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. 139-150  Cox-Rubinstein, Option Markets, 1985, 5.8, pp. 215-235 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, 1996 all rights reserved. Confidential, Proprietary Information of Financial Labs, Inc. 1 1 1

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

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

 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

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

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

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

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

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

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

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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

  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

“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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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

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

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