Computing Equity Instrument Sensitivities
Sensitivities can be reported either as dollar price changes or percentage price changes. The delta, gamma, and vega sensitivities that the toolbox computes are dollar sensitivities.
The functions crrsens, eqpsens, ittsens,
and sttsens compute the delta,
gamma, and vega sensitivities of instruments using a stock tree. They
also optionally return the calculated price for each instrument. The
sensitivity functions require the same two input arguments used by
the pricing functions (CRRTree and CRRInstSet for
CRR, EQPTree and EQPInstSet for
EQP, ITTTree and ITTInstSet for
ITT, and STTTree and STTInstSet for
STT).
As with the instrument pricing functions, the optional input argument
Options is also allowed. You would include this argument if you
want a sensitivity function to generate a price for a barrier option as one of its
outputs and want to control the method that the toolbox uses to perform the pricing
operation. See Pricing Options Structure or the
derivset function for more
information.
For path-dependent options (lookback and Asian), delta and gamma
are computed by finite differences in calls to crrprice, eqpprice, ittprice, and sttprice.
For the other options (stock option, barrier, and compound), delta
and gamma are computed from the CRR, EQP, ITT, and STT trees and the
corresponding option price tree. (See Chriss, Neil, Black-Scholes
and Beyond, pp. 308–312.)
Computing CRR Sensitivities
This example shows how to calculate the sensitivity of each instrument in the instrument set using crrsens.
Load the data into the MATLAB® workspace.
load deriv.matUse the MATLAB® whos command to display a list of the variables loaded from the MAT-file.
whos
Name Size Bytes Class Attributes BDTInstSet 1x1 28737 struct BDTTree 1x1 6289 struct BKInstSet 1x1 28727 struct BKTree 1x1 7355 struct CIRInstSet 1x1 31123 struct CIRTree 1x1 6784 struct CRRInstSet 1x1 22035 struct CRRTree 1x1 5832 struct EQPInstSet 1x1 22035 struct EQPTree 1x1 5832 struct HJMInstSet 1x1 28729 struct HJMTree 1x1 7206 struct HWInstSet 1x1 28727 struct HWTree 1x1 7355 struct ITTInstSet 1x1 22039 struct ITTTree 1x1 11367 struct STTInstSet 1x1 22039 struct STTTree 1x1 6600 struct ZeroInstSet 1x1 18203 struct ZeroRateSpec 1x1 1732 struct
CRRTree and CRRInstSet are the input arguments required to call the function crrsens.
Use the command 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
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)
Use crrsens to calculate the sensitivity of each instrument in the instrument set.
[Delta, Gamma, Vega, Price] = crrsens(CRRTree, CRRInstSet)
Delta = 8×1
0.5949
-0.3115
0.6885
-0.1203
-0.3991
-0.4200
0.6049
0.8187
Gamma = 8×1
105 ×
0.0000
0.0000
0.0000
-0.0000
-0.4593
-1.1214
0.4593
1.1214
Vega = 8×1
53.4471
67.0043
67.0043
-98.0793
88.1769
119.1890
49.2128
41.7070
Price = 8×1
8.2863
2.5016
12.1272
3.3241
7.6015
11.7772
4.1797
3.4219
You can conveniently examine the sensitivities and the prices by arranging them into a single matrix.
format bank
All = [Delta, Gamma, Vega, Price]All = 8×4
0.59 0.04 53.45 8.29
-0.31 0.03 67.00 2.50
0.69 0.03 67.00 12.13
-0.12 -0.01 -98.08 3.32
-0.40 -45926.32 88.18 7.60
-0.42 -112143.15 119.19 11.78
0.60 45926.32 49.21 4.18
0.82 112143.15 41.71 3.42
As with the prices, each row of the sensitivity vectors corresponds to the similarly indexed instrument in CRRInstSet. To view the per-dollar sensitivities, divide each dollar sensitivity by the corresponding instrument price.
All = [Delta ./ Price, Gamma ./ Price, Vega ./ Price, Price]
All = 8×4
0.07 0.00 6.45 8.29
-0.12 0.01 26.78 2.50
0.06 0.00 5.53 12.13
-0.04 -0.00 -29.51 3.32
-0.05 -6041.77 11.60 7.60
-0.04 -9522.02 10.12 11.78
0.14 10987.98 11.77 4.18
0.24 32771.92 12.19 3.42
ITT Sensitivities Example
The calling syntax for the sensitivity function is:
[Delta, Gamma, Vega, Price] = ittsens(ITTTree, ITTInstSet,
Options)
Using the example data in deriv.mat, calculate
the sensitivity of the instruments.
load deriv.mat
warning('off', 'fininst:itttree:Extrapolation');
[Delta, Gamma, Vega, Price] = ittsens(ITTTree, ITTInstSet);
You can conveniently examine the sensitivities and the prices by arranging them into a single matrix.
format bank
All = [Delta, Gamma, Vega, Price]All =
0.24 0.03 19.35 1.65
-0.43 0.02 49.69 10.68
0.35 0.04 12.29 2.41
-0.07 0.00 6.73 3.23
0.63 142945.66 38.90 0.54
0.60 22703.21 68.92 6.18
0.32 -142945.66 18.48 3.21
0.67 -22703.21 17.75 6.61As with the prices, each row of the sensitivity vectors corresponds
to the similarly indexed instrument in ITTInstSet.
Note
In this example, the extrapolation warnings are turned off before calculating the sensitivities to avoid displaying many warnings on the Command Window as the sensitivities are calculated.
If the extrapolation warnings are turned on
warning('on', 'fininst:itttree:Extrapolation');
ittsens is
rerun, the extrapolation warnings scroll as the command executes:[Delta, Gamma, Vega, Price] = ittsens(ITTTree, ITTInstSet)
Warning: The option set specified in StockOptSpec was too narrow for the
generated tree.
This made extrapolation necessary. Below is a list of the options that were
outside of the
range of those specified in StockOptSpec.
Option Type: 'call' Maturity: 01-Jan-2007 Strike=67.2897
Option Type: 'put' Maturity: 01-Jan-2007 Strike=37.1528
Option Type: 'put' Maturity: 01-Jan-2008 Strike=27.6066
Option Type: 'put' Maturity: 31-Dec-2008 Strike=20.5132
Option Type: 'call' Maturity: 01-Jan-2010 Strike=164.0157
Option Type: 'put' Maturity: 01-Jan-2010 Strike=15.2424
> In itttree>InterpOptPrices (line 680)
In itttree (line 285)
In stocktreesens>stocktreevega (line 193)
In stocktreesens (line 94)
In ittsens (line 79)
Delta =
0.24
-0.43
0.35
-0.07
0.63
0.60
0.32
0.67
Gamma =
0.03
0.02
0.04
0.00
142945.66
22703.21
-142945.66
-22703.21
Vega =
19.35
49.69
12.29
6.73
38.90
68.92
18.48
17.75
Price =
1.65
10.68
2.41
3.23
0.54
6.18
3.21
6.61These warnings are a consequence of having to extrapolate to find the option price of the tree nodes. In this example, the set of inputs options was too narrow for the shift in the tree nodes introduced by the disturbance used to calculate the sensitivities. As a consequence extrapolation for some of the nodes was needed. Since the input data is quite close the extrapolated data, the error introduced by extrapolation is fairly low.
Computing STT Sensitivities
This example shows how to calculate the sensitivity of each instrument in the instrument set using sttsens.
Load the data into the MATLAB® workspace.
load deriv.matUse the MATLAB® whos command to display a list of the variables loaded from the MAT-file.
whos
Name Size Bytes Class Attributes BDTInstSet 1x1 28737 struct BDTTree 1x1 6289 struct BKInstSet 1x1 28727 struct BKTree 1x1 7355 struct CIRInstSet 1x1 31123 struct CIRTree 1x1 6784 struct CRRInstSet 1x1 22035 struct CRRTree 1x1 5832 struct EQPInstSet 1x1 22035 struct EQPTree 1x1 5832 struct HJMInstSet 1x1 28729 struct HJMTree 1x1 7206 struct HWInstSet 1x1 28727 struct HWTree 1x1 7355 struct ITTInstSet 1x1 22039 struct ITTTree 1x1 11367 struct STTInstSet 1x1 22039 struct STTTree 1x1 6600 struct ZeroInstSet 1x1 18203 struct ZeroRateSpec 1x1 1732 struct
ITTTree and ITTInstSet are the input arguments required to call the function sttsens.
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)
Use ittsens to calculate the sensitivity of each instrument in the instrument set.
[Delta, Gamma, Vega, Price] = sttsens(STTTree, STTInstSet)
Delta = 8×1
0.5267
-0.0943
0.4726
-0.0624
0.2313
0.3266
0.5706
0.6646
Gamma = 8×1
105 ×
0.0000
0.0000
0.0000
0.0000
-1.8650
-1.9119
1.8650
1.9119
Vega = 8×1
52.8980
42.4369
25.9792
-9.5266
70.3758
92.9226
25.8122
37.8757
Price = 8×1
4.5025
3.0603
3.7977
1.7090
11.7296
12.9120
1.6905
2.6203
You can conveniently examine the sensitivities and the prices by arranging them into a single matrix.
format bank
All = [Delta, Gamma, Vega, Price]All = 8×4
0.53 0.02 52.90 4.50
-0.09 0.00 42.44 3.06
0.47 0.03 25.98 3.80
-0.06 0.00 -9.53 1.71
0.23 -186495.25 70.38 11.73
0.33 -191186.43 92.92 12.91
0.57 186495.25 25.81 1.69
0.66 191186.43 37.88 2.62
As with the prices, each row of the sensitivity vectors corresponds to the similarly indexed instrument in ITTInstSet. To view the per-dollar sensitivities, divide each dollar sensitivity by the corresponding instrument price.
All = [Delta ./ Price, Gamma ./ Price, Vega ./ Price, Price]
All = 8×4
0.12 0.01 11.75 4.50
-0.03 0.00 13.87 3.06
0.12 0.01 6.84 3.80
-0.04 0.00 -5.57 1.71
0.02 -15899.56 6.00 11.73
0.03 -14806.91 7.20 12.91
0.34 110321.92 15.27 1.69
0.25 72962.88 14.45 2.62
Computing EQP Sensitivities
This example shows how to calculate the sensitivity of each instrument in the instrument set using eqpsens.
Load the data into the MATLAB® workspace.
load deriv.matUse the MATLAB® whos command to display a list of the variables loaded from the MAT-file.
whos
Name Size Bytes Class Attributes BDTInstSet 1x1 28737 struct BDTTree 1x1 6289 struct BKInstSet 1x1 28727 struct BKTree 1x1 7355 struct CIRInstSet 1x1 31123 struct CIRTree 1x1 6784 struct CRRInstSet 1x1 22035 struct CRRTree 1x1 5832 struct EQPInstSet 1x1 22035 struct EQPTree 1x1 5832 struct HJMInstSet 1x1 28729 struct HJMTree 1x1 7206 struct HWInstSet 1x1 28727 struct HWTree 1x1 7355 struct ITTInstSet 1x1 22039 struct ITTTree 1x1 11367 struct STTInstSet 1x1 22039 struct STTTree 1x1 6600 struct ZeroInstSet 1x1 18203 struct ZeroRateSpec 1x1 1732 struct
EQPTree and EQPInstSet are the input arguments required to call the function eqpsens.
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
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)
Use eqpsens to calculate the sensitivity of each instrument in the instrument set.
[Delta, Gamma, Vega, Price] = eqpsens(EQPTree, EQPInstSet)
Delta = 8×1
0.6501
-0.2336
0.7664
-0.1251
-0.5443
-0.4516
0.7066
0.8357
Gamma = 8×1
105 ×
0.0000
0.0000
0.0000
-0.0000
-0.5435
-1.2133
0.5435
1.2133
Vega = 8×1
44.2790
73.3662
73.3662
0
101.1634
109.5524
36.7642
42.3509
Price = 8×1
8.4791
2.6375
12.2632
3.5091
8.7941
12.9577
4.7444
3.9178
You can conveniently examine the sensitivities and the prices by arranging them into a single matrix.
format bank
All = [Delta, Gamma, Vega, Price]All = 8×4
0.65 0.04 44.28 8.48
-0.23 0.02 73.37 2.64
0.77 0.02 73.37 12.26
-0.13 -0.01 0 3.51
-0.54 -54353.69 101.16 8.79
-0.45 -121330.73 109.55 12.96
0.71 54353.69 36.76 4.74
0.84 121330.73 42.35 3.92
As with the prices, each row of the sensitivity vectors corresponds to the similarly indexed instrument in EQPInstSet. To view the per-dollar sensitivities, divide each dollar sensitivity by the corresponding instrument price.
All = [Delta ./ Price, Gamma ./ Price, Vega ./ Price, Price]
All = 8×4
0.08 0.00 5.22 8.48
-0.09 0.01 27.82 2.64
0.06 0.00 5.98 12.26
-0.04 -0.00 0 3.51
-0.06 -6180.71 11.50 8.79
-0.03 -9363.61 8.45 12.96
0.15 11456.35 7.75 4.74
0.21 30968.71 10.81 3.92
See Also
crrtree | eqptree | lrtree | stockspec | crrtimespec | eqptimespec | lrtimespec | itttree | itttimespec | treepath | trintreepath | asianbycrr | barrierbycrr | compoundbycrr | crrprice | crrsens | lookbackbycrr | optstockbycrr | instasian | instbarrier | instcompound | instlookback | instoptstock | asianbyeqp | barrierbyeqp | compoundbyeqp | eqpprice | eqpsens | lookbackbyeqp | optstockbyeqp | optstockbylr | optstocksensbylr | asianbyitt | barrierbyitt | compoundbyitt | ittprice | ittsens | lookbackbyitt | optstockbyitt | asianbystt | barrierbystt | compoundbystt | sttprice | sttsens | lookbackbystt | optstockbystt
Topics
- Understanding Equity Trees
- Pricing Equity Derivatives Using Trees
- Graphical Representation of Equity Derivative Trees
- Creating Instruments or Properties
- Graphical Representation of Equity Derivative Trees
- Pricing European Call Options Using Different Equity Models
- Pricing Asian Options
- Supported Equity Derivative Functions
- Supported Energy Derivative Functions
- Supported Interest-Rate Instrument Functions
- Mapping Financial Instruments Toolbox Functions for Equity, Commodity, FX Instrument Objects
