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
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
Exercise Price - X What happens to the value of calls and puts when the strike price is higher? 100 100 5.5% 5.5% 60 12.5% 100.00 $2.00 $2.00 100 101 5.5% 5.5% 60 12.5% 100.00 $1.56 $2.55 100 102 5.5% 5.5% 60 12.5% 100.00 $1.18 $3.17 100 103 5.5% 5.5% 60 12.5% 100.00 $0.88 $3.86 100 104 5.5% 5.5% 60 12.5% 100.00 $0.64 $4.61 100 105 5.5% 5.5% 60 12.5% 100.00 $0.46 $5.42 What happens to the value of calls and puts when the strike price is lower? 100 100 5.5% 5.5% 60 12.5% 100.00 $2.00 $2.00 100 99 5.5% 5.5% 60 12.5% 100.00 $2.53 $1.54 100 98 5.5% 5.5% 60 12.5% 100.00 $3.13 $1.15 100 97 5.5% 5.5% 60 12.5% 100.00 $3.81 $0.83 100 96 5.5% 5.5% 60 12.5% 100.00 $4.55 $0.59 100 95 5.5% 5.5% 60 12.5% 100.00 $5.35 $0.40 X => Call: or Put: or Size: V X V X Call: 1.56 2.0 101 100 2.0 100 22 V X V X Put: 2.55 2.0 101 100 2.0 100 27.5 Relation: linear - nonlinear Intuition: less in the money, less likely to be exercised and less valuable Black-Scholes-Merton Model, Page 4
Sensitivity of Option Values to Changes in Strike Price $6.00 $5.00 $4.00 $3.00 $2.00 $1.00 $0.00 95 96 97 98 99 100 101 102 103 104 105 Strike Price Call Put Call Put $2.00 $2.00 95 $5.35 $0.40 96 $4.55 $0.59 97 $3.81 $0.83 98 $3.13 $1.15 Strike 99 $2.53 $1.54 Price 100 $2.00 $2.00 101 $1.56 $2.55 102 $1.18 $3.17 103 $0.88 $3.86 104 $0.64 $4.61 105 $0.46 $5.42 Black-Scholes-Merton Model, Page 5
Maturity - T What happens to the values of calls and puts when days to expiration is longer? 100 100 5.5% 5.5% 60 12.5% 100.00 $2.00 $2.00 100 100 5.5% 5.5% 90 12.5% 100.00 $2.44 $2.44 100 100 5.5% 5.5% 120 12.5% 100.00 $2.81 $2.81 100 100 5.5% 5.5% 150 12.5% 100.00 $3.12 $3.12 100 100 5.5% 5.5% 180 12.5% 100.00 $3.41 $3.41 100 100 5.5% 5.5% 360 12.5% 100.00 $4.69 $4.69 What happens to the values of calls and puts when days to expiration is shorter? 100 100 5.5% 5.5% 60 12.5% 100.00 $2.00 $2.00 100 100 5.5% 5.5% 30 12.5% 100.00 $1.42 $1.42 100 100 5.5% 5.5% 20 12.5% 100.00 $1.16 $1.16 100 100 5.5% 5.5% 10 12.5% 100.00 $0.82 $0.82 100 100 5.5% 5.5% 5 12.5% 100.00 $0.58 $0.58 100 100 5.5% 5.5% 1 12.5% 100.00 $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
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 $0.00 1 30 60 90 120 150 180 210 240 270 300 330 360 Call Put Call Put $2.00 $2.00 1 $0.26 $0.26 30 $1.42 $1.42 Days 60 $2.00 $2.00 90 $2.44 $2.44 120 $2.81 $2.81 150 $3.12 $3.12 180 $3.41 $3.41 210 $3.66 $3.66 240 $3.90 $3.90 270 $4.12 $4.12 300 $4.32 $4.32 330 $4.51 $4.51 360 $4.69 $4.69 Black-Scholes-Merton Model, Page 7
Spot Price - S What happens to the value of calls and puts when the spot price goes up? 100 100 5.5% 5.5% 60 12.5% 100.00 $2.00 $2.00 101 100 5.5% 5.5% 60 12.5% 101.00 $2.55 $1.56 102 100 5.5% 5.5% 60 12.5% 102.00 $3.17 $1.18 103 100 5.5% 5.5% 60 12.5% 103.00 $3.86 $0.88 104 100 5.5% 5.5% 60 12.5% 104.00 $4.61 $0.64 105 100 5.5% 5.5% 60 12.5% 105.00 $5.42 $0.46 What happens to the value of calls and puts when the spot price goes down? 100 100 5.5% 5.5% 60 12.5% 100.00 $2.00 $2.00 99 100 5.5% 5.5% 60 12.5% 99.00 $1.54 $2.53 98 100 5.5% 5.5% 60 12.5% 98.00 $1.15 $3.13 97 100 5.5% 5.5% 60 12.5% 97.00 $0.83 $3.81 96 100 5.5% 5.5% 60 12.5% 96.00 $0.59 $4.55 95 100 5.5% 5.5% 60 12.5% 95.00 $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
Sensitivity of Option Values to Changes in Spot Price $6.00 $5.00 $4.00 $3.00 $2.00 $1.00 $0.00 95 96 97 98 99 100 101 102 103 104 105 Spot Price Call Put Call Put $2.00 $2.00 95 $0.40 $5.35 96 $0.59 $4.55 97 $0.83 $3.81 98 $1.15 $3.13 Spot 99 $1.54 $2.53 Price 100 $2.00 $2.00 101 $2.55 $1.56 102 $3.17 $1.18 103 $3.86 $0.88 104 $4.61 $0.64 105 $5.42 $0.46 106 $6.27 $0.32 Black-Scholes-Merton Model, Page 9
Volatility - What happens to the values of calls and puts with increases in volatility? 100 100 5.5% 5.5% 60 12.5% 100.00 $2.00 $2.00 100 100 5.5% 5.5% 60 15.0% 100.00 $2.40 $2.40 100 100 5.5% 5.5% 60 17.5% 100.00 $2.80 $2.80 100 100 5.5% 5.5% 60 20.0% 100.00 $3.20 $3.20 100 100 5.5% 5.5% 60 22.5% 100.00 $3.61 $3.61 100 100 5.5% 5.5% 60 25.0% 100.00 $4.01 $4.01 What happens to the values of calls and puts with decreases in volatility? 100 100 5.5% 5.5% 60 12.5% 100.00 $2.00 $2.00 100 100 5.5% 5.5% 60 10.0% 100.00 $1.60 $1.60 100 100 5.5% 5.5% 60 7.5% 100.00 $1.20 $1.20 100 100 5.5% 5.5% 60 5.0% 100.00 $0.80 $0.80 100 100 5.5% 5.5% 60 2.5% 100.00 $0.40 $0.40 100 100 5.5% 5.5% 60 0.0% 100.00 $0.00 $0.00 => Call: or Put: or Size: V V Relation: linear or nonlinear Intuition (Vega): Black-Scholes-Merton Model, Page 10
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 $0.00 0.0% 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 $2.00 0.0% $0.00 $0.00 2.5% $0.40 $0.40 5.0% $0.80 $0.80 7.5% $1.20 $1.20 10.0% $1.60 $1.60 Volatility 12.5% $2.00 $2.00 15.0% $2.40 $2.40 17.5% $2.80 $2.80 20.0% $3.20 $3.20 22.5% $3.61 $3.61 25.0% $4.01 $4.01 Black-Scholes-Merton Model, Page 11
Cost of Funds - R What happens to the values of calls and puts when rate (cost of funds) goes up? 100 100 5.5% 5.5% 60 12.5% 100.00 $2.00 $2.00 100 100 6.5% 5.5% 60 12.5% 100.16 $2.08 $1.92 100 100 7.5% 5.5% 60 12.5% 100.33 $2.17 $1.84 100 100 8.5% 5.5% 60 12.5% 100.49 $2.25 $1.76 100 100 9.5% 5.5% 60 12.5% 100.66 $2.34 $1.69 100 100 10.5% 5.5% 60 12.5% 100.83 $2.43 $1.62 What happens to the values of calls and puts when rate (cost of funds) goes down? 100 100 5.5% 5.5% 60 12.5% 100.00 $2.00 $2.00 100 100 4.5% 5.5% 60 12.5% 99.84 $1.92 $2.09 100 100 3.5% 5.5% 60 12.5% 99.67 $1.85 $2.17 100 100 2.5% 5.5% 60 12.5% 99.51 $1.77 $2.26 100 100 1.5% 5.5% 60 12.5% 99.34 $1.70 $2.35 100 100 0.0% 5.5% 60 12.5% 99.10 $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
Sensitivity of Option Values to Changes in Rate $2.60 $2.40 $2.20 $2.00 $1.80 $1.60 $1.40 0.5% 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 $2.00 0.5% $1.63 $2.45 1.5% $1.70 $2.35 2.5% $1.77 $2.26 3.5% $1.85 $2.17 4.5% $1.92 $2.09 Rate 5.5% $2.00 $2.00 6.5% $2.08 $1.92 7.5% $2.17 $1.84 8.5% $2.25 $1.76 9.5% $2.34 $1.69 10.5% $2.43 $1.62 Black-Scholes-Merton Model, Page 13
Current Yield - Y What happens to the values of calls and puts when yield goes up? 100 100 5.5% 5.5% 60 12.5% 100.00 $2.00 $2.00 100 100 5.5% 6.5% 60 12.5% 99.84 $1.92 $2.08 100 100 5.5% 7.5% 60 12.5% 99.67 $1.84 $2.17 100 100 5.5% 8.5% 60 12.5% 99.51 $1.76 $2.25 100 100 5.5% 9.5% 60 12.5% 99.34 $1.69 $2.34 100 100 5.5% 10.5% 60 12.5% 99.18 $1.62 $2.43 What happens to the vaues of calls and puts when yield goes down? 100 100 5.5% 5.5% 60 12.5% 100.00 $2.00 $2.00 100 100 5.5% 4.5% 60 12.5% 100.16 $2.09 $1.92 100 100 5.5% 3.5% 60 12.5% 100.33 $2.17 $1.85 100 100 5.5% 2.5% 60 12.5% 100.49 $2.26 $1.77 100 100 5.5% 1.5% 60 12.5% 100.66 $2.35 $1.70 100 100 5.5% 0.0% 60 12.5% 100.91 $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
Sensitivity of Option Values to Cha nges in Yield $2.60 $2.40 $2.20 $2.00 $1.80 $1.60 $1.40 0.5% 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 $2.00 0.5% $2.45 $1.63 1.5% $2.35 $1.70 2.5% $2.26 $1.77 3.5% $2.17 $1.85 4.5% $2.09 $1.92 Yield 5.5% $2.00 $2.00 6.5% $1.92 $2.08 7.5% $1.84 $2.17 8.5% $1.76 $2.25 9.5% $1.69 $2.34 10.5% $1.62 $2.43 Black-Scholes-Merton Model, Page 15
Delta - What happens to the value of call and put deltas when the spot price goes up? 100 100 5.5% 5.5% 60 12.5% 100.00 0.506 0.485 101 100 5.5% 5.5% 60 12.5% 101.00 0.582 0.409 102 100 5.5% 5.5% 60 12.5% 102.00 0.655 0.336 103 100 5.5% 5.5% 60 12.5% 103.00 0.722 0.269 104 100 5.5% 5.5% 60 12.5% 104.00 0.781 0.210 105 100 5.5% 5.5% 60 12.5% 105.00 0.831 0.160 What happens to the value of call and put deltas when the spot price goes down? 100 100 5.5% 5.5% 60 12.5% 100.00 0.506 0.485 99 100 5.5% 5.5% 60 12.5% 99.00 0.427 0.564 98 100 5.5% 5.5% 60 12.5% 98.00 0.351 0.640 97 100 5.5% 5.5% 60 12.5% 97.00 0.280 0.711 96 100 5.5% 5.5% 60 12.5% 96.00 0.216 0.775 95 100 5.5% 5.5% 60 12.5% 95.00 0.160 0.831 S => Call delta: or Put delta: or Size: Delta s Delta Relation: linear or nonlinear Intuition (Gamma): s Black-Scholes-Merton Model, Page 16
Sensitivity of Option Deltas to Changes in Spot Price (strike =100) 1.20-1.20 1.00-1.00 0.80-0.80 0.60-0.60 0.40-0.40 0.20-0.20 0.00 0.00 90 92 94 96 98 100 102 104 106 108 110 112 Spot Price Call Put Call Put 0.51-0.49 90 0.02-0.97 92 0.05-0.94 94 0.11-0.88 96 0.22-0.78 98 0.35-0.64 100 0.51-0.49 102 0.66-0.34 104 0.78-0.21 106 0.87-0.12 108 0.93-0.06 110 0.96-0.03 112 0.98-0.01 Black-Scholes-Merton Model, Page 17
Cash % - X What happens to the value of call and put cash % when the spot price goes up? 100 100 5.5% 5.5% 60 12.5% 100.00-0.485 0.506 101 100 5.5% 5.5% 60 12.5% 101.00-0.563 0.428 102 100 5.5% 5.5% 60 12.5% 102.00-0.637 0.354 103 100 5.5% 5.5% 60 12.5% 103.00-0.705 0.286 104 100 5.5% 5.5% 60 12.5% 104.00-0.766 0.225 105 100 5.5% 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? 100 100 5.5% 5.5% 60 12.5% 100.00-0.485 0.506 99 100 5.5% 5.5% 60 12.5% 99.00-0.408 0.583 98 100 5.5% 5.5% 60 12.5% 98.00-0.333 0.658 97 100 5.5% 5.5% 60 12.5% 97.00-0.263 0.728 96 100 5.5% 5.5% 60 12.5% 96.00-0.201 0.790 95 100 5.5% 5.5% 60 12.5% 95.00-0.148 0.843 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
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 0.00-0.20 0.20-0.40 0.40 Calls -0.60 0.60 Puts -0.80 0.80-1.00 1.00-1.20 1.20 Spot Price Call Put 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 100-0.49 0.51 102-0.64 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 Black-Scholes-Merton Model, Page 19
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
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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
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
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