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Option Pricing and Dynamic Modeling of Stock Prices

Option Pricing and Dynamic Modeling of Stock Prices. Investments 2003. 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.

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Option Pricing and Dynamic Modeling of Stock Prices

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  1. Option Pricing and Dynamic Modeling of Stock Prices Investments 2003

  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. 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. 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). Value(0)=? ? 0 T Værdi(T)=[ST-X]+

  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. Pricing by arbitrage - PCP Therefore: C = P + S0 – PV(X) ..otherwise there is arbitrage!

  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. 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 = $20 Stockprice = $18

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

  10. 22D – 1 18D Constructing a riskless portfolio • Consider the portfolio: long Dstocksshort 1 call option • The portfolio is riskless if 22D – 1 = 18D ie. when D = 0.25.

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

  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. S0u ƒu S0 ƒ S0d ƒd Generalization • A contingent claim expires at timeTand payoff depends on stock price

  14. Generalization • Consider portfolio which is longDstocks and short 1 claim • Portfolio is riskless whenS0uD – ƒu = S0dD – ƒdor • Note:  is the hedgeratio, i.e. the number of stocks needed to hedge the option. S0 uD – ƒu S0– f S0dD – ƒd

  15. Generalization • Portfolio value at timeTis S0uD – ƒu. Certain! • Present value must thus be(S0uD – ƒu )e–rT • but present value is also given as S0D – f • We therefore haveƒ = S0D – (S0uD – ƒu)e–rT

  16. Generalization • Plugging in the expression forDwe get ƒ = [ q ƒu + (1 – q )ƒd ]e–rT where

  17. S0u ƒu S0 ƒ S0d ƒd Risk-neutral pricing • ƒ = [ q ƒu + (1 – q )ƒd ]e-rT = e-rT EQ{fT} • The parametersq 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. q (1 – q )

  18. Back to the example S0u = 22 ƒu = 1 q • We can deriveqby pricing the stock: 20e0.12 ´0.25 = 22q + 18(1 – q ); q = 0.6523 • This result corresponds to the result from using the formula S0 ƒ S0d = 18 ƒd = 0 (1– q )

  19. S0u = 22 ƒu = 1 0.6523 S0 ƒ S0d = 18 ƒd = 0 0.3477 Pricing the option Value of option is e–0.12´0.25 [0.6523´1 + 0.3477´0] = 0.633.

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

  21. Pricing a call option, X=21 24.2 3.2 22 • 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 B 19.8 0.0 20 1.2823 2.0257 A 18 C 0.0 16.2 0.0 f = e-2rt[q2fuu + 2q(1-q)fud + (1-q)2fdd] = e-2rt EQ{fT}

  22. General formula

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

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

  25. Delta • Delta (D) is the hedge ratio,- the change in the option value relative to the change in the underlying asset/stock price • Dchanges 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. How are u and d chosen? There are different ways. The following is the most common and the most simple wheresis p.a. volatility and dtis length of time steps measured in years. Note u=1/d. This is Cox, Ross, and Rubinstein’s approach.

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

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

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

  30. Exercise

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

  33. Properties of the Wiener process • Mean of [z (T ) – z (0)] is 0. • Variance of [z (T ) – z (0)] isT. • 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. 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. 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 letdtgo to zero. • We will see processes of this type many times! (Stock prices, interest rates, temperatures etc.)

  36. A good model for stock prices wheremis the expected return andsis the volatility. This is the Geometric Brownian Motion (GBM). The discrete time parallel:

  37. The Lognormal distribution • A consequence of the GBM specification is • The Log ofSTis normal distributed, ie. STfollows a log-normal distribution.

  38. Lognormal-density

  39. Monte Carlo Simulation • The model is best illustrated by sampling a series of values of eand plugging in…… • Suppose e.g., that m= 0.14, s= 0.20, anddt = 0.01, so that we have

  40. Monte Carlo Simulation – One path

  41. A sample path:

  42. 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 forG(t, x).

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

  44. 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!

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

  46. Examples Integrate!

  47. 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.

  48. 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! ......

  49. Derivation of the Black-Scholes equation

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