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Black-Scholes Pricing cont’d & Beginning Greeks

Black-Scholes Pricing cont’d & Beginning Greeks. Black-Scholes cont’d. Through example of JDS Uniphase Pricing Historical Volatility Implied Volatility. Beginning Greeks & Hedging. Hedge Ratios Greeks (Option Price Sensitivities) delta, gamma (Stock Price) rho (riskless rate)

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Black-Scholes Pricing cont’d & Beginning Greeks

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  1. Black-Scholes Pricing cont’d& Beginning Greeks

  2. Black-Scholes cont’d • Through example of JDS Uniphase • Pricing • Historical Volatility • Implied Volatility

  3. Beginning Greeks & Hedging • Hedge Ratios • Greeks (Option Price Sensitivities) • delta, gamma (Stock Price) • rho (riskless rate) • theta (time to expiration) • vega (volatility) • Delta Hedging

  4. Hedge Ratios • Number of units of hedging security to moderate value change in exposed position • If trading options: Number of units of underlying to hedge options portfolio • If trading underlying: Number of options to hedge underlying portfolio • For now: we will act like trading European Call Stock Options with no dividends on underlying stock.

  5. Delta, Gamma • Sensitivity of Call Option Price to Stock Price change (Delta):  = N(d1) • We calculated this to get option price. • Gamma is change in Delta measure as Stock Price changes…. we’ll get to this later!

  6. Delta Hedging • If an option were on 1 share of stock, then to delta hedge an option, we would take the overall position: +C -  S = 0 (change) • This means whatever your position is in the option, take an opposite position in the stock • (+ = bought option  sell stock) • (+ = sold option  buy stock)

  7. Recall the Pricing Example • IBM is trading for $75. Historically, the volatility is 20% (s). A call is available with an exercise of $70, an expiry of 6 months, and the risk free rate is 4%. ln(75/70) + (.04 + (.2)2/2)(6/12) d1 = -------------------------------------------- = .70, N(d1) =.7580 .2 * (6/12)1/2 d2 = .70 - [ .2 * (6/12)1/2 ] = .56, N(d2) = .7123 C = $75 (.7580) - 70 e -.04(6/12) (.7123) = $7.98 Intrinsic Value = $5, Time Value = $2.98

  8. Hedge the IBM Option • Say we bought (+) a one share IBM option and want to hedge it: + C -  S means 1 call option hedged with  shares of IBM stock sold short (-). •  = N(d1) = .758 shares sold short.

  9. Hedge the IBM Option • Overall position value: Call Option cost = -$ 7.98 Stock (short) gave = +$ 56.85 (S = .758*75 = 56.85) • Overall account value: +$ 48.87

  10. Why a Hedge? • Suppose IBM goes to $74. ln(74/70) + (.04 + (.2)2/2)(6/12) d1 = -------------------------------------------- = 0.61, N(d1) =.7291 .2 * (6/12)1/2 d2 = 0.61 - [ .2 * (6/12)1/2 ] = 0.47, N(d2) = .6808 C = $74 (.7291) - 70 e -.04(6/12) (.6808) = $7.24

  11. Results • Call Option changed: (7.24 - 7.98)/7.98 = -9.3% • Stock Price changed: (74 - 75)/75 = -1.3% • Hedged Portfolio changed: (Value now –7.24 + (.758*74) = $48.85) (48.85 - 48.87)/48.87 = -0.04%! • Now that’s a hedge!

  12. Hedging Reality #1 • Options are for 100 shares, not 1 share. • You will rarely have one option to hedge. • Both these issues are just multiples! + C -  S becomes + 100 C - 100  S for 1 actual option, or + X*100 C - X*100  S for X actual options

  13. Hedging Reality #2 • Hedging Stock more likely: + C -  S = 0 becomes algebraically - (1/) C + S • So to hedge 100 shares of long stock (+), you would sell (-) 1/ options • For example, (1/.758) = 1.32 options

  14. Hedging Reality #3 • Convention does not hedge long stock by selling call options (covered call). • Convention hedges long stock with bought put options (protective put). • Instead of - (1/) C + S - (1/P) P + S

  15. Hedging Reality #3 cont’d • P = [N(d1) - 1], so if N(d1) < 1 (always), then P < 0 • This means - (1/P) P + S actually has the same positions in stock and puts ( -(-) = + ). • This is what is expected, protective put is long put and long stock.

  16. Reality #3 Example • Remember IBM pricing: ln(75/70) + (.04 + (.2)2/2)(6/12) d1 = -------------------------------------------- = .70, N(d1) =.7580 .2 * (6/12)1/2 d2 = .70 - [ .2 * (6/12)1/2 ] = .56, N(d2) = .7123 C = $75 (.7580) - 70 e -.04(6/12) (.7123) = $7.98 Put Price = Call Price + X e-rT - S Put = $7.98 + 70 e -.04(6/12) - 75 = $1.59

  17. Hedge 100 Shares of IBM • - (1/P) P + S = - 100 * (1/P) P + 100 * S • P = N(d1) – 1 = .758 – 1 = -.242 - (1/P) = - (1/ -.242) = + 4.13 options • Thus if “ + “ of + S means bought stock, then “ + “ of +4.13 means bought put options! • That’s a protective put!

  18. Hedge Setup • Position in Stock: $75 * 100 = +$7500 • Position in Put Options: $1.59 * +4.13 * 100 = +$656.67 • Total Initial Position =+$8156.67

  19. IBM drops to $74 • Remember call now worth $7.24 • Puts now worth $1.85 * 4.13 * 100 = $ 764.05 • Total Position = $7400 + 764.05 = $8164.05 Put Price = Call Price + X e-rT - S Put = $7.24 + 70 e -.04(6/12) - 74 = $1.85

  20. Results • Stock Price changed: (74 - 75)/75 = -1.3% • Portfolio changed: (8164.05 – 8156.67) / 8156.67 = +0.09%!!!! • Now that’s a hedge!

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