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# F B E559f3 B S Formula

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### F B E559f3 B S Formula

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