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# F B E559f3 B S Formula

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### Transcript

• 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.