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F B E559f3 B S  Formula
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F B E559f3 B S Formula

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  • 1. Black and Scholes Formula For European Options
  • 2. Stock Price Dynamics
    • Suppose that the price of the stock satisfies:
      •  is the expected return.
      •  is the volatility.
      • Both are constant.
    • Value of S at moment T:
  • 3. Lognormal Distribution
    • Graphic representation:
    0
  • 4. Bond Price Dynamics
    • There is a bond or checking account that satisfies:
      • r is the continuously paid interest rate.
      • It is constant.
    • The price of the bond at moment T is:
  • 5. European Call Option Dynamics
    • Consider an European call on S with strike price X and maturity at T.
    • The price C will be a function of time (or time left to maturity) and S: C(S,t).
    • By Ito’s Lemma:
  • 6. European Call Dynamics (cont.)
    • The previous expression is equivalent to:
    • Suppose we form a portfolio with the option and the stock but without uncertainty term:
      • That portfolio would be riskless.
      • Its expected return should be the riskfree rate.
  • 7. BS differential equation
    • After constructing such portfolio we are left with:
    • Subject to the following condition at maturity:
  • 8. Black and Scholes formula
    • Solution to the previous equation:
    • Where:
    • r is the continuously compounded interest.
    •  is the volatility of the return.
  • 9. Black and Scholes (cont.)
    • N(d) is the cumulative normal distribution:
    d 0
  • 10. Black and Scholes (cont.)
    • N(d) is the “delta” or number of shares (smaller than one) needed to replicate it.
    • e -rt  X is the present value of X.
    • Price of the European put: we can get it from the put-call parity:
  • 11. Risk-neutral valuation
    • Suppose that the stock satisfies the following dynamics:
    • BS is the result of:
    • As in the binomial case.
    • This will allow simple numerical methods.
  • 12. Assumptions of BS
    • Continuous and constant interest rate.
    • Constant expected return  :
      • It does not appear in the BS formula.
    • Constant standard deviation  :
      • Very restrictive.
    • Frictionless markets.
    • Unlimited borrowing/shortselling possibilities.
  • 13. Graph: European, American call Call option price X Stock price S-X
  • 14. Graph: American put Put option price X Stock price X-S Early exercise
  • 15. Computing volatility
    •  is the only parameter not directly observable.
    • Typically, estimated from past data.
    • Volatility of the return, not of the price:
  • 16. Computing volatility (cont.)
    • We compute the standard deviation of previous expression (say s ).
    • We then derive  by adjusting the time period.
    • For example, if we have considered daily returns:
  • 17. Implied volatility
    • Concept:
      • Consider all the observed values.
      • Including the price of the option.
      • It is the volatility for which the BS formula would yield that price.
    • In some markets, implied volatility quoted (instead of price of option).
    • Provide information about the market:
      • Different options on same stock can differ.
  • 18. European options with dividends
    • We assume the dividend and date of payment are known.
    • Dividend is a “riskless component” of price of stock.
    • We subtract the present value of the dividend and apply BS to the rest.
  • 19. American options with dividends
    • For put options, it could be optimal to exercise before maturity, with or without dividends:
      • With dividends, only after dividend is paid, if around dividend date.
    • For calls, only can be before dividend is paid, but, if dividend is too small, it is not optimal:
      • From put-call parity, if:
      • It will not be optimal to exercise early.
  • 20. Black’s approximation for calls
    • We need:
      • Estimate of the dividend.
      • Date to be paid.
    • Two different prices are computed:
      • Value if held until maturity.
      • Value if early exercise.
    • We pick the maximum of them.
  • 21. Black’s approximation (cont.)
    • If held until maturity:
    • Compute:
    • Compute Black and Scholes with S * instead of S.
  • 22. Black’s approximation (cont.)
    • If early exercise:
    • Compute S * (as before).
    • Use the Black and Scholes formula but:
      • With S * instead of S.
      • With the time to dividend payment instead of time to maturity.
      • With strike price X-D.

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