Valuing Stock Options The BlackScholes Merton Model Chapter
Valuing Stock Options: The Black–Scholes– Merton Model Chapter 13 13. 1
Goals of Chapter 13 l l l The stock price distribution in Black–Scholes– Merton (BSM) model and the estimation of its parameters The risk-neutral valuation relationship (RNVR) and BSM option pricing formulae Extension of RNVR on pricing forward contracts Implied volatilities (隱含波動度) of options Effects of cash dividend payments on option prices 13. 2
13. 1 Distribution of the Stock Price in Black– Scholes–Merton Model 13. 3
Distribution of The Stock Price l In 1973, Fischer Black, Myron Scholes, and Robert Merton achieved a major breakthrough in pricing European options – They developed pricing formulae for European options, which are known as the Black–Scholes– Merton (or simply Black–Scholes) model – Merton and Scholes won Nobel Prize in 1997 with this achievement – This chapter presents the Black–Scholes–Merton model for valuing European calls and puts on a nondividend-paying stock first 13. 4
Distribution of The Stock Price l 13. 5
Distribution of The Stock Price l 13. 6
Distribution of The Stock Price l 13. 7
Distribution of The Stock Price l 13. 8
Distribution of The Stock Price l 13. 9
Distribution of The Stock Price l 13. 10
Distribution of The Stock Price l 13. 11
Distribution of The Stock Price l 13. 12
Distribution of The Stock Price l 13. 13
13. 2 Risk-Neutral Valuation Relationship and Black –Scholes–Merton option pricing formulae 13. 14
RNVR and BSM model l 13. 15
RNVR and BSM model l 13. 16
RNVR and BSM model l 13. 17
RNVR and BSM model l 13. 18
RNVR and BSM model l 13. 19
RNVR and BSM model l 13. 20
RNVR and BSM model l A full proof of the BSM formulae is beyond the scope of this course – In fact, Black and Scholes derived the option pricing formulae in another way which is similar to what the binomial tree model does n n Basic idea is to construct a riskless portfolio by determining proper weights for the option and its underlying asset This is because the option price and the underlying asset price share the same source of uncertainty The portfolio is instantaneously riskless and must instantaneously earn the risk-free rate Thus, a partial differential equation (偏微分方程式) is derived, and the solution of it is the BSM option pricing 13. 21 formula
RNVR and BSM model l 13. 22
13. 3 Apply Risk-Neutral Valuation Relationship to Pricing Forward Contracts 13. 23
RNVR for Forward Contracts l 13. 24
RNVR for Forward Contracts l 13. 25
13. 4 Implied Volatility 13. 26
Implied Volatility l 13. 27
Implied Volatility – The implied volatility reflects the consensus of traders in the option market on the volatility of the underlying asset price for a future period n n The implied volatility is a forward-looking estimation, which is the expected volatility about a future period over which the option will exist The historical volatility works well only if the price behavior of the underlying asset in the immediate future is the same as that in the recent past – The implied volatility of an option DOES depend on its strike price and time to maturity – Traders and brokers often quote implied volatilities rather than dollar prices (to facilitate identifying 13. 28 overvalued or undervalued options)
Implied Volatility l VIX (volatility index) – The CBOE publishes indices of implied volatilities – The most popular index, the S&P VIX, is an index of the implied volatility of 30 -day S&P 500 index options calculated from a wide range of calls and puts – The S&P VIX, with a normal range between 15% and 25%, can be interpreted as the expectation of the volatility of the S&P 500 index in the future one month – Trading in futures on the VIX started in 2004 and trading in options on the VIX started in 2006 n Note that the underlying asset of those derivatives is the VIX 13. 29 index, which is not a tradable asset
13. 5 Effects of Cash Dividend Payments on Option Prices 13. 30
Effects of Cash Dividend Payments on Option Prices l 13. 31
Effects of Cash Dividend Payments on Option Prices l 13. 32
Effects of Cash Dividend Payments on Option Prices l 13. 33
Effects of Cash Dividend Payments on Option Prices l Fischer Black proposed an approximation for the value of an American call based on the BSM model if there are dividend payments during the life of the option – The well-known early exercise behavior of American calls n n An American call on a non-dividend-paying stock should never be exercised early An American call on a dividend-paying stock should only be exercised immediately prior to an ex-dividend date 13. 34
Effects of Cash Dividend Payments on Option Prices – Approximate the American call equal to the maximum of two European option prices: 1. The 1 st European option price is for an option maturing at the same time as the American option 2. The 2 nd European option price is for an option maturing just before the final ex-dividend date (The strike price, initial stock price, risk-free interest rate, and the volatility are the same for the options under consideration) ※Note that the binomial tree model can evaluate American calls or puts accurately with proper modifications for dealing with the cash dividend payments (beyond the scope of this course) 13. 35
(Material from Ch. 15) European Options on Stocks Paying Dividend Yields l 13. 36
(Material from Ch. 15) European Options on Stocks Paying Dividend Yields l 13. 37
(Material from Ch. 15) European Options on Stocks Paying Dividend Yields l 13. 38
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