London SPAN for the LME
Volume 3
Specification for Calculation of Risk Arrays
Department: Risk Management
Document Type: Guide
Issue no.: 2.0
Issue Date: April 2008
,Document History
Date Version Author Summary of Changes
10/09/96 1.0 L Vosper First Issue
01/11/96 1.1 L Vosper Pricing Calls: rounding of interest rate to 6 dps;
and e to a minimum of 8 dps.
Change to calculation of dt, dtn and E_4 in
derivation of TAPO Volatility.
Expressions for expired option intrinsic value
specified in calculation theoretical option prices.
Expiry Group Derivation Methods specified for
TAPOS.
April 2008 2.0 Paul Kirkwood Updated format
, Volume 3
CONTENTS
1 PURPOSE OF DOCUMENT..........................................................................................1
2 GLOSSARY OF TERMS................................................................................................1
3 CALCULATION OF RISK ARRAYS............................................................................3
3.1 Array Structure....................................................................................................3
3.2 Overview of Array Calculation...........................................................................4
3.2.1 For Options......................................................................................................................4
3.2.2 For Forwards....................................................................................................................4
3.2.3 Rounding..........................................................................................................................4
3.2.4 Risk Array Elements........................................................................................................4
3.3 Risk Array Scenarios...........................................................................................4
3.3.1 Modification of Underlying Contract Series Price.........................................................4
3.3.2 Modification of Time to Expiry......................................................................................5
3.3.3 Modification of Volatility...............................................................................................5
3.3.4 Recalculation of Option’s Price......................................................................................5
3.3.5 Forward’s Price................................................................................................................5
3.4 TAPOS Risk Array Scenarios.............................................................................7
3.5 Calculating the Risk Array Elements..................................................................7
3.5.1 Sign Convention..............................................................................................................8
3.6 Calculating Composite Delta...............................................................................8
3.6.1 Options.............................................................................................................................8
3.6.2 Forwards..........................................................................................................................8
3.7 Delta Weights......................................................................................................9
3.8 Expiry Group Derivation Methods....................................................................10
3.8.1 Expiry Group Derivation Method 1..............................................................................10
3.8.2 Expiry Group Derivation Method 2..............................................................................10
3.8.3 Expiry Group Derivation Method 3..............................................................................10
3.8.4 Expiry Group Derivation Method 4..............................................................................10
4 CALCULATION OF THEORETICAL OPTIONS PRICES........................................12
4.1 Modified Black Option Pricing Formula...........................................................12
4.2 Pricing Calls......................................................................................................12
4.2.1 Note on Continuously Compounded Interest Rates......................................................13
4.3 Pricing Puts........................................................................................................14
4.4 Deltas.................................................................................................................14
5 PRICING TRADED AVERAGE PRICE OPTIONS (TAPOS)...................................15
5.1 Black Option Pricing Formula..........................................................................15
5.2 TAPO Volatility................................................................................................16
5.2.1 Derivation of TAPO Volatility......................................................................................16
Volume 3
Specification for Calculation of Risk Arrays
Department: Risk Management
Document Type: Guide
Issue no.: 2.0
Issue Date: April 2008
,Document History
Date Version Author Summary of Changes
10/09/96 1.0 L Vosper First Issue
01/11/96 1.1 L Vosper Pricing Calls: rounding of interest rate to 6 dps;
and e to a minimum of 8 dps.
Change to calculation of dt, dtn and E_4 in
derivation of TAPO Volatility.
Expressions for expired option intrinsic value
specified in calculation theoretical option prices.
Expiry Group Derivation Methods specified for
TAPOS.
April 2008 2.0 Paul Kirkwood Updated format
, Volume 3
CONTENTS
1 PURPOSE OF DOCUMENT..........................................................................................1
2 GLOSSARY OF TERMS................................................................................................1
3 CALCULATION OF RISK ARRAYS............................................................................3
3.1 Array Structure....................................................................................................3
3.2 Overview of Array Calculation...........................................................................4
3.2.1 For Options......................................................................................................................4
3.2.2 For Forwards....................................................................................................................4
3.2.3 Rounding..........................................................................................................................4
3.2.4 Risk Array Elements........................................................................................................4
3.3 Risk Array Scenarios...........................................................................................4
3.3.1 Modification of Underlying Contract Series Price.........................................................4
3.3.2 Modification of Time to Expiry......................................................................................5
3.3.3 Modification of Volatility...............................................................................................5
3.3.4 Recalculation of Option’s Price......................................................................................5
3.3.5 Forward’s Price................................................................................................................5
3.4 TAPOS Risk Array Scenarios.............................................................................7
3.5 Calculating the Risk Array Elements..................................................................7
3.5.1 Sign Convention..............................................................................................................8
3.6 Calculating Composite Delta...............................................................................8
3.6.1 Options.............................................................................................................................8
3.6.2 Forwards..........................................................................................................................8
3.7 Delta Weights......................................................................................................9
3.8 Expiry Group Derivation Methods....................................................................10
3.8.1 Expiry Group Derivation Method 1..............................................................................10
3.8.2 Expiry Group Derivation Method 2..............................................................................10
3.8.3 Expiry Group Derivation Method 3..............................................................................10
3.8.4 Expiry Group Derivation Method 4..............................................................................10
4 CALCULATION OF THEORETICAL OPTIONS PRICES........................................12
4.1 Modified Black Option Pricing Formula...........................................................12
4.2 Pricing Calls......................................................................................................12
4.2.1 Note on Continuously Compounded Interest Rates......................................................13
4.3 Pricing Puts........................................................................................................14
4.4 Deltas.................................................................................................................14
5 PRICING TRADED AVERAGE PRICE OPTIONS (TAPOS)...................................15
5.1 Black Option Pricing Formula..........................................................................15
5.2 TAPO Volatility................................................................................................16
5.2.1 Derivation of TAPO Volatility......................................................................................16
