Black-Scholes (1973) Option Pricing Formula



|Black-Scholes (1973) Option Pricing Formula | |

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|Explained: | |

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|Black-Scholes (1973) option pricing formula | |

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|In 1973, Fischer Black and Myron Scholes published their groundbreaking paper the pricing of options and corporate | |

|liabilities. Not only did this specify the first successful options pricing formula, but it also described a general | |

|framework for pricing other derivative instruments. That paper launched the field of financial engineering. Black and | |

|Scholes had a very hard time getting that paper published. Eventually, it took the intersession of Eugene Fama and Merton | |

|Miller to get it accepted by the Journal of Political Economy. In the mean time, Black and Scholes had published in the | |

|Journal of Finance a more accessible (1972) paper that cited the as-yet unpublished (1973) option pricing formula in an | |

|empirical analysis of current options trading. | |

|The Black-Scholes (1973) option pricing formula prices European put or call options on a stock that does not pay a | |

|dividend or make other distributions. The formula assumes the underlying stock price follows a geometric Brownian motion | |

|with constant volatility. It is historically significant as the original option pricing formula published by Black and | |

|Scholes in their landmark (1973) paper. | |

|Values for a call price c or put price p are: | |

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|[1] | |

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|[2] | |

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|where: | |

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|[3] | |

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|[4] | |

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|Here, log denotes the natural logarithm, and: | |

|[pic]s = the price of the underlying stock | |

|[pic]x = the strike price | |

|[pic]r = the continuously compounded risk free interest rate | |

|[pic]t = the time in years until the expiration of the option | |

|[pic]σ = the implied volatility for the underlying stock | |

|[pic]Φ = the standard normal cumulative distribution function. | |

|Consider a European call option on 100 shares of non-dividend-paying stock ABC. The option is struck at USD 55 and expires| |

|in .34 years. ABC is trading at USD 56.25 and has 28% (that is .28) implied volatility. The continuously compounded risk | |

|free interest rate is .0285. Applying formula [1], the option's market value per share of ABC is USD 4.56. Since the call | |

|is for 100 shares, its total value is USD 456. Of this, USD 125 is intrinsic value, and USD 331 is time value. | |

|The Greeks—delta, gamma, vega, theta and rho—for a call are: | |

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|delta = Φ(d1) | |

|[5] | |

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|gamma = [pic] | |

|[6] | |

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|vega = [pic] | |

|[7] | |

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|theta = [pic] | |

|[8] | |

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|[9] | |

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|where [pic]denotes the standard normal probability density function. For a put, the Greeks are: | |

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|delta = Φ(d1) – 1 | |

|[10] | |

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|gamma = [pic] | |

|[11] | |

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|vega = [pic] | |

|[12] | |

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|theta = [pic] | |

|[13] | |

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|[14] | |

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|Note that gamma formulas [6] and [11]are identical for puts and calls, as are vega formulas [7] and [12]. | |

|Related Internal Links | |

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|[pic]Merton (1973) option pricing formula Used to price European options on dividend paying stocks or stock indexes. | |

|[pic]Black (1976) option pricing formula Used to price options on forwards. | |

|[pic]Garman and Kohlhagen (1983) option pricing formula Used for pricing foreign exchange options. | |

|[pic]option pricing theory The body of financial theory used by financial engineers to value options and other derivative | |

|instruments. | |

|[pic]put-call parity A formula that relates the price of a put to the price of a corresponding call. | |

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|Sponsored Links | |

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|Related Books | |

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|To learn about the historical origins of the Black-Scholes formula, see Mehrling (2005) or Bernstein (1993). Haug (1997) | |

|is a handy encyclopedia of published option pricing formulas, including Black-Scholes. Natenberg (1994) is an excellent | |

|introduction to options trading. Cox and Rubinstein (1985) is a classic. Pretty much everyone who works with options has | |

|read it at some point. Hull (2005) is a standard introduction to financial engineering that covers the derivation of | |

|Black-Scholes in detail. | |

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|Fischer Black | |

|And the Revolutionary Idea of Finance | |

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|Perry Mehrling | |

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|quality | |

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|technical | |

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|2005 | |

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|Capital Ideas | |

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|Peter L. Bernstein | |

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|technical | |

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|1993 | |

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|Complete Guide to | |

|Option Pricing Formulas | |

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|Espen G. Haug | |

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|quality | |

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|technical | |

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|1997 | |

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|Option Volatility & Pricing | |

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|Sheldon Natenberg | |

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|quality | |

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|technical | |

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|1994 | |

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|Options Markets | |

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|John C. Cox and Mark Rubinstein | |

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|quality | |

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|technical | |

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|1985 | |

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|Cited Papers | |

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|[pic]Black, Fischer and Myron S. Scholes (1972) The valuation of option contracts and a test of market efficiency, Journal| |

|of Finance, 27 (2), 399–418. | |

|[pic]Black, Fischer and Myron S. Scholes (1973). The pricing of options and corporate liabilities, Journal of Political | |

|Economy, 81 (3), 637-654. | |

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|Sponsored Links | |

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|Related Forum Discussions | |

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|Black-Scholes 07 Oct 1998 | |

|Intuitive understanding of the Black-Scholes formula. | |

|Options probability distributions 02 Sep 1998 | |

|Risk neutrality and the probability of an option expiring in-the-money. | |

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