Option pricing: pre-analytics Lecture 5

1. Option pricing:pre-analyticsLecture 5

2. c : European call option price p : European put option price S0: Stock price today X : Strike price T: Life of option : Volatility of stock price C : American Call option price P : American Put option price ST:Stock price at time T D : Present value of dividends during option’s life r: Risk-free rate for maturity Twith cont comp Notation

3. Calls: An Arbitrage Opportunity? • Suppose that c = 3 S0= 20 T = 1 r = 10% X = 18 D= 0 • Is there an arbitrage opportunity?

4. Lower Bound for European Call Option Prices; No Dividends c + Xe -rTS0 consider 2 portfolios:a. buy a call and a ZCB worth X at Tb. buy the stock in fact, at maturity:if S>X, identical resultif S<X, superior result

5. Lower Bound for European Call Option Prices; No Dividends c + Xe -rTS0 cS0 -Xe -rT c 20 -18*0.9048 c 3.71

6. arbitrage if c<3.71 TODAY • sell the stock at 20, and with the proceeds: • buy the call at 3 and a ZCB maturing in 1 year time for 16,28 today (i.e. 18=X) • invest the diff=Y=20-3-16,28=0.72 in a ZCB maturing in 1 year AT MATURITY • if S<18, you have 18+0.79 (so better off than keeping the stock • if S>18, you get: • S-X from the call (=S-18) • X from the ZCB (=18) • 0.79 from the residual • sum up the terms and you will be happier than having kept the stock

7. Puts: An Arbitrage Opportunity? • Suppose that p = 1 S0 = 37 T = 0.5 r =5% X = 40 D = 0 • Is there an arbitrage opportunity?

8. Lower Bound for European Put Prices; No Dividends p + S0Xe -rT pXe -rT - S0 consider 2 portfolios:a. buy a put and a stock b. buy the ZCB worth X at maturity

9. Put-Call Parity; No Dividends • Consider the following 2 portfolios: • Portfolio A: European call on a stock + PV of the strike price in cash • Portfolio B: European put on the stock + the stock • Both are worth MAX(ST, X ) at the maturity of the options • They must therefore be worth the same today • This means that c + Xe -rT = p + S0

10. Arbitrage Opportunities • Suppose that c = 3 S0= 31 T = 0.25 r = 10% X =30 D= 0 • What are the arbitrage possibilities when p = 2.25 ? p = 1 ?

11. arbitrage • c + Xe -rT = p + S0 3+30*0.97531=p+31 3+29.2593=p+31 p=1.2593 • if p=1, then • TODAY: • sell the call at 3 and buy the put at 1. You have 2 euro remaining. • buy the stock at 31, by borrowing 30.23 and using 0.77 from the options proceeds • invest 1.23 (out of the original 2) in a ZCB • in 3 months time: • if S=33, for example, put=0, call=3. So you are at -3. • sell the stock at 33 and repay your debt. So you are at +2 • so you are at -1, but in reality you still have 1.23*exp(0.25*10%)

12. The Impact of Dividends on Lower Bounds to Option Prices

13. Variable S0 X + + T  r D Effect of Variables on Option Pricing c p C P – – + + – – + + + + + + + + – – + + – – + +

14. American vs European Options An American option is worth at least as much as the corresponding European option CcPp

15. The Black-ScholesModel

16. The Stock Price Assumption • Consider a stock whose price is S • In a short period of time of length Dt the change in then stock price is assumed to be normal with mean mSdt and standard deviation • m is expected return and s is volatility

17. The Lognormal Distribution

18. The Concepts Underlying Black-Scholes • The option price & the stock price depend on the same underlying source of uncertainty • We can form a portfolio consisting of the stock and the option which eliminates this source of uncertainty • The portfolio is instantaneously riskless and must instantaneously earn the risk-free rate • This leads to the Black-Scholes differential equation

19. 1 of 3:The Derivation of theBlack-Scholes Differential Equation

20. 2 of 3:The Derivation of theBlack-Scholes Differential Equation • there are no stochastic terms inside

21. 3 of 3:The Derivation of theBlack-Scholes Differential Equation

22. Risk-Neutral Valuation • The variable m does not appear in the Black-Scholes equation • The equation is independent of all variables affected by risk preference • The solution to the differential equation is therefore the same in a risk-free world as it is in the real world • This leads to the principle of risk-neutral valuation

23. Applying Risk-Neutral Valuation 1. Assume that the expected return from the stock price is the risk-free rate 2. Calculate the expected payoff from the option 3. Discount at the risk-free rate

24. The Black-Scholes Formulas

25. Implied Volatility • The implied volatility of an option is the volatility for which the Black-Scholes price equals the market price • The is a one-to-one correspondence between prices and implied volatilities • Traders and brokers often quote implied volatilities rather than dollar prices

26. Dividends • European options on dividend-paying stocks are valued by substituting the stock price less the present value of dividends into Black-Scholes • Only dividends with ex-dividend dates during life of option should be included • The “dividend” should be the expected reduction in the stock price expected

27. European Options on StocksPaying Continuous Dividendscontinued We can value European options by reducing the stock price to S0e–qTand then behaving as though there is no dividend

28. Formulas for European Options

29. The Foreign Interest Rate • We denote the foreign interest rate by rf • When a European company buys one unit of the foreign currency it has an investment of S0euro • The return from investing at the foreign rate is rf S0euro • This shows that the foreign currency provides a “dividend yield” at rate rf

30. Valuing European Currency Options • A foreign currency is an asset that provides a continuous “dividend yield” equal to rf • We can use the formula for an option on a stock paying a continuous dividend yield : Set S0 = current exchange rate Set q = rƒ

31. Formulas for European Currency Options