Brownian Motion

Binomial Asset Pricing Model

We already have learned about one period binomial model. Now we will slowly move towards the binomial asset pricing model. The actual stock pricing much more complicated than the Binomial asset pricing model.

Then why are we studying this ?

In the Binomial asset pricing model the concept of arbitrage pricing and risk neutral probability is clearly visible. We are going to explain this terms a little bit later. Binomial asset pricing model can be viewed as simplified and computationally easier version of continuous time models. Most importantly, in binomial asset pricing model we can develop the theory of conditional expectations and martingales, which lies at the heart of continuous-time models.

Now, what is arbitrage pricing ?

In finance, the practice of taking advantage of difference in prices in two or more market is called an arbitrage. Let us see this using an example. Suppose an item is sold at two different prices in two different markets, $\$10$ in market A and $\$12$ in market B. Then if you buy one unit from market A and sell it to market B, you have a gain of $\$2$.

What is Risk neutral probability?

Risk-neutral probabilities are probabilities of possible future outcomes that have been adjusted for risk. To mathematically understand Risk neutral probability we need to understand change of measure, which we will see later.

Before we go bit technical, let us see some useful definitions and examples. First-

What is an Option?

In finance, the Option refers to an instrument that gives the buyer of the instrument the right, but not the obligation to buy or sell the underlying asset at a predefined time on a predefined date.

Options can be of two types- Call option and Put option.

Call Option: A call is an option contract giving the owner the right, but not the obligation, to buy a specified amount of an underlying security at a predefined price within a specified time.

Put Option: A Put option gives holder of the option the right, but not the obligation, to sell a specified amount of an underlying security at a specified price within a specified time frame.

Let us go through a simple example of European Option, which will help us understand how options work; what is the pay-off of the options etc.

Example: Suppose today an investor buys a call options on Tesla Inc. with strike price, $K= \$50$. The premium is- $\$5/\text{share}$, 100 shares for total of $\$500$. At expiration Tesla Inc. is trading at $\$75/\text{stock}$. In this case, the owner of the call option has the right to purchase the stock at $\$50/\text{stock}$, by exercising their option and making a profit of $\$(75-50)/\text{share} = \$25/\text{share}$. So in this case, total profit will be-

\[\begin{aligned} \text{Total Profit} &= \text{(Current price of the stock - Strike price - Premium Price) x No. of stocks} \\ &= \$ (75-50-5)*100 \\ &= \$ 2000 \end{aligned}\]

Now let us consider another scenario- where at the stock price of Tesla fell to $\$30/\text{stock}$ at the time of the expiration. Since the stock price is below the strike price(it’s a right no obligation, in this case- since he is going to lose money for the option- he will opt NOT to exercise the option) the option is not exercised, and expires worthless. In this case total amount of loss would be- the Premium paid to buy the option, which is $\$500$.

(For further knowledge about European option go to- https://www.investopedia.com/terms/e/europeanoption.asp)

Let us recap the terminologies again from one period binomial model and then generalize the idea.

Suppose $S_t$ is the stock price at time $t$. At time $t=0$, we start with stock price $S_0$, a positive value. At $t=1$, the price of one share of that stock would be one of the two positive values. How to decide? We toss a coin, not necessarily a fair one (i.e. $P[H]=p \neq \frac{1}{2}$). If it’s Head $S_1 = S_1(H)$, if it’s Tail $S_1 = S_1(T)$

\[\begin{equation} S_1 = \begin{cases} S_1(H) & \text{if $\omega_1$ = H} \\ S_1(T) & \text{if $\omega_1$ = T} \\ \end{cases} \end{equation}\]

Here, $S_1(H) = u.S_0$ and $S_1(T) = d.S_0$. $u$ is the upward factor and $d$ is the downward factor. Here we assume- $d = \frac{1}{u}$ . As we previously proved- $0 < d < 1+r <u$.

Here we will assume that- $S_1(T)<K< S_1(H)$.

Now the process goes like this- you toss a coin. If you get a tail, the option expires worthless, i.e. if you have tail you will not activate the call option, because if you activate the call option, you will be at loss. $[S_1(T)-K < 0]$

But, if you get a head, the option should be exercised and the profit will be- $S_1(H)-K>0$. So this profit can be represented as- $(S_1-K)^+$, where- $(x)^+ = \max(0,x)$.

Let us go through a simple example.

Example: Suppose we consider a one period binomial model. The amount we are investing in the stock at time t=0 is 4 unit, i.e. $S_0=4$. In this binomial model we also have- $u = \frac{1}{d} = 2$ and $r = \frac{1}{4}$. [So we are also following the previous condition we logically proved- $0 < d < (1+r) < u$].

Then we have- $S_1(H) = u.S_0= 8$ and $S_1(T) = d.S_0= 2$ as in the figure below.

We started with an European call option. Let us define some notations relevant for an European call option.

$K$ = The strike price of the European call option
$X_t$ = The amount of wealth you have at time t. So $X_0$ is the initial wealth (the amount of money you started with)
$\Delta_t$ = The share of stock at time t

Now back to the example. Suppose we are starting with initial wealth of 1.2 units, i.e. $X_0 = 1.2$ and at time t=0 we want to buy 1/2 units of stock, i.e. we have- $\Delta_0 = \frac{1}{2}$.

We know, Stock price at time t=0, i.e. $S_0 = 4$. So, to buy 1/2 share of that stock we require- $S_0.\Delta_0 = 2$ units. But we initially have 1.2 units; so we need to borrow 0.80 unit.

From where we can this money ?

We can borrow this money from bank at a risk free rate r per unit time. In this story we can borrow unlimited amount of money at a risk-free rate (r/unit time).

Similarly we can also invest money in the bank. The interest rate of investing the money in the bank is same as interest rate of borrowing the money from the bank

So our cash position is $X_0 - \Delta_0S_0 = 1.2-2= -0.80$. The negative sign in the cash position means we are borrowing money from the bank at a risk free rate. Instead if the cash position was positive we would invest the extra money after investing in stocks in the bank at the same risk free rate.

So, at time t=0, if you have invested (or borrowed) $(X_0-\Delta_0S_0)$; at time t=1, you will get back (or pay back) $(1+r)(X_0-\Delta_0S_0)$ unit of money. $(1+r)(X_0-\Delta_0S_0) = (1+0.25)(1.2-2)= -1$, i.e. at time t = 1 you will have 1 unit debt to the bank.

On the other hand, at time t = 1, the value of the stock available to you is either $\Delta_0S_1(H)$ or $\Delta_0S_1(T)$ depending on heads or tails in the toss. In this case- we would have- $\frac{1}{2}S_1(H) = 4$ or $\frac{1}{2}S_1(T) = 1$.

So the amount of wealth will be- \[\begin{aligned} X_1(H) &= \Delta_0S_0(H)+ (1+r)(X_0-\Delta_0S_0) \qquad \text{if H appears} \\ X_1(T) &= \Delta_0S_0(T)+ (1+r)(X_0-\Delta_0S_0) \qquad \text{if T appears} \\ \end{aligned}\]

So in this example we have-

\[\begin{aligned} X_1(H) &= \frac{1}{2}S_0(H)+ (1+r)(X_0-\Delta_0S_0) \qquad \text{if H appears} \\ &= 3 \\ X_1(T) &= \frac{1}{2}S_0(T)+ (1+r)(X_0-\Delta_0S_0) \qquad \text{if T appears} \\ &= 0 \\ \end{aligned}\]

Now if you remember we started with European call option. Let us define $V_t$ = the value of the call option at time t. Then we have-

\[\begin{aligned} V_1(H) &= (S_1(H)-K)^+ \qquad \text{if H appears} \\ &= (8-5)^+ = 3 \\ V_1(T) &= (S_1(T)-K)^+ \qquad \text{if T appears} \\ &= (1-5)^+ = 0 \end{aligned}\]

So, we have replicated the option by trading in the stocks and the money market. Now-

\[\begin{aligned} &X_1 = \Delta_0S_1+(1+r)(X_0-\Delta_0S_0) = (1+r)X_0 + \Delta_0(S_1- (1+r)S_0) \end{aligned}\]

We want to choose $X_0$ and $\Delta_0$ such that we have- $X_1(H)= V_1(H)$ and $X_1(T)= V_1(T)$ Now, we already know the value of $V_1(H)$ and $V_1(T)$. What we don’t know is the value of P[Head] = p. We start with comparing the above equations to get-

\[X_0 + \Delta_0(\frac{1}{1+r}S_1(H)- S_0) = \frac{1}{1+r}V_1(H)\] \[X_0 + \Delta_0(\frac{1}{1+r}S_1(T)- S_0) = \frac{1}{1+r}V_1(T)\]

One way to solve this is- multiply the first equation by $\tilde p$ and second equation by $\tilde q = 1- \tilde p$ and add them to get- \[X_0+ \Delta_0(\frac{1}{1+r}[\tilde p S_1(H)+ \tilde q S_1(T)]- S_0) = \frac{1}{1+r}[\tilde p V_1(H)+ \tilde q V_1(T)]\] So we want to choose $\tilde p$ such that- the term multiplied to to $\Delta_0$ is zero. i.e.

$\begin{aligned} &\frac{1}{1+r}[\tilde p S_1(H)+ \tilde q S_1(T)]- S_0 = 0 \\ = \ &\frac{1}{1+r}[\tilde p S_1(H)+ \tilde q S_1(T)] = S_0 \\ = \ &\frac{1}{1+r}[\tilde p uS_0+ \tilde q dS_0] = S_0 \\ = \ &\frac{1}{1+r}[\tilde p u+ \tilde q d]S_0 = S_0 \\ = \ &(\tilde p u+ \tilde q d) = 1+r \\ = \ &\tilde p u+ (1-\tilde p) d = 1+r \\ = \ &\tilde p (u-d) = 1+r-d \\ = \ &\tilde p = \frac{1+r-d}{u-d} \\ \end{aligned}$

Hence we have- $\tilde q = 1- \frac{1+r-d}{u-d} = \frac{u-1-r}{u-d}$.

From this we have- \[\begin{aligned} &\Delta_0(\frac{1}{1+r}S_1(H)- S_0) - \Delta_0(\frac{1}{1+r}S_1(T)- S_0) = \frac{1}{1+r}V_1(H)-\frac{1}{1+r}V_1(T) \\ = \ & \Delta_0 = \frac{V_1(H)-V_1(T)}{S_1(H)- S_1(T)} \end{aligned}\]

So, in the end with estimated $X_0$ and $\Delta_0$ an investor can hedge a short position in the derivative security or Delta hedging. That derivative security that pays $V_1$ at time t=1, which should be priced $V_0 = \frac{1}{1+r}[\tilde pV_1(H)+\tilde qV_1(T)]$, at t=0.

Now, why are studying all about these?

All of these is because of $\tilde p$ and $\tilde q$, which is called risk neutral probabilities. $\tilde p$ and $\tilde q$ will be useful to evaluate the limiting distribution of the binomial model.

How do we prove that? You can find out in my next blog.