CapeTools Generic MonteCarlo Pricer

General Description

Functions to price generic Equity/FX/Commodity structures under a MonteCarlo framework. If your product's payoff does not refer to asset prices in the future (ie - Bermudan / American options) you can use this framework.

Here are some general rules to keep in mind regarding how the payoff grid will be processed :

Rows are time points. As you read down the grid you are moving forward in time.

Columns represent your payoff formulas. You can create as many columns as you like. Columns are there to help you split up your payoff, or for a payoff function within a cell to refer to other cells within the grid.

For the MonteCarlo pricer, the pricing starts at the first timestep and prices from left to right.

Formula expressions within a cell cannot refer to formula cells to the right of itself.

Formula expressions within a cell cannot refer to formula cells below itself (future time points).

If you bear the 5 points in mind, this should aid you in constructing payoff grids.

Below is an example of a payoff table. The Step column is not required.

Step EventDate Repeat? PayOff Col PayOff Col PayOff Col PayOff Col
1 #19/07/2005# FALSE Asset(1, cRow) + _a + 5pct Asset(2, cRow) + Interp1D(1,cRow) - 100bp GridRowAvg(cRow, cCol-2, cCol-1)
2 #19/01/2006# FALSE Asset(1, cRow) + _a + 5pct Asset(2, cRow) + Interp1D(1,cRow) - 100bp GridRowAvg(cRow, cCol-2, cCol-1)
3 #19/07/2006# FALSE Asset(1, cRow) + _a + 5pct Asset(2, cRow) + Interp1D(1,cRow) - 100bp GridRowAvg(cRow, cCol-2, cCol-1)
4 #19/01/2007# FALSE Asset(1, cRow) + _a + 5pct Asset(2, cRow) + Interp1D(1,cRow) - 100bp GridRowAvg(cRow, cCol-2, cCol-1)
5 #19/07/2007# FALSE Asset(1, cRow) + _a + 5pct Asset(2, cRow) + Interp1D(1,cRow) - 100bp GridRowAvg(cRow, cCol-2, cCol-1)
6 #19/01/2008# FALSE Asset(1, cRow) + _a + 5pct Asset(2, cRow) + Interp1D(1,cRow) - 100bp GridRowAvg(cRow, cCol-2, cCol-1)
7 #19/07/2008# FALSE Asset(1, cRow) + _a + 5pct Asset(2, cRow) + Interp1D(1,cRow) - 100bp GridRowAvg(cRow, cCol-2, cCol-1)
8 #19/01/2009# FALSE Asset(1, cRow) + _a + 5pct Asset(2, cRow) + Interp1D(1,cRow) - 100bp GridRowAvg(cRow, cCol-2, cCol-1)
9 #19/07/2009# FALSE Asset(1, cRow) + _a + 5pct Asset(2, cRow) + Interp1D(1,cRow) - 100bp GridRowAvg(cRow, cCol-2, cCol-1)
10 #19/01/2010# FALSE Asset(1, cRow) + _a + 5pct Asset(2, cRow) + Interp1D(1,cRow) - 100bp GridRowAvg(cRow, cCol-2, cCol-1)
11 #19/07/2010# FALSE Asset(1, cRow) + _a + 5pct Asset(2, cRow) + Interp1D(1,cRow) - 100bp GridRowAvg(cRow, cCol-2, cCol-1)
12 #19/01/2011# FALSE Asset(1, cRow) + _a + 5pct Asset(2, cRow) + Interp1D(1,cRow) - 100bp GridRowAvg(cRow, cCol-2, cCol-1) max(GridColMin(cCol-1, 1, cRow) - AssetMin(1, 1, cRow), 0.0) * DCF(1, cRow)

The 'Repeat' column indicates that the instruction at the current step will be repeated for all the timesteps between this and the next.

Even though there are only 12 timesteps above, you can instruct the function to insert additional steps between each step.

Financial variables (case sensitive) :

cRow - refers to the current Row with the payoff table. Within all the Financial functions listed below, we recommend that you always use this variable (with perhaps offset values, ie - cRow, cRow-1, cRow+2 etc...) instead of a fixed value (ie - 1, 2 etc...). This is important especially if you enable American type handling from one timestep to another. Using static integer values will generate incorrect results.

cCol - refers to the current Column with the payoff table. Within all the financial functions listed below, we recommend that you always use this variable (with perhaps offset values, ie - cCol, cCol-1, cCol+2 etc...).

pct - When an integer value is placed in front of this expression (no spaces), the function will interp the whole expression as a percent. (ie - 5pct => 0.05)

bp - When an integer value is placed in front of this expression (no spaces), the function will interp the whole expression as basis points. (ie - 5bp => 0.0005)

_XXX - Constant variables can be defined by filling in the 'ConstParams' parameter of the GenericMCPricer() function. All variables must be defined with a leading underscore character ( ie - _XXX = 0.5)

Financial functions (case sensitive) :

Asset(AssetNo, TimeStep)) - Retrieves the value of an asset value of an asset at the requested 'TimeStep'. If a CorrArrayProcesses process object was passed into the MonteCarlo pricer, the 'AssetNo is the index into this asset array. If a single Process object was passed in, the 'AssetNo' should equal one (1).

Cell(row, col) - Retrieves a computed value within the PayOff Table. You can only refer to rows and columns below and to the left of the current cell location. You can also refer to the current cell location. This is because the MonteCarlo pricer prices forwards in time and thus future payoff values have not been computed as yet. The expression 'Cell(cRow, cCol-1)' retrieves the payoff value for the current timestep but at the previous column.

DCF(AssetNo, TimeStep)) - Retrieves the discount value of an asset at the requested 'TimeStep'. . If a CorrArrayProcesses process object was passed into the MonteCarlo pricer, the 'AssetNo is the index into this asset array. If a single Process object was passed in, the 'AssetNo' should equal one (1). Unlike some of the other pricers, the MonteCarlo pricer does not discount values. This function is needed in order to compute the discount factor needed for discounting. The discount factor curve will be retrieved from the Process object indicated by the 'AssetNo' parameter.

DCF2(AssetNo, FromTimeStep, ToTimeStep)) - Retrieves the discount value between two timeSteps (ie - discount factor between step 'FromTimeStep' and 'ToTimeStep'). If a CorrArrayProcesses process object was passed into the MonteCarlo pricer, the 'AssetNo is the index into this asset array. If a single Process object was passed in, the 'AssetNo' should equal one (1). Unlike some of the other pricers, the MonteCarlo pricer does not discount values. This function is needed in order to compute the discount factor needed for discounting. The discount factor curve will be retrieved from the Process object indicated by the 'AssetNo' parameter.

AssetMin(AssetNo, startRow, endRow) - Retrieves the minimum asset value from a starting to an ending 'step' range. For the 'startRow' and 'endRow' parameters you can only refer to rows above or equal to the current cell location (past dates).

AssetMax(AssetNo, startRow, endRow) - Retrieves the maximum asset value from a starting to an ending 'step' range. For the 'startRow' and 'endRow' parameters you can only refer to rows above or equal to the current cell location (past dates).

AssetSum(AssetNo, startRow, endRow) - Retrieves the sum of the asset's value from a starting to an ending 'step' range. For the 'startRow' and 'endRow' parameters you can only refer to rows above or equal to the current cell location (past dates).

AssetAvg(AssetNo, startRow, endRow) - Retrieves the average of the asset's value from a starting to an ending 'step' range. For the 'startRow' and 'endRow' parameters you can only refer to rows above or equal to the current cell location (past dates).

GridRowMin(Row, startCol, endCol) - Retrieves the minimum value across the columns of the PayOff table (given a 'TimeStep' or row value). For the 'startCol' and 'endCol' parameters you can only refer to columns to the left or equal to the current cell location.

GridRowMax(Row, startCol, endCol) - Retrieves the maximum value across the columns of the PayOff table (given a 'TimeStep' or row value). For the 'startCol' and 'endCol' parameters you can only refer to columns to the left or equal to the current cell location.

GridRowAvg(Row, startCol, endCol) - Retrieves the average value across the columns of the PayOff table (given a 'TimeStep' or row value). For the 'startCol' and 'endCol' parameters you can only refer to columns to the left or equal to the current cell location.

GridRowSum(Row, startCol, endCol) - Retrieves the sum across the columns of the PayOff table (given a 'TimeStep' or row value). For the 'startCol' and 'endCol' parameters you can only refer to columns to the left or equal to the current cell location.

GridColMin(Col, startRow, endRow) - Retrieves the minimum value across the rows of the PayOff table (given a column value). For the 'startRow' and 'endRow' parameters you can only refer to rows above or equal to the current cell location (past dates).

GridColMax(Col, startRow, endRow) - Retrieves the maximum value across the rows of the PayOff table (given a column value). For the 'startRow' and 'endRow' parameters you can only refer to rows above or equal to the current cell location (past dates).

GridColAvg(Col, startRow, endRow) - Retrieves the average value across the rows of the PayOff table (given a column value). For the 'startRow' and 'endRow' parameters you can only refer to rows above or equal to the current cell location (past dates).

GridColSum(Col, startRow, endRow) - Retrieves the sum value across the rows of the PayOff table (given a column value). For the 'startRow' and 'endRow' parameters you can only refer to rows above or equal to the current cell location (past dates).

Interp1D(nIndex, XValue ) - If any 1D interpolation objects have been passed into GenericMCPricer() function (a maximum of 2 objects can be defined), then you can execute this function within your payoff grid. The 'nIndex' parameter refers to the intepolation object (1-based). 'XValue' is the value where the interpolation is to be performed. The interpolation object will not extrapolate values.

Interp2D(nIndex, XValue, YValue) - If any 2D interpolation objects have been passed into GenericMCPricer() function (a maximum of 2 objects can be defined), then you can execute this function within your payoff grid. The 'nIndex' parameter refers to the intepolation object (1-based). 'XValue' and 'YValue' are the values where the interpolation is to be performed. The interpolation object will not extrapolate values.

iff(condition, a, b) - Tests for the condition and returns the value of 'a' if true or 'b' if false.

Predefined functions (case sensitive) :

sin(x) ( sine function )

cos(x) ( cosine function )

tan(x) ( tangens function )

asin(x) ( arcus sine function )

acos(x) ( arcus cosine function )

atan(x) ( arcus tangens function )

sinh(x) ( hyperbolic sine function )

cosh(x) ( hyperbolic cosine function )

tanh(x) ( hyperbolic tangens function )

asinh(x) ( hyperbolic arcus sine function )

acosh(x) ( hyperbolic arcus cosine function )

atanh(x) ( hyperbolic arcur tangens function )

log2(x) ( logarithm to the base 2 )

log10(x) ( logarithm to the base 10 )

log(x) ( logarithm to the base 10 )

ln(x) ( logarithm to the base e (2.71828...)

exp(x) ( e raised to the power of x )

sqrt(x) ( square root of a value )

sign(x) ( sign function ) -1=(if x<0), 1=(if x>0 )

rint(x) ( round to nearest integer )

abs(x) ( absolute value )

min(a,b,c,d,e,...) ( min of all arguments ie - min(-2,1,5,-6) )

max(a,b,c,d,e,...) ( max of all arguments ie - max(-2,1,5,-6) )

sum(a,b,c,d,e,...) ( sum of all arguments ie - sum(-2,1,5,-6) )

avg(a,b,c,d,e,...) ( mean value of all arguments ie - avg(-2,1,5,-6) )

ite(x,y,z) ( if ... then ... else ... : ie - ite(a<b, 5.0, -3.0) )

Predefined operators :

+ ( addition )

- ( subtraction )

* ( multiplication )

/ ( division )

^ ( raise x to the power of y )

Predefined operators that can be used within the first parameter of the ite() or iff() function :

and ( logical and )

or ( logical or )

<= ( less or equal )

>= ( greater or equal )

!= ( not equal )

== ( equal to )

> ( greater than )

< ( less than )

Brackets, '(' and ')', can also be used to simplify expressions.

Some possible examples of what is possible now follow (Today = 19/07/2005):

Vanilla Option. (The Step column is not required).

Step EventDate Repeat? PayOff Col
1 #19/07/2006# FALSE max(Asset(1, cRow) - 10, 0) * DCF(1, cRow)

Two Vanilla Options (Call and Put). (The Step column is not required).

Step EventDate Repeat? PayOff Col PayOff Col PayOff Col
1 #19/07/2006# FALSE 100 max(Asset(1, cRow) - Cell(cRow, cCol-1), 0) * DCF(1, cRow)
2 #19/07/2007# FALSE 90 max(Cell(cRow, cCol-1) - Asset(1, cRow), 0) * DCF(1, cRow) GridColSum(cCol-1, 1, cRow)

Asian Lookback Option. (The Step column is not required).

Step EventDate Repeat? PayOff Col PayOff Col
1 #19/07/2005# FALSE Asset(1, cRow)
2 #19/01/2006# FALSE Asset(1, cRow)
3 #19/07/2006# FALSE Asset(1, cRow)
4 #19/01/2007# FALSE Asset(1, cRow)
5 #19/07/2007# FALSE Asset(1, cRow) max(GridColAvg(cCol-1, 1, cRow) - GridColMax(cCol-1, 1, cRow), 0) * DCF(1, cRow)

Asian Option with fixed strike. (The Step column is not required).

Step EventDate Repeat? PayOff Col PayOff Col
1 #19/07/2005# FALSE Asset(1, cRow)
2 #19/01/2006# FALSE Asset(1, cRow)
3 #19/07/2006# FALSE Asset(1, cRow)
4 #19/01/2007# FALSE Asset(1, cRow)
5 #19/07/2007# FALSE Asset(1, cRow) max(GridColAvg(cCol-1, 1, cRow) - 10, 0) * DCF(1, cRow)

Asian Option with fixed strike (daily averaging). (The Step column is not required).

Step EventDate Repeat? PayOff Col
1 #19/07/2005# TRUE
2 #19/01/2006# TRUE
3 #19/07/2006# TRUE
4 #19/01/2007# TRUE
5 #19/07/2007# FALSE max(AssetAvg(1, 1, cRow) - 10, 0) * DCF(1, cRow)

Asian Option with floating strike. (The Step column is not required).

Step EventDate Repeat? PayOff Col PayOff Col
1 #19/07/2005# FALSE Asset(1, cRow)
2 #19/01/2006# FALSE Asset(1, cRow)
3 #19/07/2006# FALSE Asset(1, cRow)
4 #19/01/2007# FALSE Asset(1, cRow)
5 #19/07/2007# FALSE Asset(1, cRow) max(GridColAvg(cCol-1, 1, cRow) - Cell(cRow, cCol), 0) * DCF(1, cRow)

Asian Option with floating strike (daily averaging). (The Step column is not required).

Step EventDate Repeat? PayOff Col
1 #19/07/2005# TRUE
2 #19/01/2006# TRUE
3 #19/07/2006# TRUE
4 #19/01/2007# TRUE
5 #19/07/2007# FALSE max(AssetAvg(1, 1, cRow) - Cell(cRow, cCol), 0) * DCF(1, cRow)

Quanto Asian Basket Option with floating strike. (The Step column is not required).

This example was picked up from one of the QuantLib's mailing list (with some modifications).

Let us consider a quanto asian basket average rate option, which has as underlying 2 equities quoted in currencies AAA and BBB (the values are not "real word" data).

Underlying #1: current price = 75 AAA, dividend yield = 3.1%, volatility = 20%
Underlying #2: current price = 100 BBB, dividend yield = 2%, volatility = 15%

AAA risk free rate: 5.1%
BBB risk free rate: 3.5%
Exchange rate AAA/BBB: 1.5
Volatility of exchange rate AAA/BBB: 15%
Correlation between Underlying #2 and the exchange rate = 0.4

The option is paid in currency AAA and has the payoff of a quanto asian
Basket (note: in the payoff there is no exchange rate):

Basket1 = underlying1[samplingDate1] + underlying2[samplingDate1]
Basket2 = underlying1[samplingDate2] + underlying2[samplingDate2]
.
.
.
BasketN = underlying1[samplingDateN] + underlying2[samplingDateN]

Average = (basket1 + basket2 + ... + basketN) / N
Strike: 175
Sampling dates: 19/07/2005, 19/01/2006, 19/07/2006, 19/01/2007, 19/07/2007
Expiry date: 19/07/2007
Payoff at expiry = max(0; Average - Strike)

One can use the following payoff table :

Step EventDate Repeat? PayOff Col PayOff Col PayOff Col PayOff Col
1 #19/07/2005# FALSE Asset(1, cRow) Asset(2, cRow) Cell(cRow, cCol-2)+Cell(cRow, cCol-1)
2 #19/01/2006# FALSE Asset(1, cRow) Asset(2, cRow) Cell(cRow, cCol-2)+Cell(cRow, cCol-1)
3 #19/07/2006# FALSE Asset(1, cRow) Asset(2, cRow) Cell(cRow, cCol-2)+Cell(cRow, cCol-1)
4 #19/01/2007# FALSE Asset(1, cRow) Asset(2, cRow) Cell(cRow, cCol-2)+Cell(cRow, cCol-1)
5 #19/07/2007# FALSE Asset(1, cRow) Asset(2, cRow) Cell(cRow, cCol-2)+Cell(cRow, cCol-1) max(GridColAvg(cCol-1, 1, cRow) - 175, 0) * DCF(1, cRow)

In terms of the assets themselves, we would construct a CorrArrayProcesses() object and fill it with just 2 process objects created via the GBSProcess() function.
The correlation matrix used in the simulation is consequently just 2x2 and has only the correlations between the assets.

For the first process, GBSProcess(), we would consider the following arguments.

Asset price = 75

Volatility = 0.2

Rate = 0.051

dividend yield = 0.031

However for the second process, GBSProcess(), we would consider the following arguments.

Asset price = 100

Volatility = 0.15

Rate = 0.035 - (correlation * volAsset2 * volExRate)

dividend yield = 0.02

Thus the Rate value for the second process will equal : 0.035 - (0.4 * 0.15 * 0.15).

The reason why the second process's rate value is modified is because the currency of this asset is not equal to the payment currency. If you do not want to use a FX rate in the expression then a Quanto adjustment must be made to the 'rate' parameter. You also see that we use the DCF(1, cRow) function within the payoff table. The first parameter indicates which asset's discounting curve are we using. In this case, Asset #1's discounting curve.

When pricing Quanto products, there is no need to construct another process object to simulate the exchange rate.

Step	EventDate	Repeat?	PayOff Col	PayOff Col	PayOff Col	PayOff Col
1	#19/07/2005#	FALSE	Asset(1, cRow) + _a + 5pct	Asset(2, cRow) + Interp1D(1,cRow) - 100bp	GridRowAvg(cRow, cCol-2, cCol-1)
2	#19/01/2006#	FALSE	Asset(1, cRow) + _a + 5pct	Asset(2, cRow) + Interp1D(1,cRow) - 100bp	GridRowAvg(cRow, cCol-2, cCol-1)
3	#19/07/2006#	FALSE	Asset(1, cRow) + _a + 5pct	Asset(2, cRow) + Interp1D(1,cRow) - 100bp	GridRowAvg(cRow, cCol-2, cCol-1)
4	#19/01/2007#	FALSE	Asset(1, cRow) + _a + 5pct	Asset(2, cRow) + Interp1D(1,cRow) - 100bp	GridRowAvg(cRow, cCol-2, cCol-1)
5	#19/07/2007#	FALSE	Asset(1, cRow) + _a + 5pct	Asset(2, cRow) + Interp1D(1,cRow) - 100bp	GridRowAvg(cRow, cCol-2, cCol-1)
6	#19/01/2008#	FALSE	Asset(1, cRow) + _a + 5pct	Asset(2, cRow) + Interp1D(1,cRow) - 100bp	GridRowAvg(cRow, cCol-2, cCol-1)
7	#19/07/2008#	FALSE	Asset(1, cRow) + _a + 5pct	Asset(2, cRow) + Interp1D(1,cRow) - 100bp	GridRowAvg(cRow, cCol-2, cCol-1)
8	#19/01/2009#	FALSE	Asset(1, cRow) + _a + 5pct	Asset(2, cRow) + Interp1D(1,cRow) - 100bp	GridRowAvg(cRow, cCol-2, cCol-1)
9	#19/07/2009#	FALSE	Asset(1, cRow) + _a + 5pct	Asset(2, cRow) + Interp1D(1,cRow) - 100bp	GridRowAvg(cRow, cCol-2, cCol-1)
10	#19/01/2010#	FALSE	Asset(1, cRow) + _a + 5pct	Asset(2, cRow) + Interp1D(1,cRow) - 100bp	GridRowAvg(cRow, cCol-2, cCol-1)
11	#19/07/2010#	FALSE	Asset(1, cRow) + _a + 5pct	Asset(2, cRow) + Interp1D(1,cRow) - 100bp	GridRowAvg(cRow, cCol-2, cCol-1)
12	#19/01/2011#	FALSE	Asset(1, cRow) + _a + 5pct	Asset(2, cRow) + Interp1D(1,cRow) - 100bp	GridRowAvg(cRow, cCol-2, cCol-1)	max(GridColMin(cCol-1, 1, cRow) - AssetMin(1, 1, cRow), 0.0) * DCF(1, cRow)

Step	EventDate	Repeat?	PayOff Col	PayOff Col	PayOff Col
1	#19/07/2006#	FALSE	100	max(Asset(1, cRow) - Cell(cRow, cCol-1), 0) * DCF(1, cRow)
2	#19/07/2007#	FALSE	90	max(Cell(cRow, cCol-1) - Asset(1, cRow), 0) * DCF(1, cRow)	GridColSum(cCol-1, 1, cRow)

Step	EventDate	Repeat?	PayOff Col	PayOff Col
1	#19/07/2005#	FALSE	Asset(1, cRow)
2	#19/01/2006#	FALSE	Asset(1, cRow)
3	#19/07/2006#	FALSE	Asset(1, cRow)
4	#19/01/2007#	FALSE	Asset(1, cRow)
5	#19/07/2007#	FALSE	Asset(1, cRow)	max(GridColAvg(cCol-1, 1, cRow) - GridColMax(cCol-1, 1, cRow), 0) * DCF(1, cRow)

Step	EventDate	Repeat?	PayOff Col
1	#19/07/2005#	TRUE
2	#19/01/2006#	TRUE
3	#19/07/2006#	TRUE
4	#19/01/2007#	TRUE
5	#19/07/2007#	FALSE	max(AssetAvg(1, 1, cRow) - 10, 0) * DCF(1, cRow)