OPTION PRICING: BASICS

Transcription

1 1 OPTION PRICING: BASICS

2 The ingredients that make an op?on 2 An op?on provides the holder with the right to buy or sell a specified quan?ty of an underlying asset at a fixed price (called a strike price or an exercise price) at or before the expira?on date of the op?on. There has to be a clearly defined underlying asset whose value changes over?me in unpredictable ways. The payoffs on this asset (real op?on) have to be con?ngent on an specified event occurring within a finite period. 2

3 A Call Op?on 3 A call op?on gives you the right to buy an underlying asset at a fixed price (called a strike or an exercise price). That right may extend over the life of the op?on (American op?on) or may apply only at the end of the period (European op?on). 3

4 Payoff Diagram on a Call 4 Net Payoff on Call Strike Price Price of underlying asset 4

5 A Put Op?on 5 A put op?on gives you the right to buy an underlying asset at a fixed price (called a strike or an exercise price). That right may extend over the life of the op?on (American op?on) or may apply only at the end of the period (European op?on). 5

6 Payoff Diagram on Put Op?on 6 Net Payoff On Put Strike Price Price of underlying asset 6

7 Determinants of op?on value 7 Variables Rela?ng to Underlying Asset Value of Underlying Asset; as this value increases, the right to buy at a fixed price (calls) will become more valuable and the right to sell at a fixed price (puts) will become less valuable. Variance in that value; as the variance increases, both calls and puts will become more valuable because all op?ons have limited downside and depend upon price vola?lity for upside. Expected dividends on the asset, which are likely to reduce the price apprecia?on component of the asset, reducing the value of calls and increasing the value of puts. Variables Rela?ng to Op?on Strike Price of Op?ons; the right to buy (sell) at a fixed price becomes more (less) valuable at a lower price. Life of the Op?on; both calls and puts benefit from a longer life. Level of Interest Rates; as rates increase, the right to buy (sell) at a fixed price in the future becomes more (less) valuable. 7

8 8 The essence of op?on pricing models: The Replica?ng poruolio & Arbitrage Replica?ng poruolio: Op?on pricing models are built on the presump?on that you can create a combina?on of the underlying assets and a risk free investment (lending or borrowing) that has exactly the same cash flows as the op?on being valued. For this to occur, The underlying asset is traded - this yield not only observable prices and vola?lity as inputs to op?on pricing models but allows for the possibility of crea?ng replica?ng poruolios An ac?ve marketplace exists for the op?on itself. You can borrow and lend money at the risk free rate. Arbitrage: If the replica?ng poruolio has the same cash flows as the op?on, they have to be valued (priced) the same. 8

9 Crea?ng a replica?ng poruolio 9 The objec?ve in crea?ng a replica?ng poruolio is to use a combina?on of riskfree borrowing/lending and the underlying asset to create the same cashflows as the op?on being valued. Call = Borrowing + Buying D of the Underlying Stock Put = Selling Short D on Underlying Asset + Lending The number of shares bought or sold is called the op?on delta. The principles of arbitrage then apply, and the value of the op?on has to be equal to the value of the replica?ng poruolio. 9

10 The Binomial Op?on Pricing Model 10 Stock Price Call Option Details K = $ 40 t = 2 r = 11% 100 D B = D B = 10 D = 1, B = Call = 1 * = D B = D B = 4.99 D = , B = Call = * = Call = Call = Call = D B = D B = 0 D = 0.4, B = 9.01 Call = 0.4 * =

11 The Limi?ng Distribu?ons. 11 As the?me interval is shortened, the limi?ng distribu?on, as t - > 0, can take one of two forms. If as t - > 0, price changes become smaller, the limi?ng distribu?on is the normal distribu?on and the price process is a con?nuous one. If as t- >0, price changes remain large, the limi?ng distribu?on is the poisson distribu?on, i.e., a distribu?on that allows for price jumps. The Black- Scholes model applies when the limi?ng distribu?on is the normal distribu?on, and explicitly assumes that the price process is con?nuous and that there are no jumps in asset prices. 11

12 Black and Scholes to the rescue 12 The version of the model presented by Black and Scholes was designed to value European op?ons, which were dividend- protected. The value of a call op?on in the Black- Scholes model can be wricen as a func?on of the following variables: S = Current value of the underlying asset K = Strike price of the op?on t = Life to expira?on of the op?on r = Riskless interest rate corresponding to the life of the op?on σ 2 = Variance in the ln(value) of the underlying asset 12

13 The Black Scholes Model 13 Value of call = S N (d1) - K e - rt N(d2) where d 1 =! ln S # " K$ + (r + σ 2 σ t 2 ) t d2 = d1 - σ t The replica?ng poruolio is embedded in the Black- Scholes model. To replicate this call, you would need to Buy N(d1) shares of stock; N(d1) is called the op?on delta Borrow K e - rt N(d2) 13

14 The Normal Distribu?on 14 N(d1) d1 d N(d) d N(d) d N(d)

15 Adjus?ng for Dividends 15 If the dividend yield (y = dividends/ Current value of the asset) of the underlying asset is expected to remain unchanged during the life of the op?on, the Black- Scholes model can be modified to take dividends into account. Call value = S e - yt N(d1) - K e - rt N(d2) where, d 1 =! ln S # " K$ d2 = d1 - σ t σ2 + (r - y + 2 ) t σ t The value of a put can also be derived from put- call parity (an arbitrage condi?on): Put value = K e - rt (1- N(d2)) - S e - yt (1- N(d1)) 15

16 Choice of Op?on Pricing Models 16 Some prac??oners who use op?on pricing models to value op?ons argue for the binomial model over the Black- Scholes and jus?fy this choice by no?ng that Early exercise is the rule rather than the excep?on with real op?ons Underlying asset values are generally discon?nous. In prac?ce, deriving the end nodes in a binomial tree is difficult to do. You can use the variance of an asset to create a synthe?c binomial tree but the value that you then get will be very similar to the Black Scholes model value. 16