Derivative and Financial Markets Concepts Module 7 Black-Scholes-Merton

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, 4 th ed. , pp. 139 -150 Cox-Rubinstein, Option Markets, 1985, 5. 8, pp. 215 -235 Options 9 th: Chapters 15 and 17; optional Chapter 19 Options 8 th: Chapters 14 and 16; optional Chapter 18 Options 7 th: Chapters 13 and 15; optional Chapter 17 Options 6 th: Chapters 13 and 14; optional Chapter 15 Options 5 th: Chapters 12 and 13; optional Chapter 14 Options 4 th: Chapters 11 and 12; 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. Black-Scholes-Merton Model, Page 1

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

II) Option Price/Value Sensitivity Changes in: Value - V Influence for Relative Call Put Size? Relation? ( or ) (big, medium, small) (linear, non-linear) 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. Black-Scholes-Merton Model, Page 3

Exercise Price - X X => Call: or Put: Size: Relation: linear - nonlinear Intuition: less in the money, less likely to be exercised and less valuable Black-Scholes-Merton Model, Page 4

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Maturity - T T => Call: or Put: or Size: Relation: linear or nonlinear Intuition: Black-Scholes-Merton Model, Page 6

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Spot Price - S S => Call: or Put: or Size: Relation: linear or nonlinear Intuition (delta): Black-Scholes-Merton Model, Page 8

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Volatility - s s => Call: or Put: or Size: Relation: linear or nonlinear Intuition (Vega): Black-Scholes-Merton Model, Page 10

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Cost of Funds - R R => Call: or Put: or Size: Relation: linear or nonlinear Intuition (Rho): Black-Scholes-Merton Model, Page 12

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Current Yield - Y Y => Call: or Put: or Size: Relation: linear or nonlinear Intuition (Rho): Black-Scholes-Merton Model, Page 14

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Delta - D S => Call delta: or Put delta: or Size: Relation: linear or nonlinear Intuition (Gamma): Black-Scholes-Merton Model, Page 16

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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. 506 101 100 5. 5% 60 12. 5% 101. 00 -0. 563 0. 428 102 100 5. 5% 60 12. 5% 102. 00 -0. 637 0. 354 103 100 5. 5% 60 12. 5% 103. 00 -0. 705 0. 286 104 100 5. 5% 60 12. 5% 104. 00 -0. 766 0. 225 100 5. 5% 60 12. 5% 105. 00 -0. 818 0. 173 What happens to the value of call and put cash % when the spot price goes down? SPOT STRIKE RATE YIELD DAYS VOL FWD CALL PUT 100 5. 5% 60 12. 5% 100. 00 -0. 485 0. 506 99 100 5. 5% 60 12. 5% 99. 00 -0. 408 0. 583 98 100 5. 5% 60 12. 5% 98. 00 -0. 333 0. 658 97 100 5. 5% 60 12. 5% 97. 00 -0. 263 0. 728 96 100 5. 5% 60 12. 5% 96. 00 -0. 201 0. 790 95 100 5. 5% 60 12. 5% 95. 00 -0. 148 0. 843 S => Call cash %: or Put cash %: or Size: Relation: linear or nonlinear Intuition (Risk Neutral Exercise Likelihood): Black-Scholes-Merton Model, Page 18

Sensitivity of Option Cash %'s to Changes in Spot Price (strike =100) 0. 00 90 92 94 96 98 100 102 104 106 108 110 112 0. 40 -0. 60 -0. 80 -1. 00 -1. 20 Puts 0. 20 -0. 40 Calls -0. 20 Spot Price Call Spot Price 90 92 94 96 98 100 102 104 106 108 110 112 Put Call -0. 49 -0. 02 -0. 05 -0. 11 -0. 20 -0. 33 -0. 49 -0. 64 -0. 77 -0. 86 -0. 92 -0. 96 -0. 98 Put 0. 51 0. 97 0. 94 0. 89 0. 79 0. 66 0. 51 0. 35 0. 23 0. 13 0. 07 0. 03 0. 01 Black-Scholes-Merton Model, Page 19

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

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Why Do Yield and Cost of Funds Matter in Time Value of Bond Options? Long Position Repo Out a Bond Buy a Call Do not Earn + Earn Yield Carry - 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 Black-Scholes-Merton Model, Page 22

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

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