1 / 18

Black-Scholes Pricing & Related Models

Black-Scholes Pricing & Related Models. Option Valuation. Black and Scholes Call Pricing Put-Call Parity Variations. Option Pricing: Calls. Black-Scholes Model:. C = Call S = Stock Price N = Cumulative Normal Distrib. Operator X = Exercise Price e = 2.71..... r = risk-free rate

Download Presentation

Black-Scholes Pricing & Related Models

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Black-Scholes Pricing & Related Models

  2. Option Valuation • Black and Scholes • Call Pricing • Put-Call Parity • Variations

  3. Option Pricing: Calls • Black-Scholes Model: C = Call S = Stock Price N = Cumulative Normal Distrib. Operator X = Exercise Price e = 2.71..... r = risk-free rate T = time to expiry = Volatility

  4. Call Option 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

  5. - = - - - rT P Xe N ( d ) SN ( d ) 2 1 Put Option Pricing • Put priced through Put-Call Parity: Put Price = Call Price + X e-rT - S (or : ) From Last Example of IBM Call: Put = $7.98 + 70 e -.04(6/12) - 75 = $1.59 Intrinsic Value = $0, Time Value = $1.59

  6. Black-Scholes Variants • Options on Stocks with Dividends • Futures Options (Option that delivers a maturing futures) • Black’s Call Model (Black (1976)) • Put/Call Parity • Options on Foreign Currency • In text (Pg. 375-376, but not req’d) • Delivers spot exchange, not forward!

  7. 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

  8. - - = - * * * 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):

  9. Futures Option Pricing Model • Black’s futures option pricing model for European call options:

  10. Futures Option Pricing Model (cont’d) • Black’s futures option pricing model for European put options: • Alternatively, value the put option using put/call parity:

  11. 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 (Merton model) • Interest rates and volatility remain constant, but are unknown

  12. Interest Rates Remain Constant • There is no real “riskfree” interest rate • Often use the closest T-bill rate to expiry

  13. Calculating Volatility Estimates • 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).

  14. Intro to Implied Volatility • 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

  15. 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 .

  16. 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

  17. Volatility Smiles (cont’d)

  18. 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

More Related