Option Pricing Using BlackScholes Model

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Option Pricing Using Black-Scholes Model
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Black-Scholes Model

• Black-Scholes option pricing model was developed
  in 1970 by
• Fischer Balck
• Myron Scholes
• Robert Merton
• Their work has had huge influence on the way in
  which market participants price and hedge options.

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Blacks Formula

• The formulas for options on futures are known as
  Blacks formulas

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• c call price
• p put price
• Ffutures price
• Xstrike price
• r risk free interest rate
• T time to maturity
• ? volatility

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• The function N(x) is a cumulative probability
  function for a standardized normal variable. In
  other words, it is the probability that a
  variable with a standard normal distribution will
  be less than x.

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Estimating Volatility from Historical Data

• 1. Take observations S 0, S 1, . . . , Sn at
  intervals of t years
• 2. Define the continuously compounded return as
• 3. Calculate the standard deviation of the ui s
  (s)
• 4. The volatility estimate (s) is

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Standard Deviation

• Standard deviation of ui
• SQRT((?u2i )/(n-1)- (?ui)2/n(n-1))

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Properties of Black-Scholes Formula

• As F becomes very large c tends to (F-X)e-rT and
  p tends to zero
• As F becomes very small c tends to zero and p
  tends to e-rT (X-F)

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Problem

• A futures price is currently 25, its volatility
  is 30 percent per annum, and the risk-free
  interest rate is 10 percent per annum. What is
  the value of a nine-month European call on the
  futures with a strike price of 26?

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Problem

• The futures price, nine months from the
  expiration of a put option, is 40, the exercise
  price of the option is 39, the risk-free
  interest rate is 12 percent per annum, and the
  volatility is 23 percent per annum. Calculate
  the put option price using Black-Scholes option
  pricing model.

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Problem

• Consider a put futures option on crude oil. The
  time to maturity is four months, the current
  future price is 20, the exercise price is 20,
  the risk free interest rate is 9 percent per
  annum and the volatility of the futures price is
  25 percent per annum. Calculate the premium for
  the put option.