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Pricing Cont’d & Beginning Greeks

Pricing Cont’d & Beginning Greeks. Assumptions of the Black-Scholes Model. European exercise style Markets are efficient No transaction costs The stock pays no dividends during the option’s life Interest rates and volatility remain constant.

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Pricing Cont’d & Beginning Greeks

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  1. Pricing Cont’d &Beginning Greeks

  2. Assumptions of the Black-Scholes Model • European exercise style • Markets are efficient • No transaction costs • The stock pays no dividends during the option’s life • Interest rates and volatility remain constant

  3. The Stock Pays no Dividends During the Option’s Life • If you apply the BSOPM to two securities, one with no dividends and the other with a dividend yield, the model will predict the same call premium • Robert Merton developed a simple extension to the BSOPM to account for the payment of dividends

  4. - - = - * * * T RT C e SN ( d ) Xe N ( d ) 1 2 where æ ö s 2 æ ö S ç ÷ + - + ç ÷ ln R  T ç ÷ X 2 è ø è ø = * d 1 s T and = - s * * d d T 2 1 The Stock Pays Dividends During the Option’s Life (cont’d) Adjust the BSOPM by following (=continuous dividend yield):

  5. Interest Rates and Volatility Remain Constant • There is no real “riskfree” interest rate • Often use the closest T-bill rate to expiry • Volatility expectations change constantly. That’s why option prices can change when everything else remains constant!

  6. Calculating Black-Scholes Prices • from Historical Data: S, R, T that just was, and  as standard deviation of historical returns from some arbitrary past period • from Actual Data: S, R, T that just was, and  implied from pricing of nearest “at-the-money” option (termed “implied volatility).

  7. Implied Volatility • Introduction • Calculating implied volatility • Volatility smiles

  8. Introduction • Instead of solving for the call premium, assume the market-determined call premium is correct • Then solve for the volatility that makes the equation hold • This value is called the implied volatility

  9. Calculating Implied Volatility • Setup spreadsheet for pricing “at-the-money” call option. • Input actual price. • Run SOLVER to equate actual and calculated price by varying .

  10. Volatility Smiles • Volatility smiles are in contradiction to the BSOPM, which assumes constant volatility across all strike prices • When you plot implied volatility against striking prices, the resulting graph often looks like a smile

  11. Volatility Smiles (cont’d)

  12. Problems Using the Black-Scholes Model • Does not work well with options that are deep-in-the-money or substantially out-of-the-money • Produces biased values for very low or very high volatility stocks • Increases as the time until expiration increases • May yield unreasonable values when an option has only a few days of life remaining

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

  14. 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.

  15. 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!

  16. 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)

  17. 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

  18. 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.

  19. 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

  20. 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

  21. 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!

  22. 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

  23. 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

  24. 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

  25. 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.

  26. 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

  27. 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!

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

  29. 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

  30. 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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