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 T = time to expiry = Volatility

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

- = - - - 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

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!

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

- - = - * * * 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):

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

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:

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

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

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).

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

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 .

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

Volatility Smiles (cont’d)

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

Black-Scholes Pricing & Related Models