The Black-Scholes Model

Transcription

1

2 The Black-Scholes Model Inputs Spot Price Exercise Price Time to Maturity Rate-Cost of funds & Yield Volatility Process The Black Box Output "Fair Market Value" For those interested in looking inside the process... C Se yt rt N( x t ) Xe N( x) Black-Scholes-Merton Model, Page 2

3 II) Option Price/Value Sensitivity Changes in: Value - V Influence for Relative Call Put Size? Relation? ( or ) ( or ) (big, medium, small) (linear, non-linear) Contract Terms Exercise Price - X Increase Maturity - T Longer Markets and Position: Current Price - S Increase Volatility - Up Rate-Cost of Funds - R (term currency rate) Yield - Y (commodity currency rate) Time to Maturity -T Increase Up 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

4 Exercise Price - X What happens to the value of calls and puts when the strike price is higher? % 5.5% % $2.00 $ % 5.5% % $1.56 $ % 5.5% % $1.18 $ % 5.5% % $0.88 $ % 5.5% % $0.64 $ % 5.5% % $0.46 $5.42 What happens to the value of calls and puts when the strike price is lower? % 5.5% % $2.00 $ % 5.5% % $2.53 $ % 5.5% % $3.13 $ % 5.5% % $3.81 $ % 5.5% % $4.55 $ % 5.5% % $5.35 $0.40 X => Call: or Put: or Size: V X V X Call: V X V X Put: Relation: linear - nonlinear Intuition: less in the money, less likely to be exercised and less valuable Black-Scholes-Merton Model, Page 4

5 Sensitivity of Option Values to Changes in Strike Price $6.00 $5.00 $4.00 $3.00 $2.00 $1.00 $ Strike Price Call Put Call Put $2.00 $ $5.35 $ $4.55 $ $3.81 $ $3.13 $1.15 Strike 99 $2.53 $1.54 Price 100 $2.00 $ $1.56 $ $1.18 $ $0.88 $ $0.64 $ $0.46 $5.42 Black-Scholes-Merton Model, Page 5

6 Maturity - T What happens to the values of calls and puts when days to expiration is longer? % 5.5% % $2.00 $ % 5.5% % $2.44 $ % 5.5% % $2.81 $ % 5.5% % $3.12 $ % 5.5% % $3.41 $ % 5.5% % $4.69 $4.69 What happens to the values of calls and puts when days to expiration is shorter? % 5.5% % $2.00 $ % 5.5% % $1.42 $ % 5.5% % $1.16 $ % 5.5% % $0.82 $ % 5.5% % $0.58 $ % 5.5% % $0.26 $0.26 T => Call: or Put: or V V Size: T T Relation: linear or nonlinear Intuition: Black-Scholes-Merton Model, Page 6

7 Sensitivity of Option Values to Changes in Maturity $5.00 $4.50 $4.00 $3.50 $3.00 $2.50 $2.00 $1.50 $1.00 $0.50 $ Call Put Call Put $2.00 $ $0.26 $ $1.42 $1.42 Days 60 $2.00 $ $2.44 $ $2.81 $ $3.12 $ $3.41 $ $3.66 $ $3.90 $ $4.12 $ $4.32 $ $4.51 $ $4.69 $4.69 Black-Scholes-Merton Model, Page 7

8 Spot Price - S What happens to the value of calls and puts when the spot price goes up? % 5.5% % $2.00 $ % 5.5% % $2.55 $ % 5.5% % $3.17 $ % 5.5% % $3.86 $ % 5.5% % $4.61 $ % 5.5% % $5.42 $0.46 What happens to the value of calls and puts when the spot price goes down? % 5.5% % $2.00 $ % 5.5% % $1.54 $ % 5.5% % $1.15 $ % 5.5% % $0.83 $ % 5.5% % $0.59 $ % 5.5% % $0.40 $5.35 S => Call: or Put: or V V Size: S S Relation: linear or nonlinear Intuition (delta): Black-Scholes-Merton Model, Page 8

9 Sensitivity of Option Values to Changes in Spot Price $6.00 $5.00 $4.00 $3.00 $2.00 $1.00 $ Spot Price Call Put Call Put $2.00 $ $0.40 $ $0.59 $ $0.83 $ $1.15 $3.13 Spot 99 $1.54 $2.53 Price 100 $2.00 $ $2.55 $ $3.17 $ $3.86 $ $4.61 $ $5.42 $ $6.27 $0.32 Black-Scholes-Merton Model, Page 9

10 Volatility - What happens to the values of calls and puts with increases in volatility? % 5.5% % $2.00 $ % 5.5% % $2.40 $ % 5.5% % $2.80 $ % 5.5% % $3.20 $ % 5.5% % $3.61 $ % 5.5% % $4.01 $4.01 What happens to the values of calls and puts with decreases in volatility? % 5.5% % $2.00 $ % 5.5% % $1.60 $ % 5.5% % $1.20 $ % 5.5% % $0.80 $ % 5.5% % $0.40 $ % 5.5% % $0.00 $0.00 => Call: or Put: or Size: V V Relation: linear or nonlinear Intuition (Vega): Black-Scholes-Merton Model, Page 10

11 Sensitivity of Option Va lues to Changes in Vola tility $4.50 $4.00 $3.50 $3.00 $2.50 $2.00 $1.50 $1.00 $0.50 $ % 2.5% 5.0% 7.5% 10.0% 12.5% 15.0% 17.5% 20.0% 22.5% 25.0% Call Put Call Put $2.00 $ % $0.00 $ % $0.40 $ % $0.80 $ % $1.20 $ % $1.60 $1.60 Volatility 12.5% $2.00 $ % $2.40 $ % $2.80 $ % $3.20 $ % $3.61 $ % $4.01 $4.01 Black-Scholes-Merton Model, Page 11

12 Cost of Funds - R What happens to the values of calls and puts when rate (cost of funds) goes up? % 5.5% % $2.00 $ % 5.5% % $2.08 $ % 5.5% % $2.17 $ % 5.5% % $2.25 $ % 5.5% % $2.34 $ % 5.5% % $2.43 $1.62 What happens to the values of calls and puts when rate (cost of funds) goes down? % 5.5% % $2.00 $ % 5.5% % $1.92 $ % 5.5% % $1.85 $ % 5.5% % $1.77 $ % 5.5% % $1.70 $ % 5.5% % $1.59 $2.49 R => Call: or Put: or V V Size: R R Relation: linear or nonlinear Intuition (Rho): Black-Scholes-Merton Model, Page 12

13 Sensitivity of Option Values to Changes in Rate $2.60 $2.40 $2.20 $2.00 $1.80 $1.60 $ % 1.5% 2.5% 3.5% 4.5% 5.5% 6.5% 7.5% 8.5% 9.5% 10.5% Rate (cost of funds) Call Put Call Put $2.00 $ % $1.63 $ % $1.70 $ % $1.77 $ % $1.85 $ % $1.92 $2.09 Rate 5.5% $2.00 $ % $2.08 $ % $2.17 $ % $2.25 $ % $2.34 $ % $2.43 $1.62 Black-Scholes-Merton Model, Page 13

14 Current Yield - Y What happens to the values of calls and puts when yield goes up? % 5.5% % $2.00 $ % 6.5% % $1.92 $ % 7.5% % $1.84 $ % 8.5% % $1.76 $ % 9.5% % $1.69 $ % 10.5% % $1.62 $2.43 What happens to the vaues of calls and puts when yield goes down? % 5.5% % $2.00 $ % 4.5% % $2.09 $ % 3.5% % $2.17 $ % 2.5% % $2.26 $ % 1.5% % $2.35 $ % 0.0% % $2.49 $1.59 Y => Call: or Put: or V V Size: Y Y Relation: linear or nonlinear Intuition (Rho): Black-Scholes-Merton Model, Page 14

15 Sensitivity of Option Values to Cha nges in Yield $2.60 $2.40 $2.20 $2.00 $1.80 $1.60 $ % 1.5% 2.5% 3.5% 4.5% 5.5% 6.5% 7.5% 8.5% 9.5% 10.5% Yield Call Put Call Put $2.00 $ % $2.45 $ % $2.35 $ % $2.26 $ % $2.17 $ % $2.09 $1.92 Yield 5.5% $2.00 $ % $1.92 $ % $1.84 $ % $1.76 $ % $1.69 $ % $1.62 $2.43 Black-Scholes-Merton Model, Page 15

16 Delta - What happens to the value of call and put deltas when the spot price goes up? % 5.5% % % 5.5% % % 5.5% % % 5.5% % % 5.5% % % 5.5% % What happens to the value of call and put deltas when the spot price goes down? % 5.5% % % 5.5% % % 5.5% % % 5.5% % % 5.5% % % 5.5% % S => Call delta: or Put delta: or Size: Delta s Delta Relation: linear or nonlinear Intuition (Gamma): s Black-Scholes-Merton Model, Page 16

17 Sensitivity of Option Deltas to Changes in Spot Price (strike =100) Spot Price Call Put Call Put Black-Scholes-Merton Model, Page 17

18 Cash % - X What happens to the value of call and put cash % when the spot price goes up? % 5.5% % % 5.5% % % 5.5% % % 5.5% % % 5.5% % % 5.5% % What happens to the value of call and put cash % when the spot price goes down? % 5.5% % % 5.5% % % 5.5% % % 5.5% % % 5.5% % % 5.5% % S => Call cash %: or Put cash %: or Size: Cash% Cash% s s Relation: linear or nonlinear Intuition (Risk Neutral Exercise Likelihood): Black-Scholes-Merton Model, Page 18

19 Sensitivity of Option Cash %'s to Changes in Spot Price (strike =100) Calls Puts Spot Price Call Put Call Put Spot Price Black-Scholes-Merton Model, Page 19

20 The Greeks DELTA Sensitivity of Option Value to Changes in Price of Underlying GAMMA Sensitivity of Delta to 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 (lambda, kappa, or sigma) Sensitivity of Option Value to Changes in Volatility. Black-Scholes-Merton Model, Page 20

21 This page is intentionally left blank. Black-Scholes-Merton Model, Page 21

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

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 Rate Differential Up Call worth relatively MORE 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 Rate Differential Up Put worth relatively LESS Put worth relatively MORE Black-Scholes-Merton Model, Page 23

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