Fins5535 Lecture 4 -sc.pdf

■

Binomial Trees Options Pricing Model & Stochastic Processes （Part 1) Later, we will go over quadratic weeds

Lecture 4: 1

You are expected to know after this lecture ■ ■ ■ ■ ■ ■ ■ ■ ■

Single step binomial trees options pricing model Risk natural valuation Multi steps binomial trees options pricing model (call/put) Binomial trees pricing model for American Options Stochastic Process: Markov process Wiener process Generalized Wiener process Ito process

Lecture 4: 2

Binomial trees ■ A useful and very popular and simple technique for pricing an option involves constructing a binomial tree, i.e., a diagram representing different possible paths for the stock price over the life of an option. ■ The underlying assumption is that the stock price follows a random walk. This implies that in each time interval, the stock price will move up or down by a certain amount with a certain probability. ■ Given the stock price movement in each time interval, the value of option written on the stock can be computed at different point in time and level of stock prices described in the tree. ■ In the limit, as the time interval becomes smaller, this model converges to the Black-Scholes-Merton model.

Lecture 4: 3

A Simple Binomial Model • A stock price is currently $30 • In three months it will be either $32 or $27

Stock Price = $32 Stock price = $30 Stock Price = $27

Lecture 4: 4

Pricing assumption

■ We try to form a risk-free portfolio using the stock and its derivative and then use the value of the portfolio to price the derivative => No Arbitrage Pricing

■ There is a risk neutral probability which can be used to price any derivative by taking the expected value of its cash flows discounted at the risk-free rate => Risk Neutral Pricing

■ Question: Risk-Neutral Pricing= Real World Pricing? Lecture 4: 5

A Call Option •

A 3-month call option on the stock has a strike price of 31. Stock Price = $32 Option Price = $1 Stock price = $30 Option Price=?

Stock Price = $27 Option Price = $0

Lecture 4: 6

Thinking Path ■ First, establish a portfolio of stock and option such

that there is no uncertainty about the value of the portfolio in the next period (i.e., at maturity in our example as it has only one period.) ■ Because the portfolio has no risk, it must earn the

risk free rate of return. ■ From this we deduce the value of the portfolio today,

and hence the value of the option today. Lecture 4: 7

Setting Up a Riskless Portfolio • Consider the Portfolio:

long Δ shares short 1 call option

32Δ – 1

27Δ • Portfolio is riskless when 32Δ – 1 = 27Δ or Δ = 0.20

Lecture 4: 8

Hedge ratio Δ ■ Hedge Ratio Δ of the stock options represents the ratio of the change in the price of the stock option to the change in the price of the stock. ■ It is the number of units of the stock we shall hold for each option on hand in order to create a riskless portfolio.

Lecture 4: 9

Valuing the Portfolio (Risk-Free Rate is 10%)

• The riskless portfolio is: ▪ long 0.20 shares ▪ short 1 call option

• The value of the portfolio in 3 months is 32 × 0.20 – 1 = 5.40 • The value of the portfolio today is 5.4e – 0.10×0.25 = 5.26 Lecture 4: 10

Deduce the value of the portfolio ■ Regardless of how the stock price moves, the portfolio is always worth 5.4 at maturity. This means that there is no uncertainty with the value of portfolio. ■ Since the portfolio is riskless, it must earn the riskless rate r; otherwise there would be an arbitrage opportunity (why?) ■ Assume that r = 10% (continuous compounding) ■ Hence, the present value of the portfolio must be 5.4e – 0.10×0.25 = 5.26 Lecture 4: 11

Valuing the Option • The portfolio that is ▪ long 0.20 shares ▪ short 1 option ▪ is worth 5.26

• The value of the shares is 6.000 = (0.20 × 30 ) • The value of the option is therefore 0.74 =(6.000 – 5.26 )

Lecture 4: 12

Arbitrage opportunities may occur ■ If the option value exceeded 0.74, the portfolio would cost less than 5.26 to construct but still return 5.4 at maturity. Hence, the portfolio would earn more than the risk-free rate. An arbitrager would borrow at riskfree rate and buy the portfolio. ■ Similarly, if the option value were less than 0.74, shorting the portfolio would provide a way of borrowing at less than the risk-free rate. An arbitrager would simply short the portfolio and invest the proceeds at the risk-free rate. Lecture 4: 13

Generalization

■ Consider a non-dividend-paying stock with current price S0 and an option on this stock with the current price f

S0 ƒ

S 0u ƒu S 0d ƒd Lecture 4: 14

Generalization (continued)

• Consider the portfolio that is long Δ shares and short 1 derivative

S0uΔ – ƒu S0dΔ – ƒd • The portfolio is riskless when S0uΔ – ƒu = S0dΔ – ƒd or

• Where have we seen this before?

Lecture 4: 15

Generalization

Lecture 4: 16

Generalization ■ Value of the portfolio at time T is ■ S0uΔ – ƒu ■ Value of the portfolio today is ■ (S0uΔ – ƒu)e–rT ■ Another expression for the portfolio value today is ■ S0Δ – f ■ Hence ƒ = S0Δ – (S0uΔ – ƒu )e–rT

Lecture 4: 17

No arbitrage option price ■ Let’s recall

and

ƒ = S0Δ – (S0uΔ – ƒu )e–rT

■ Substituting for Δ we obtain

ƒ = [ pƒu + (1 – p)ƒd ]e–rT ■ Where

Lecture 4: 18

p represents the probability

Lecture 4: 19

Risk-neutral pricing ■ {p, 1-p} corresponds to a risk neutral probability measure. This probability has nothing to do with the “true” probability which we may or may not know ■ Then, represents the expected value of the derivative’s payoffs discounted at the risk-free rate under this “risk neutral” probability {p, 1-p}. ■ This is called the risk neutral valuation. ■ In this world, what is the expected return on a stock? Lecture 4: 20

Risk neutral valuation ■ Assumption of risk-neutral means: investors do not increase the expected return they require from an investment to compensate for increased risk. ■ Risk Neutral has two features: ➢ The expected return on a stock (or other investment asset） is the risk-free rate ➢ The discount rate used for the expected payoff on an option (or other derivatives) is the risk-free rate

■

■

When we are valuing an option in terms of the price of the underlying asset, the probability of up and down movements in the real world are irrelevant This is an example of a more general result stating that the expected return on the underlying asset in the real world is irrelevant Lecture 4: 21

Call option revisit

Lecture 4: 22

Call option revisit

Lecture 4: 23

Multi-step binomial trees

Lecture 4: 24

A Two-Step Example

42.75

52.21 35

35 28.65

23.46

• Each time step is 3 months • K=35, r=5% Lecture 4: 25

American options and early exercise ■ Let’s consider a one-step tree for an American put option. As usual, let p be the risk-neutral probability of an “up” state, and 1 – p be the risk-neutral probability of a “down” state. ■ Let fE represent the price of a European put, and fA be the price of an American put. Here’s the tree:

Lecture 4: 26

American put option ■ For a European option, we would simply use risk-neutral pricing:

■ We can think of this price as the value of the option if we continue to hold it. ■ For an American option, however, we have the possibility of early exercise. If we exercise at time t=0, our payoff for the put option is K–S.

Lecture 4: 27

American put option ■ If we exercise at date 0, our payoff for the put option is K–S. So, the price of the American option is the maximum of these two values:

■ For a multi-period binomial tree, we make this calculation at each node of the tree, and the price of an American option will be at least as much as its European equivalent. Lecture 4: 28

Binomial trees’ parameters ■ Binomial trees are frequently used to approximate the movements in the price of a stock or other asset ■ RECALL: In each small interval of time the stock price is assumed to move up by a proportional amount u or to move down by a proportional amount d. ■ How are we going to choose the parameters u and d? ■ We choose the tree parameters u and d so that the tree gives correct values for the mean & standard deviation of the stock price changes in a risk-neutral world erΔt = pu + (1– p)d σ2Δt = pu2 + (1– p)d 2 – [pu + (1– p)d ]2 ■ A further condition often imposed is u=1/d

Lecture 4: 29

Binomial trees’ parameters ■ For a small Δt, a solution to the equations is:

Lecture 4: 30

Backwards induction & option on other assets ■ We know the value of the option at the final nodes ■ We work back through the tree using risk-neutral valuation to calculate the value of the option at each node, testing for early exercise when appropriate. ■ One-Step Binomial Model

where

Lecture 4: 31

Wiener Processes and Itô’s Lemma Is this a judgemental hot dog Lecture 4: 32

Wiener processes and Itô’s Lemma (Ch. 13) ■ Stochastic Process: ➢Discrete vs continuous time ➢Discrete vs continuous variable ■ Markov process ➢Properties ➢Subsets of Markov processes ● Wiener process ● Generalized Wiener process ● Ito process ■ Calculus review and Taylor series expansion ■ Ito’s Lemma (important theorem) ■ Applications of Ito’s Lemma Lecture 4: 33

Stochastic processes ■ Any variable whose value changes over time in an uncertain way is said to follow a stochastic process. ■ A stochastic process can be classified as discrete time or continuous time: ➢Discrete time – the value of the variable can change only at certain fixed points in time ➢Continuous time – the value of the variable can change at any time ■ Alternatively a stochastic process can be classified as discrete variable or continuous variable

Lecture 4: 34

Discrete-time stochastic processes ■ Discrete time with discrete variable ➢Example: the multi-period binomial model ■ Discrete time with continuous variable ➢Example: A stock price is currently at $40. At the end of 1 year, the stock price is a random value that follows a probability distribution of N(40,10), where N(µ,σ) is the normal distribution with mean µ and standard deviation σ. Note that in the textbook, the normal distribution is described by N(µ,v), where v is variance rather than standard deviation. Lecture 4: 35

Continuous-time stochastic processes ■ Continuous time with discrete variable Example: the variable can take five possible values {1, 2, 3, 4, 5} but switching from one value to the other can occur at any time (Regime switching models) ■ Continuous time with continuous variable. The variable follows a normal distribution and the value of the variable can change at any time, e.g., Markov process ■ Stock price is assumed to follow a continuous-variable and continuous-time stochastic process. Is this true? Lecture 4: 36

Examples Discrete time discrete variable

Continuous time, Continuous variable

Continuous time, discrete variable

Lecture 4: 37

Continuous-time and continuous Variable stochastic process We’ll review the following stochastic process. ■ Markov process, only the present value is relevant for predicting the future. ■ Wiener process, Markov with mean change of 0, and variance equals to T. ■ Generalized Wiener process, with mean change and variance change proportional to time. ■ Ito process, like a generalized Wiener process except that the mean and variance changes are functions of the time and the variable being modelled. Lecture 4: 38

Relation of the processes

Markov processes

Generalized Wiener and Ito processes

Wiener processes (Brownian Motion)

Lecture 4: 39

Markov processes ■ In a Markov process, future movements in a variable depend only on where they are now, not on the history of how they got there. ■ We assume that stock prices follow a Markov Process. This is consistent with a weak form of market efficiency. ■ Mathematically, this means that for any “reasonable” function, f, the conditional expectation satisfies: E[f(St)|all past prices] = E[f(St)|today’s price only] ■ This implies that the changes in value at different time intervals are independent. Lecture 4: 40

Markov process – stock returns and variances ■ Therefore, a stock’s return on day t+1, Rt+1, is independent of the stock’s return on day t, Rt.

■ If they’re independent, then they are uncorrelated. So, variances of stock return over a period of time are additive:

Lecture 4: 41

Markov process – stock returns ■ If we use continuously compounded returns, then

■ So that the two-period return is

Lecture 4: 42

Markov process – variances ■ In general, the T-period return is ■ taking variances (and assuming independence)

■ if we assume that variances are equal from period to period

Lecture 4: 43

Markov process – variances ■ If we know the variance of daily returns and if we assume that there are 252 trading days in a year, then ■ Var(annual returns) = 252×Var(daily returns), ■ And standard deviations are (taking square roots): St.Dev(annual returns) = 252×St.Dev(daily returns) ■ This means that variances are additive, while standard deviations are not additive.

Lecture 4: 44

Weak-form market efficiency

■ This asserts that it is impossible to produce consistently superior returns with a trading rule based on the past history of stock prices. In other words technical analysis does not work. ■ Does technical analysis work? ■ A Markov process for stock prices is clearly consistent with weak-form market efficiency. ■ Why? Lecture 4: 45

Wiener process (brownian motion) ■ We consider a variable Z whose value changes continuously over time. For a small change in time, ∆t, we’ll call the change as the random (stochastic) process, ∆Z. ■ The process {Z} follows a Wiener process if 1. , where ε follows a standard normal distribution, and εt and εt + 1 are independent. 2. The values of ∆Z for any two non-overlapping periods of time are independent. 3. Z has continuous paths. Lecture 4: 46

Plot of wiener process

0

t Lecture 4: 47

Properties of Wiener process ■ ■ ■ ■

For each t, Z is normally distributed. The mean of [ZT − Z0] is E[ZT − Z0] = 0. The variance of [ZT − Z0] is Var[ZT − Z0] = T The standard deviation of [ZT − Z0] is

■ NOTE: it is usually assumed that Z0 = 0. ■ Taking the limit as ∆t approaches zero, we replace

Lecture 4: 48

Generalized wiener processes ■ A Wiener process has a drift rate of 0 and a variance rate of 1. ■ Note that the mean change per unit time for a stochastic process is known as the drift rate and the variance per unit time is known as the variance rate. ■ In a generalized Wiener process the drift rate and the variance rate can be set equal to any chosen constants. Lecture 4: 49

More specifically: ■ A variable X follows a generalized Wiener process with drift rate a and variance rate b2 (with a and b constant) if ■ or

➢Mean change in X over is ➢Variance of change in X over is ➢Standard deviation of change in X over

is

Lecture 4: 50

Drift and Variance rates ■ X is just a simple function of the Brownian motion, Z. ■ How do we interpret the drift rate, a, and the variance rate, b2? ■ We can see that the mean change in X in the time interval T is

■ The variance of change in X in the time interval T is

■ The standard deviation of change in X in time interval T is

■ Note that initial time t is assumed to be zero. Lecture 4: 51

Distribution of XT ■ It follows that XT is normally distributed with mean X0 + aT, and standard deviation

■ We sometimes write that XT has the probability distribution N(X0 + aT, ) which is written as XT ~ N(X0 + aT,

)

Lecture 4: 52

The Example Revisited

• A stock price starts at 40 and has a probability distribution of φ(40,10) at the end of the year • If we assume the stochastic process is Markov with no drift then the process is dS = 10dz • If the stock price were expected to grow by $8 on average during the year, so that the year-end distribution is φ(48,10), the process would be dS = 8dt + 10dz Lecture 4: 53

Questions • What is the probability distribution of the stock price at the end of 2 years? • ½ years? • ¼ years? • Δt years? • Excel

Lecture 4: 54

Generalized wiener process: is this a reasonable process to describe dynamics of stock prices? ■ There are two principal reasons of why a generalized Wiener process is NOT appropriate. ■ A generalized Wiener process is normally distributed, => it can become negative. But stock prices can’t be negative. A more reasonable model would assume that continuously compounded returns are normally distributed ■ If stock prices follow a generalized Wiener process, then the variance of stock’s prices is constant, regardless of price level. It’s more reasonable to assume that returns have a constant variance, regardless of price level. Lecture 4: 55

Itô process ■ In an Itô process the drift rate and the variance rate are functions of time and underlying variable:

dx=a(x,t) dt+b(x,t) dz ■ The discrete time equivalent

is only true in the limit as Δt tends to zero

Lecture 4: 56

An Itô process for stock prices

■ where μ is the expected return and σ is the volatility. ■ The discrete time equivalent is

Lecture 4: 57

Monte-carlo simulation ■ We can sample random paths for the stock price by sampling values for ε ■ Suppose μ= 0.14, σ= 0.20, and Δt = 0.01, then

Lecture 4: 58

Monte-Carlo simulation of stock prices

Lecture 4: 59

Sampling from a Normal Distribution • One simple way to obtain a sample from φ(0,1) is to generate 12 random numbers between 0.0 & 1.0, take the sum, and subtract 6.0 • In Excel =NORMSINV(RAND()) gives a random sample from φ(0,1) • Excel Stochastic process examples Lecture 4: 60