FE520 – 0225 – Introduction

Introduction

This course presents an introduction to stochastic models in financial decision making. Wefirst introduce discrete time models: Martingales, risk-neutral probability measure andchange-of-measure in discrete time. Then, we discuss Brownian motion, Ito’s formula andBlack-Scholes-Merton equation. We present various applications of risk-neutral pricingequation.

Reference Materials

References: Students are strongly encouraged to attend classes and take notes. Lecture notesare mostly based on Shreve Volume I and Volume II.

1. Shreve, S. E. (2004). Stochastic Calculus for Finance I. Springer.

2. Shreve, S. E. (2004). Stochastic Calculus for Finance II. Springer.

3. Baxter, M. and Rennie, A. (1996), Financial Calculus: an introduction to derivativepricing, Cambridge University Press.

Lecture Content

FE 520 Lecture Notes Part 1

The Binomial Asset Pricing Model

Feynmans Explanation

I'd be happy to explain the Binomial Asset Pricing Model using the Feynman Technique!

1. Choose a concept: The Binomial Asset Pricing Model

2. Pretend to teach the concept to a student:

Okay, imagine that you're an investor who wants to price a stock. You want to know what the stock is worth so that you can decide whether to buy it or not. The Binomial Asset Pricing Model is one way to calculate the value of a stock, assuming that the stock's price can either go up or down over a period of time.

To use the model, we start with the current stock price and the probability that the stock's price will either go up or down in the future. We then calculate the expected value of the stock price at the end of the period by multiplying the possible future stock prices by their respective probabilities and adding the products together. Finally, we discount this expected value back to the present using a risk-free interest rate.

This process of calculating the expected future value of an asset based on different possible future outcomes is known as a "binomial tree." Each node on the tree represents a possible future price for the stock, and the probabilities associated with each branch of the tree represent the likelihood of each outcome.

3. Identify gaps in your knowledge:

To explain the Binomial Asset Pricing Model more fully, I would need to clarify a few things. For example, I would need to explain how we determine the probabilities of the stock price going up or down, how we select the time period for the model, and how we choose the risk-free interest rate to discount the expected future value. I would also need to provide some examples of how to apply the model in practice.

4. Review and simplify:

In summary, the Binomial Asset Pricing Model is a way to calculate the value of a stock by estimating the possible future stock prices and their probabilities over a given time period, and then discounting the expected future value back to the present using a risk-free interest rate. It uses a binomial tree to represent the different possible outcomes and their probabilities. While the model can be complex in practice, the basic idea is relatively simple and provides a useful tool for investors to make informed decisions about buying and selling stocks.

More detailed explanation of the Binomial Asset Pricing Model

1. Choose a concept: The Binomial Asset Pricing Model

2. Pretend to teach the concept to a student:

Let's start with a basic example. Suppose you want to price a stock that currently trades at $S_0$$S_0$, and you want to calculate its price at the end of a period of time T. Let's assume that the stock can either go up by a factor of u (with probability p) or go down by a factor of d (with probability 1-p) over this period. We can represent the possible future stock prices using a binomial tree with two possible branches, as shown below:

4
1
                S0
2
               /  \
3
              /    \
4
            Su      Sd

where Su = uS0 and Sd = dS0.

To calculate the expected future price of the stock at the end of the period, we use the following formula:

This formula represents the weighted average of the possible future prices, where the weight is given by the probability of each outcome.

To calculate the present value of the expected future stock price, we discount it back to the present using a risk-free interest rate r. The present value of the stock is given by:

3. Identify gaps in your knowledge:

To fully understand the Binomial Asset Pricing Model, we need to discuss how to determine the values of u, d, and p, as well as how to choose the risk-free interest rate and the time period for the model. We also need to provide some examples of how to apply the model in practice.

4. Review and simplify:

In summary, the Binomial Asset Pricing Model is a way to calculate the value of a stock by estimating the possible future stock prices and their probabilities over a given time period, and then discounting the expected future value back to the present using a risk-free interest rate. It uses a binomial tree to represent the different possible outcomes and their probabilities, and can be represented using the following formulas:

The expected future stock price at the end of the period:

$S_T=p\cdot S_u+(1-p)\cdot S_d$$S_T = p \cdot S_u + (1-p) \cdot S_d$

The present value of the stock:

$PV=\frac{1}{(1+r{)}^T}\cdot E(S_T)=\frac{1}{(1+r{)}^T}\cdot [p\cdot S_u+(1-p)\cdot S_d]$$PV = \frac{1}{(1+r)^T} \cdot E(S_T) = \frac{1}{(1+r)^T} \cdot [p \cdot S_u + (1-p) \cdot S_d]$

The calculation of u and d in the Cox-Ross-Rubinstein (CRR) model:

$u=e^{\sigma \sqrt{\mathrm{\Delta }t}}$$u = e^{\sigma \sqrt{\Delta t}}$

$d=\frac{1}{u}$$d = \frac{1}{u}$ or $d=e^{-\sigma \sqrt{\mathrm{\Delta }t}}$$d = e^{-\sigma \sqrt{\Delta t}}$

The calculation of the probability of the stock price going up (p) or going down (1-p):

$p=\frac{e^{r\mathrm{\Delta }t}-d}{u-d}$$p = \frac{e^{r\Delta t} - d}{u - d}$

where:

$S_T$$S_T$ is the expected future stock price at the end of the period,

$p$$p$ is the probability of the stock price going up,

$S_u$$S_u$ is the possible future stock price if the stock goes up by a factor of u,

$S_d$$S_d$ is the possible future stock price if the stock goes down by a factor of d,

$PV$$PV$ is the present value of the stock,

$r$$r$ is the risk-free interest rate,

$T$$T$ is the length of the time period,

$\sigma$$\sigma$ is the standard deviation of the stock returns, and

$\mathrm{\Delta }t$$\Delta t$ is the length of each time step in the binomial tree.

The binomial asset pricing model example

Suppose you own a stock that is currently worth $100. You want to calculate its expected value at the end of a 2-year period. The Binomial Asset Pricing Model can help you do this.

To start, we assume that the stock price can either go up by a factor of u or down by a factor of d each year. Let's say that u is 1.2 and d is 0.8.

We also assume that there is a 50% chance of the stock price going up each year, and a 50% chance of it going down. We can represent the possible future stock prices using a binomial tree:

11
1
                 100
2
                /   \
3
              /       \
4
           120         80
5
          /   \       /   \
6
        /       \   /       \
7
     144         96        64
8
    /   \       /   \     /   \
9
  /       \   /       \ /       \
10
172.8    115.2       76.8     51.2
11

Each node in the tree represents a possible future stock price at the end of the 2-year period, and each branch represents a possible movement in the price of the stock.

To calculate the expected future stock price at the end of the 2-year period, we take a weighted average of the possible future stock prices. We do this by multiplying each possible future stock price by its probability, and then adding up the products. The formula for the expected future stock price is:

where S_0 is the current stock price and p is the probability of the stock price going up each year. In this case, S_0 is 100, u is 1.2, d is 0.8, and p is 0.5. Therefore, the expected future stock price is:

To calculate the present value of the expected future stock price, we need to discount it back to the present using the risk-free interest rate. The formula for the present value of the expected future stock price is:

where r is the risk-free interest rate and T is the time period. Let's say that r is 0.05 and T is 2 years. Therefore, the present value of the expected future stock price is:

In summary, the Binomial Asset Pricing Model uses a binomial tree to represent the different possible outcomes and their probabilities, and can help you calculate the expected value of a stock at a future point in time. By discounting the expected future value back to the present using a risk-free interest rate, you can make informed decisions about buying and selling stocks.