Important Concepts LECTURE 3.2: OPTION PRICING MODELS: THE BLACK-SCHOLES-MERTON MODEL. Applications of Logarithms and Exponentials in Finance

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1 Important Concepts The Black Scholes Merton (BSM) option pricing model LECTURE 3.2: OPTION PRICING MODELS: THE BLACK-SCHOLES-MERTON MODEL Black Scholes Merton Model as the Limit of the Binomial Model Origins of the Black Scholes Merton Formula A Nobel Formula : Financial Modelling Nattawut Jenwittayaroje, PhD, CFA NIDA Business School National Institute of Development Administration How to adjust the model to accommodate dividends The concepts of historical and implied volatility Estimating the Volatility BSM option pricing model for put options. 1 2 Black Scholes Merton Model as the Limit of the Binomial Model Applications of Logarithms and Exponentials in Finance Recall the binomial model and the notion of a dynamic risk free hedge in which no arbitrage opportunities are available. See the value of $1 invested for one year at 6% with various compounding frequencies.. Compounding period per year 1 (annually) 2 (semiannually) $1 invested for one year at 6% Consider the DCRB June 125 call option. Figure 5.1 shows the model price for an increasing number of time steps. The binomial model is in discrete time. As you decrease the length of each time step, it converges to continuous time. Conversion between discrete compounding and continuous compounding. 4 (quarterly) 12 (monthly) 52 (weekly) 365 (yearly) 10,000 times a year 100,000 times a year times a year The binomial model converges to the Black Scholes Merton Model as the number of time periods increases. 3 4

2 Origins of the Black Scholes Merton Formula Black, Scholes, Merton and the 1997 Nobel Prize F. Black and M. S. Scholes. The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81 (May June 1973), R. C. Merton. The Theory of Rational Option Pricing. Bell Journal of Economics and Management Science, 4 (Spring 1973), A Nobel Formula The Black Scholes Merton model gives the correct formula for a European call under certain assumptions. The general idea of the Black Scholes Merton model is the same as that of the binomial model, except that in BSM model, trading occurs continuously. So in BSM, a hedge portfolio is established and maintained by constantly adjusting the relative proportions of stock and options, a process called dynamic trading. The model is derived with complex mathematics but is easily understandable. The Black Scholes Merton formula 5 6 where A Nobel Formula (continued) N(d 1 ), N(d 2 ) = cumulative normal probability A Digression on Using the Normal Distribution The familiar normal, bell shaped curve (Figure 5.5) See Table 5.1 for determining the normal probability for d 1 and d 2. This gives you N(d 1 ) and N(d 2 ). σ = annualized standard deviation (volatility) of the continuously compounded (log) return on the stock r c = continuously compounded riskfree rate S 0 = current stock price X = exercise price T = time to expiration in years 7 8

3 The cumulative probabilities of the standard normal distribution A Nobel Formula (continued) A Numerical Example Price the DCRB June 125 call S 0 = , X = 125, r c = ln(1.0456) = , where 4.56% is simple risk free rate, and 4.46% is continuously compounded risk free rate T = , = Black Scholes Merton Model When the Stock Pays Dividends Known Discrete Dividends Assume a single dividend of D t where the ex dividend date is time t during the option s life. Subtract present value of dividends from stock price. Adjusted stock price, S, is inserted into the B S M model. See Table 5.3 for example. Black-Scholes-Merton Model When the Stock Pays Discrete Dividends Continuous Dividend Yield Assume the stock pays dividends continuously at the rate of. Subtract present value of dividends from stock price. Adjusted stock price, S, is inserted into the B S model. See Table 5.4 for example. This approach could also be used if the underlying is a foreign currency, where the yield is replaced by the continuously compounded foreign risk free rate

4 Black-Scholes-Merton Model When the Stock Pays Continuous Dividends Put Option Pricing Models Restate put call parity with continuous discounting Substituting the B S M formula for C above gives the B S M put option pricing model The put option can also be represented as; where N(d 1 ) and N(d 2 ) are the same as in the call model. Note calculation of put price: Estimating the Volatility Estimating the Historical Volatility There are two approaches to estimating volatility: 1) the historical volatility, and 2) the implied volatility. Historical Volatility The historical volatility estimate is based on the assumption that the volatility that prevailed over the recent past will continue to hold in the future. The historical volatility is estimated from a sample of recent continuously compounded returns on the stock. This is the volatility over a recent time period. Collect daily (weekly, or monthly) returns on the stock. Convert each return to its continuously compounded equivalent by taking ln(1 + return). Calculate variance. Annualize by multiplying by 250 (daily returns), 52 (weekly returns) or 12 (monthly returns). Take square root. See Table 5.6 for example with DCRB

5 Estimating the Implied Volatility Implied Volatility This is the volatility implied when the market price of the option is set to the model price. Figure 5.17 illustrates the procedure. Substitute estimates of the volatility into the B S M formula until the market price converges to the model price. Interpreting the Implied Volatility The CBOE has constructed indices of implied volatility of one month at themoney options based on the S&P 500 (VIX) and Nasdaq (VXN). See Figure A number of studies have examined whether implied volatility is a good predictor of the future volatility of a stock. A general consensus is that implied volatility is a better reflection of the appropriate volatility for pricing an option than is the historical volatility Interpreting the Implied Volatility S&P500 from Sep 2007 to May 2011 VIX is usually referred to as the fear index. It represents one measure of the market s expectation of stock market (annualized) volatility over the next 30 day period. Summary Figure 5.21 shows the relationship between call, put, underlying asset, risk free bond, put call parity, and Black Scholes Merton call and put option pricing models. VIX from Sep 2007 to May 2011 For example, if VIX is 20%, the S&P500 is expected to move up or down about 20/sqrt(12) = 5.8% over the next 30 day period. Or there is a 68% likelihood (one SD) that the magnitude of the S&P500 s 30-day return will be less than 5.8% (either up or down). Source: