1. Module 7: Black-Scholes-Merton Model Sensitivities
Derivative and Financial Markets Concepts
Module 7: Black-Scholes-Merton Model Sensitivities
Objectives:
To understand what causes changes in option values derived with the Black-Scholes-Merton model
To develop an intuition of option value sensitivities
Structure:
Analysis of value sensitivity tables and graphs
Option Sensitivity Analysis [OPTPRICE.XLS]
Discuss the logic of the value sensitivities
Chance, D., An Introduction to Derivatives, 4th ed., pp  Cox-Rubinstein, Option Markets, 1985, 5.8, pp Options 9th: Chapters 15 and 17; optional Chapter 19 Options 8th: Chapters 14 and 16; optional Chapter 18  Options 7th: Chapters 13 and 15; optional Chapter 17  Options 6th: Chapters 13 and 14; optional Chapter 15  Options 5th: Chapters 12 and 13; optional Chapter 14  Options 4th: Chapters 11 and12; optional Chapter 13
Jointly-developed module licensed to James Bodurtha Copyright Ó Financial Labs, Inc., 1993, 1994, 1995, all rights reserved. Confidential, Proprietary Information of Financial Labs, Inc. 1 1 1

2. The Black-Scholes Model
Time to
Rate-Cost of funds & Yield
Inputs
Exercise Price
Maturity
Spot Price
Volatility
Process
The Black Box
Output
"Fair Market Value"
For those interested in looking inside the process... 2 2 2

3. II) Option Price/Value Sensitivity
Value - V Influence for Relative
Call Put Size? Relation? ( or ) ( or ) (big, medium, small) (linear, non-linear)
After they have proceeded through the case, OPFACTOR, put this overhead up and ask them to fill this out in preparation for a discussion on Factor Sensitivity of options.
Changes in:
Contract Terms
• Exercise Price - X Increase
• Maturity - T Longer
Markets and Position:
• Current Price - S Increase
• Volatility - s Up
• Rate-Cost of Funds - R Increase
(term currency rate)
• Yield - Y Up
(commodity currency rate)
• Time to Maturity -T Shorter
On the following pages, two pages of supporting information and questions are provided for each option pricing factor. Review these pages and then complete the grid above. We will discuss your analysis. 3 3 3

4.  or  Exercise Price - X X  => Call: Put: Call: Put: Size:
Relation: linear - nonlinear
Intuition: less in the money, less likely to be exercised and less valuable 4 4 4

5. 5 5 5

6. Relation: linear or nonlinear
Maturity - T
T  =>
Call:
 or 
Put:
 or 
Size:
Relation: linear or nonlinear
Intuition: 6 6 6

7. 7 7 7

8. Relation: linear or nonlinear
Spot Price - S
S  =>
Call:
 or 
Put:
 or 
Size:
Relation: linear or nonlinear
Intuition (delta): 8 8 8

9. 9 9 9

10. s  => Volatility - s Call: Put: Size:
 or 
Put:
 or 
Size:
Relation: linear or nonlinear
Intuition (Vega): 10 10 10

11. 11 11 11

12. Relation: linear or nonlinear
Cost of Funds - R
R  =>
Call:
 or 
Put:
 or 
Size:
Relation: linear or nonlinear
Intuition (Rho): 14 14 14

13. 15 15 15

14. Relation: linear or nonlinear
Current Yield - Y
Y  =>
Call:
 or 
Put:
 or 
Size:
Relation: linear or nonlinear
Intuition (Rho): 12 12 12

15. 13 13 13

16. Relation: linear or nonlinear
Delta - D
S  =>
Call delta:
 or 
Put delta:
 or 
Size:
Relation: linear or nonlinear
Intuition (Gamma): 16 16 16

17. 17 17 17

18. Relation: linear or nonlinear
Cash % - DX
What happens to the value of call and put cash % when the spot price goes up?
SPOT
STRIKE
RATE
YIELD
DAYS
VOL
FWD
CALL
PUT
100
5.5% 60
12.5%
100.00
-0.485
0.506
101
101.00
-0.563
0.428
102
102.00
-0.637
0.354
103
103.00
-0.705
0.286
104
104.00
-0.766
0.225
105
105.00
-0.818
0.173
What happens to the value of call and put cash % when the spot price goes down? 99
99.00
-0.408
0.583 98
98.00
-0.333
0.658 97
97.00
-0.263
0.728 96
96.00
-0.201
0.790 95
95.00
-0.148
0.843
S  =>
Call cash %:
 or 
Put cash %:
 or 
Size:
Relation: linear or nonlinear
Intuition (Risk Neutral Exercise Likelihood): 16 16 16

19. Call
Put
-0.49
0.51 90
-0.02
0.97 92
-0.05
0.94 94
-0.11
0.89 96
-0.20
0.79
Spot 98
-0.33
0.66
Price
100
102
-0.64
0.35
104
-0.77
0.23
106
-0.86
0.13
108
-0.92
0.07
110
-0.96
0.03
112
-0.98
0.01

20. “The Greeks” DELTA GAMMA THETA RHO VEGA
Sensitivity of Option Value to Changes in Price of Underlying
Sensitivity of Delta to Changes in Price of Underlying (Convexity)
Sensitivity of Option Value to Changes (or Differences) in Maturity.
Sensitivity of Option Value to Changes in Interest Rates and Yields
Sensitivity of Option Value to Changes in Volatility.
DELTA
GAMMA
THETA
RHO
VEGA
(lambda, kappa,
or sigma) 5 13

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22. Why Do Yield and Cost of Funds Matter in Time Value of Bond Options?
Long Position
Buy a Call
Repo Out a Bond
Do not Earn
Carry
+ Earn Yield
- Pay Repo
= Earn Carry
Carry Up Call worth relatively LESS
Carry Down Call worth relatively MORE
Short Position
Buy a Put
Reverse Repo a Bond
Do not Pay
Carry
+ Earn Repo
- Give Up Yield
= Pay Carry
Carry Up Put worth relatively MORE
Carry Down Put worth relatively LESS 18 18 18

23. Why Do Interest Rates Matter in Time Value of Currency Options?
Long Position
Buy a Pound Call
Borrow $ to Buy Pounds
Do not Earn
Rate Differential
+ Earn Europound Rate
- Pay Eurodollar Rate
= Earn Rate Differential
Rate Differential Down Call worth relatively MORE
Rate Differential Up Call worth relatively LESS
Short Position
Buy a Pound Put
Borrow Pounds to Buy $
Do not Pay
Rate Differential
+ Earn Eurodollar Rate
- Pay Europound Rate
= Pay Rate Differential
Rate Differential Down Put worth relatively LESS
Rate Differential Up Put worth relatively MORE 19 19 19

24. Interim Cash Flows on Underlying Assets
Foreign Exchange:
Bond:
Stock:
Broker Loan
Dividend yield
Loan rate - yield
Cost of Funds
Current Yield
on the
Underlying
Cost of Carry
Repurchase (or Repo) Rate
Current Yield
Current Yield -Repurchase Rate
Eurodollar
Eurocurrency Rate
Interest Rate Differential 20 20 20