derivset function.

As an example, consider an Options structure with ConstRate 'off' and Diagnostics 'on'.

Options = derivset('ConstRate', 'off', 'Diagnostics', 'on')

Options = 

  Diagnostics: 'on'
     Warnings: 'on'
    ConstRate: 'off'
BarrierMethod: 'unenhanced'

To obtain the value of a specific property from the Options structure, use derivget.

CR = derivget(Options, 'ConstRate')

CR =
Off

Note

Use derivset and derivget to construct the Options structure. These functions are guaranteed to remain unchanged, while the implementation of the structure itself may be modified in the future.

Now observe the effects of setting ConstRate 'off'. Obtain the tree dates from the HJM tree.

TreeDates = [HJMTree.TimeSpec.ValuationDate;... 
HJMTree.TimeSpec.Maturity]

TreeDates =

     730486
     730852
     731217
     731582
     731947

datedisp(TreeDates)

01-Jan-2000 
01-Jan-2001 
01-Jan-2002 
01-Jan-2003 
01-Jan-2004 

All instruments in HJMInstSet settle on January 1, 2000, and all have cash flows once a year, except for the second bond, which features a period of 2. This bond has cash flows twice a year, with every other cash flow consequently falling between tree dates. You can extract this bond from the portfolio to compare how its price differs by setting ConstRate to 'on' and 'off'.

BondPort = instselect(HJMInstSet, 'Index', 2);

instdisp(BondPort)

Index Type CouponRate Settle      Maturity     Period Basis... 
1     Bond 0.04       01-Jan-2000 01-Jan-2004  2      NaN...

First price the bond with ConstRate 'on' (default).

format long
[BondPrice, BondPriceTree] = hjmprice(HJMTree, BondPort)
Warning: Not all cash flows are aligned with the tree. Result will 
be approximated.

BondPrice =

  97.52801411736377

BondPriceTree = 
FinObj: 'HJMPriceTree'
 PBush: {1x5 cell}
AIBush: {[0]  [1x1x2 double] ... [1x4x2 double]  [1x8 double]}
  tObs: [0 1 2 3 4]

Now recalculate the price of the bond setting ConstRate 'off'.

OptionsNoCR = derivset('ConstR', 'off')

OptionsNoCR = 

Diagnostics: 'off'
   Warnings: 'on'
  ConstRate: 'off'

[BondPriceNoCR, BondPriceTreeNoCR] = hjmprice(HJMTree,... 
BondPort, OptionsNoCR)
Warning: Not all cash flows are aligned with the tree. Rebuilding 
tree.

BondPriceNoCR =

  97.53342361674437

BondPriceTreeNoCR = 

FinObj: 'HJMPriceTree'
 PBush: {1x9 cell}
AIBush: {1x9 cell}
  tObs: [0 0.5000 1 1.5000 2 2.5000 3 3.5000 4]

As indicated in the last warning, because the cash flows of the bond did not align with the tree dates, a new tree was generated for pricing the bond. This pricing method returns more accurate results since it guarantees that the process is arbitrage-free. It also takes longer to calculate and requires more memory. The tObs field of the price tree structure indicates the increased memory usage. BondPriceTree.tObs has only five elements, while BondPriceTreeNoCR.tObs has nine. While this may not seem like a large difference, it has a dramatic effect on the number of states in the last node.

size(BondPriceTree.PBush{end})

ans =

     1 8

size(BondPriceTreeNoCR.PBush{end})

ans =

     1 128

The differences become more obvious by examining the price trees with treeviewer.

treeviewer(BondPriceTree, BondPort)

treeviewer(BondPriceTreeNoCR, BondPort)

All = [Delta ./ Price, Gamma ./ Price, Vega ./ Price, Price]

All =

         -2.76         10.43          0.00         98.72
         -3.56         16.64         -0.00         97.53
       -166.18      13235.59        700.96          0.05
         -2.76         10.43          0.00         98.72
         -0.01          0.03             0        100.55
         46.95       1090.63         14.91          6.28
       -969.85     173969.77       1926.72          0.05
        -76.39        287.00          0.00          3.690