The Binomial Asset Pricing Model Part-1(MFin-1)

In this article we would be discussing one period binomial asset pricing model. Why? Because this is one of the simplest models out there that explains arbitrage, derivative pricing and it is the basis for random walk. In the simple terms, arbitrage means there is no free lunch (there is also a No Free Lunch(NFL) theorem in machine learning), that is arbitrage is a trading strategy that begins with no money, has a positive probability of making money and has zero probability of losing money, basically you can make money out of thin air, the best kind of money heist. The traders known as arbitrageurs specialize in finding arbitrage opportunities in the market, for example if traders see that stock is undervalued in some exchange but overvalued in other exchange by small amount, they buy the stock in undervalued exchange while shorting (borrowing some number of stocks with a view that the prices would go down, immediately selling them and then buying and returning the same number of stocks after the period of contract to the lender and in time making a profit if stock price goes down or loss otherwise, ignoring the transaction charges) the stock on the overvalued exchange to gain the difference between two prices. As the trading takes place the difference between prices becomes zero and arbitrage opportunity vanishes. This is one of the assumptions of Efficient Market Hypothesis that there is no arbitrage. From the perspective of studying mathematical models it is essential to have no arbitrage in the model otherwise we wouldn’t be able to arrive at any conclusion. We’ll see other assumptions in the course of this article.

Let’s start with our one period binomial model. We call the beginning of the period as time zero and the end of the period as time one. Let $S_0$ be the price(all the stock prices are assumed to be greater than zero) of the stock per share at time zero and time one stock price per share can be either $S_1(H)$ or $S_1(T)$ depending on the result of a coin toss where $p$ is the probability of head and $q=1-p$ is the probability of tail on the toss of the coin, $p$ may not necessarily be one-half, that is the coin can be unfair. These are also called the actual probabilities underlying the movement of the given stock. So $S_1$ is a random variable taking on two values because it is unknown at time zero. Let $u$ and $d$ be such that $S_1(H)=uS_0$ and $S_1(T)=dS_0$ . Also we assume that $d<u$ otherwise we can interchange them to represent $d$ as down factor while $u$ as up factor of the given stock. We also introduce interest rate $r>0$ , that is $1$ unit invested in the money market at time zero will yield $1+r$ at time one. Similarly, one unit borrowed from money market at time zero will result in the debt of $1+r$ at time one. We also assume that interest rate for borrowing is same as interest rate for investing.

In this model to rule out arbitrage we must have, $0<d<1+r<u$ . $d>0$ is obvious from the assumption of positivity of the stock prices. Now suppose $u >d \geq 1+r >0$ , in this case one can begin with zero wealth and borrow money from money market to buy one unit of stock. In the worst case of a tail on the toss, the investor would have enough money to pay the debt of money market and has a positive probability of making a non zero profit as $u>d \geq 1+r$ , hence arbitrage. Now consider that $0<d<u\leq 1+r$ , in this case the investor can short the stock and invest the money into the money market. Even in the worst case of up movement the investor would have enough money to buyback the stock and return it and also there is a positive probability of non zero profit because $d<u \leq 1+r$ , hence arbitrage. Here we showed that for no arbitrage we must have $0<d<1+r<u$ . Now we would show that if this inequality holds we must have no arbitrage. Suppose $X_0$ denotes the initial wealth of the investor and $X_1$ denotes the wealth at time 1. Also the investor buys $\Delta_0$ shares of the stocks at time zero by borrowing/investing remaining amount from/into money market at the interest rate r. For example the price of $\Delta_0$ stocks at time zero is $\Delta_0 S_0$ and if $X_0 < \Delta_0 S_0$ the investor must buy from the money market otherwise the investor must invest the surplus amount. So the value of the portfolio at time 1 is, $X_1=\Delta_0 S_1 + (1+r)(X_0-\Delta_0 S_0)$ . For coin toss resulting in head and for initial wealth $X_0=0$ , $X_1(H)=\Delta_0 S_1(H) -(1+r)\Delta_0 S_0=\Delta_0 S_0u - (1+r)\Delta_0 S_0=\Delta_0 S_0(u-(1+r))$ . For coin toss resulting in tail and for initial wealth $X_0=0$ , $X_1(T)=\Delta_0 S_1(T) -(1+r)\Delta_0 S_0=\Delta_0 S_0d - (1+r)\Delta_0 S_0=\Delta_0 S_0(d-(1+r))$ . For any fixed value of $\Delta_0$ both $X_1(H)$ and $X_1(T)$ both have opposite signs with positive probabilities when $0<d<1+r<u$ . This implies there is no arbitrage when the preceding inequality is satisfied. In general one period model we define a derivative security to be a security that pays some amount $V_1(H)$ at time one if the coin toss results in head and possibly different amount $V_1(T)$ at time one if the coin toss results in tail. We would be using this kind of portfolio strategy to replicate derivative securities.

Let us consider a European call option which is a derivative security which confers on its owner the right but not the obligation to buy one share (depends on the number of call options bought, for one call option, right for one share and so on) of the stock at time one for the strike price K. The interesting case is when $S_1(T)<K<S_1(H)$ which is what we are going to consider. So the payoff from the this call option at time one is, $V_1=(S_1-K)^+=max\{S_1-K,0\}$ , meaning maximum of the quantities inside the parentheses. That is if the price of the stock is $S_1(H)$ we make a profit of $S_1(H)-K>0$ otherwise option expires worthless because $S_1(T)-K<0$ . Let us replicate a call option using the portfolio strategy described above. Let $S_0=4,u=2,d=0.5,$ and $r=0.25$ . Then $S_1(H)=8,$ and $S_1(T)=2$ . Suppose the strike price of the European call option is 5. And also assume that we started with initial wealth of $X_0=1.20$ and buy $\Delta_0=0.5$ shares of stock (assumption here is that stocks can be bought in fractions, in real scenarios we would be buying half of let’s say 100 stocks so not a big deal) at time zero. So to buy half the stocks at time zero we need 2 units of money but we have only 1.20 so for that we need to borrow from money market 0.80 units. Therefore our current cash position is $X_0-\Delta_0 S_0=-0.80$ i.e. a debt of 0.80 to the money market. At time one our cash position will be $(1+r)(X_0-\Delta_0S_0)=-1$ , i.e. we will have a debt of 1 at time one to the money market. Also we will have stock valued at $0.5S_1(H)=4$ and $0.5S_1(T)=1$ depending on the result of the coin toss as a head or tail respectively. So we have value of the portfolio when head shows up as, $X_1(H)=0.5S_1(H)+(1+r)(X_0-\Delta_0S_0)=4-1=3$ and the value of the portfolio when tail shows up as, $X_1(T)=0.5S_1(T)+(1+r)(X_0-\Delta_0S_0)=1-1=0$ . We have value of the European call option in this case as $(S_1-K)^+$ so when the head comes up the value of the option is 3 and when the tail comes up the value of the option is 0. In any case value of the portfolio at time one matches with the value of the option at time one, that is we have replicated the option using the portfolio of the stocks and the money market.

In the next part of the article we will show that this initial wealth is right price for this option otherwise there will be arbitrage and also we will discuss general one period model for derivative security and risk neutral probabilities.

To be continued…

Reference:

Steven Shreve, Stochastic Calculus for Finance I, The Binomial Asset Pricing Model, Springer

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The Binomial Asset Pricing Model Part-2(MFin-1) – Learn, think and do

September 26, 2021

[…] the part-1 of this post we talked about arbitrage, replicating call option using portfolio of the stocks and […]

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