Interest Rates & Present Value. 1. Introduction to Options. Outline

Transcription

1 1. Introduction to Options 1.2 stock option pricing preliminaries Math4143 W08, HM Zhu Outline Continuously compounded interest rate More terminologies on options Factors affecting option prices 2 Interest Rates & Present Value For most of our course, assume that the short-term interest rate is a known function of time, not necessarily constant. For valuing options, an important concept concerning interest rates is that of present value or discounting. Ask the question: How much would I pay now to receive a guaranteed amount E at the future time T? 3 1

2 Group Discussion Assume that the constant annual interest rate is 10% and you deposit $100 in the bank today. 1.Assume your savings is compounded annually, how much money would I have in the bank after one years? 2.Assume your savings is compounded semiannually, how much money would I have in the bank after one years? 3.Assume your savings is compounded every three months, how much money would I have in the bank after one years? 4 Discrete Compounding In general, suppose that an amount M 0 is invested for t years at an annual interest rate r. If the rate is compounded m times per year, the final value of the investment is M 0 1 mt r + m 5 Discrete Compounding Compounding Frequency Annually (m=1) Value of $100 at end of one year Semiannually (m=2) Quarterly (m=4) Weekly (m=52) Daily (m=365)

3 Continuous Compounding With continuous compounding, it can be shown that an amount M 0 invested for t years at an annual rate r grows to mt r rt lim M 0 1+ = M 0e m m 01. For example, 100e = $ For practical purpose, continuous compounding can be thought of as being equivalent to daily compounding. 7 Which offer to take? With continuous compounding, if someone were to make you the offer of a) $100 immediately ( time t=0), or b) $ 100e rt at time t, which one do you choose? We regard both offers a) and b) as being of equal value: a) M 0 immediately ( time t=0), or rt b) M t = M e at time t () 0 8 How much would I pay now to receive a guaranteed amount $100 at the future time t? Similarly, a deal that is guaranteed to produce exactly $100 at time t is worth exactly $ 100e rt at time zero. Transferring from $100 to $ 100e rt discounting for interest is called ( ) () = Or if M T = E, the value at time t of the certain payoff E is rt ( t) M t Ee 9 3

4 Present Value or Discounting Compounding a sum of money at a continuously compounded rate r for t years involves multiplying it rt by e Discounting it at a continuously compounded rate r for t years involves multiplying it by e rt 10 More Terminology Moneyness : In-the-money option would give its holder a positive cash flow if it were exercised immediately At-the-money option would give its holder a zero cash flow if it were exercised immediately Out-of-the-money option would give its holder a negative cash flow if it were exercised immediately 11 Option value Option value = intrinsic value + time value Intrinsic value of an option is defined as the maximum of zero and the value the option would have if it were exercised immediately. For a call option, it is max (S K, 0) For a put option, it is max (K S, 0) 12 4

5 Option value An option's time value captures the possibility, however remote, that the option may increase in value due to volatility in the underlying asset In-the-money American options >= its intrinsic value 13 Notations c : European call option price p : European put option price S 0 :Stock price today K : Strike price T : Life of option σ: Volatility of stock price C : American Call option price P : American Put option price S T : Stock price at time T D : Present value of dividends during option s life r : Risk-free rate for maturity T with continuous compounding Note: risk-free rate is the rate of the interest that can be earned without assuming any risks 14 Reading Financial Press: Prices of options on Intel, May 29, 2003 CALLS PUTS Option Strike Price ($) June July Oct. June July Oct. Intel (20.83) What can you observe from the prices of the options vs. strike price or expiry date? 15 5

6 The effect on the price of a stock option of increasing one variable while keeping all others fixed (Hull, page 168) Variable S 0 K T σ r D c p C P