1. Option Pricing and Dynamic Modeling of Stock Prices Investments 2004

2. 2 Motivation We must learn some basic skills and set up a general framework which can be used for option pricing. The ideas will be used for the remainder of the course. Important not to be lost in the beginning. Option models can be very mathematical. We/I shall try to also concentrate on intuition.

3. 3 Overview/agenda Intuition behind Pricing by arbitrage Models of uncertainty The binomial-model. Examples and general results The transition from discrete to continuous time Pricing by arbitrage in continuous time The Black-Scholes model General principles Monte Carlo simulation, vol. estimation. Exercises along the way

4. 4 Here is what it is all about! Options are contingent claims with future payments that depend on the development in key variables (contrary to e.g. fixed income securities). 0T ? Value(0)=? Value(T)=[S T -X] +

5. 5 The need for model-building The Payoff at the maturity date is a well- specified function of the underlying variables. The challenge is to transform the future value(s) to a present value. This is straightforward for fixed income, but more demanding for derivatives. We need to specifiy a model for the uncertainty. Then pricing by arbitrage all the way home!

6. 6 Pricing by arbitrage - PCP Transaction Time 0 priceTime T flow S T >X Time T flow S T <X Long stock-S 0 STST STST Long put(X)-P0X-S T Loan(X)PV(X)-X SumPV(X)-P-S 0 S T -X0 Long call(X)-C-CS T -X0 Therefore: C = P + S 0 – PV(X)..otherwise there is arbitrage!

7. 7 Pricing by arbitrage So if we know price of underlying asset riskless borrowing/lending (the rate of interest) put option then we can uniquely determine price of otherwise identical call If we do not know the put price, then we need a little more structure......

8. 8 The World’s simplest model of undertainty – the binomial model Example: Stockprice today is $20 In three months it will be either $22 or $18 (+-10%) Stockprice = $22 Stockprice = $18 Stockprice = $20

9. 9 Stockprice = $22 Option payoff = $1 Stockprice = $18 Option payoff = $0 Stockprice = $20 Option price=? A call option Consider 3-month call option on the stock and with an exercise price of 21.

10. 10 Consider the portfolio:long  stocks short 1 call option The portfolio is riskless if 22  – 1 = 18  ie. when  = 0.25. 22  – 1 18  Constructing a riskless portfolio

11. 11 Valuing the portfolio Suppose the rate of interest is 12% p.a. (continuously comp.) The riskless portfolio was: long 0.25 stocks short 1 call option Portfolio value in 3 months is 22  0.25 – 1 = 4.50. So present value must be 4.5 e – 0.12  0.25 = 4.3670.

12. 12 Valuing the option The portfolio which was long 0.25 stocks short 1 option was worth 4.367. Value of stocks 5.000 (= 0.25  20 ). Therefore option value must be 0.633 (= 5.000 – 4.367 ),...otherwise there are arbitrage opportunities.

13. 13 Generalization A contingent claim expires at time T and payoff depends on stock price S0uƒuS0uƒu S 0 d ƒ d S0ƒS0ƒ

14. 14 Generalization Consider portfolio which is long  stocks and short 1 claim Portfolio is riskless when S 0 u  – ƒ u = S 0 d  – ƒ d or Note:  is the hedgeratio, i.e. the number of stocks needed to hedge the option. S 0 u  – ƒ u S 0 d  – ƒ d  S 0 – f

15. 15 Generalization Portfolio value at time T is S 0 u  – ƒ u. Certain! Present value must thus be (S 0 u  – ƒ u )e –rT but present value is also given as S 0  – f We therefore have ƒ = S 0  – ( S 0 u  – ƒ u )e –rT

16. 16 Generalization Plugging in the expression for  we get ƒ = [ q ƒ u + (1 – q )ƒ d ]e –rT where

17. 17 Risk-neutral pricing ƒ = [ q ƒ u + (1 – q )ƒ d ]e -rT = e -rT E Q {f T } The parameters q and (1  – q ) can be interpreted as risk-neutral probabilities for up- and down- movements. Value of contingent claim is expected payoff wrt. q- probabilities (Q-measure) discounted with riskless rate of interest. S0u ƒuS0u ƒu S0d ƒdS0d ƒd S0ƒS0ƒ q (1 – q )

18. 18 Back to the example We can derive q by pricing the stock: 20 e 0.12  0.25 = 22q + 18(1 – q ); q = 0.6523 This result corresponds to the result from using the formula S 0 u = 22 ƒ u = 1 S 0 d = 18 ƒ d = 0 S0 ƒS0 ƒ q (1  – q )

19. 19 Pricing the option Value of option is e –0.12  0.25 [0.6523  1 + 0.3477  0] = 0.633. S 0 u = 22 ƒ u = 1 S 0 d = 18 ƒ d = 0 S0ƒS0ƒ 0.6523 0.3477

20. 20 Two-period example Each step represents 3 months,  t=0.25 20 22 18 24.2 19.8 16.2

21. 21 Pricing a call option, X=21 Value in node B = e –0.12  0.25 (0.6523  3.2 + 0.3477  0) = 2.0257 Value in node A = e –0.12  0.25 (0.6523  2.0257 + 0.3477  0) = 1.2823 20 1.2823 22 18 24.2 3.2 19.8 0.0 16.2 0.0 2.0257 0.0 A B C f = e -2r  t [q 2 f uu + 2q(1-q)f ud + (1-q) 2 f dd ] = e -2r  t E Q {f T }

22. 22 General formula

23. 23 Put option; X=52 50 4.1923 60 40 72 0 48 4 32 20 1.4147 9.4636 u=1.2, d=0.8, r=0.05,  t=1, q=0.6282

24. 24 American put option – early exercise 50 5.0894 60 40 72 0 48 4 32 20 1.4147 12.0 A B C Node C: max(52-40, exp(-0.05)*(q*4+(1-q)*20)) 9.4636

25. 25 Delta Delta (  ) is the hedge ratio,- the change in the option value relative to the change in the underlying asset/stock price  changes when moving around in the binomial lattice It is an instructive exercise to determine the self-financing hedge portfolio everywhere in the lattice for a given problem.

26. 26 How are u and d chosen? There are different ways. The following is the most common and the most simple where  is p.a. volatility and  t is length of time steps measured in years. Note u=1/d. This is Cox, Ross, and Rubinstein’s approach.

27. 27 Few steps => few states. A coarse model

28. 28 Many steps => many states. A ”fine” model

29. 29 Call, S=100,  =0.15, r=0.05, T=0.5, X=105

30. 30

31. 31 Alternative intertemporal models of uncertainty Discrete time; discrete variable (binomial) Discrete time; continuous variable Continuous time; discrete variable Continuous time; continuous variable All can be used, but we will work towards the last type which often possess the nicest analytical properties

32. 32 The Wiener Process – the key element/the basic building block Consider a variable z, which takes on continuous values. The change in z is  z over time interval of length  t. z is a Wiener proces, if 1. 2. Realization/value of  z for two non- overlapping periods are independent.

33. 33 Properties of the Wiener process Mean of [ z ( T ) – z (0)] is 0. Variance of [ z ( T ) – z (0)] is T. Standarddeviation of [ z (T ) – z (0)] is A continuous time model is obtained by letting  t approach zero. When we write dz and dt it is to be understood as the limits of the corresponding expressions with  t and  z, when  t goes to zero.

34. 34 The generalized Wiener-process The drift of the standard Wiener-process (the expected change per unit of time) is zero, and the variance rate is 1. The generalized Wienerprocess has arbitrary constant drift and diffusion coefficients, i.e. dx=adt+bdz. This model is of course more general but it is still not a good model for the dynamics of stock prices.

35. 35

36. 36 Ito Processes The drift and volatility of Ito processes are general functions dx=a(x,t)dt+b(x,t)dz. Note: What we really mean is where we let  t go to zero. We will see processes of this type many times! (Stock prices, interest rates, temperatures etc.)

37. 37 A good model for stock prices where  is the expected return and  is the volatility. This is the Geometric Brownian Motion (GBM). The discrete time parallel:

38. 38 The Lognormal distribution A consequence of the GBM specification is The Log of S T is normal distributed, ie. S T follows a log-normal distribution. We will show this shortly!!

39. 39 Lognormal-density

40. 40 Monte Carlo Simulation The model is best illustrated by sampling a series of values of  and plugging in…… Suppose e.g., that  = 0.14,  = 0.20, and  t = 0.01, so that we have Methods for sampling  ’s…

41. 41 Monte Carlo Simulation – One path You MUST go home and try this…

42. 42 A sample path:

43. 43 Moving further: Ito’s Lemma We need to be able to analyze functions of S since derivates are functions of eg. a stock price. The tool for this is Ito’s lemma. More generally: If we know the stochastic process for x, then Ito’s lemma provides the stochastic process for G ( t, x ).

44. 44 Ito’s lemma in brief Let G(t,x) and dx=a(x,t)dt + b(x,t)dz

45. 45 Why the extra term? Because so But has expected value of 1 and variance of term is of order (  t) 2 So it is deterministic in the limit…..

46. 46 Ito’s lemma Substituting the expression for dx we get: THIS IS ITO’S LEMMA! The option price/the price of the contingent claim is also a diffusion process!

47. 47 Application of Ito’s lemma to functions of GBM

48. 48 Examples Integrate!

49. 49 The Black-Scholes model We consider a stock price which evolves as a GBM, ie. dS =  Sdt +  Sdz. For the sake of simplicity there are no dividends. The goal is to determine option prices in this setup.

50. 50 Pre-Nobel prize methodology Calculate expected payoff. See note… Discount using r… or  …. or something else…??

51. 51 The idea behind the Black- Scholes derivation The option and the stock is affected by the same uncertainty generating factor. By constructing a clever portfolio we can get rid of this uncertainty. When the portfolio is riskless the return must equal the riskless rate of interest. This leads to the Black-Scholes differential equation which we will then find a solution to. Let’s do it!......

52. 52 Derivation of the Black-Scholes equation

53. 53 Derivation of the Black-Scholes differential equation The uncertainty/risk of these terms cancel, cf. previous slide.

54. 54 Derivation of the Black-Scholes differential equation

55. 55 The differential equation Any asset the value of which depends on the stock price must satisfy the BS-differential equation. There are therefore many solutions. To determine the pricing functional of a particular derivative we must impose specific conditions. Boundary/terminal conditions. Eg: For a forward contrakt the boundary condition is ƒ = S – K when t =T The solution to the pde is thus ƒ = S – K e –r (T – t ) Check the pde!

56. 56 Risk-neutral pricing The parameter  does not appear in the BS- differential equation! The equation contains no parameters with relation to the investors’ preferences for risk. The solution to the equation is therefore the same in ”the real World” as in a World where all investors are risk-neutral. This observation leads to the concept of risk- neutral pricing!

57. 57 Risk-neutral pricing in practice 1. Assume the expected stock return is equal to the riskless rate of interest, ie. use  =r in the GBM. 2. Calculate the expected risk-neutral payoff for the option. 3. Perform discounting with riskless rate of interest, i.e.

58. 58 Black-Scholes formulas

59. 59 The Monte Carlo idea General pricing relation: For example: These expressions are the basis of Monte Carlo simulation. The expectation is approximated by:

60. 60 The market price of risk The fundamental pde. holds for all derivatives written on a GBM-stock. If the underlying is not traded (eg. a ”rate of interest”, a temperature, a snow depth, a Richter- number etc.) we can derive a similar pde, but there will be a term for the market price of risk of this factor. For example we can use Ito’s lemma to show that derivatives will follow where

61. 61 The market price of risk can not be determined from arbitrage arguments alone. It must be estimated using market data. When simulating the risk neutralized underlying variable the drift must be adjusted with a term which includes the market price of risk. The market price of risk

62. 62 Example of a non-priced underlying variable

63. 63 Historical volatility 1. Observe S 0, S 1,..., S n with interval length  years. 2. Calculate continuous returns in every interval: 3. Estimate standard deviation, s, of the u i ´s. 4. The historical annual volatility:

64. 64 Implied volatility The implied volatility is the volatility which – when plugged into the BS-formula – creates correspondence between model- and market price of the option. The BS-formula is inverted. This is done numerically. In the market volatility is often quoted in stead of price.

65. 65 Exercises/homework! Simulate a GBM and show the result graphically using a spread sheet. Compare the Black-Scholes price with the price of options found using the binomial approximation. How big must N be in order to obtain a ”good result”? Try to estimate the volatility using a series of stock prices which you have simulated (so that you know the true volatility). Try to determine some implied volatilities by inverting the BS formula. Try to determine a call price using Monte Carlo simulation and compare your result with the exact price obtained from the BS formula.