Traders usually don’t learn stochastic calculus. Why should they? Stochastic Calculus is very advanced. But do you know this fact that stochastic calculus is used extensively when it comes to pricing options, futures, swaps, forwards and other derivative products. Today stochastic calculus rules Wall Street. Many people blame quantitative models developed by quants for the financial crisis of 2008. But there are many who dismiss this claim as non sense. Without understanding stochastic calculus your cannot understand how to build options pricing models. Black Scholes is one such model that is based on a stochastic differential equation. But before you learn stochastic calculus, you will need to understand the modern theory of probability. Traditional probability theory cannot handle stochastic calculus. You should understand probability theory if you want to understand modern financial models. Did you read my post on the **Stock Market Fama French Three Factor Model**.

Markowitz in 1952 was the first person to apply statistics to develop his portfolio diversification models. His portfolio diversification model was build on covariance between the different stocks in the portfolio. Capital Asset Pricing Model came in 1960s. But it could not survive the test of financial markets despite its elegance and simplicity. Concepts like alpha, beta, standard deviation, R-Squared and Sharpe Ratio were developed in that period and are still being used in building alortihmic trading models. Since then models of how the financial markets work have become very sophisticated. Most of these models use continuous time stochastic partial differential equations to model market microstructure. In 1970s, Black Scholes Merton Options Pricing Model. Before you understand these sophisticated financial models you should be familiar with the modern probability theory. Learn **how you can use Neural Networks in stock trading**.

Let’s start with the Binomial Stock Pricing Model. This is a very important stock pricing model and is used quite often in modeling options pricing. Binomial Stock Pricing Model is a powerful tool that you should understand. We start with a 1 period model. We start at time zero. At time zero, stock ABC price is ^{. We know this price . After one time period we have price . This price can have two values. Either it is or it is assuming we toss a coin to determine whether it is or . This is what we are assuming in the Binomial Stock Pricing Model, at each time step we toss a coin and the coin toss result determines whether the stock price goes up or down. So at the second time step we again toss a coin and the stock price can be or it is depending on whether it is a head or tail. Now we don’t assume a fair coin with a probability . We just assume that the probability of head is and the probability of tail is . Read how one home based trader crashed DJIA.}

Now as said above, we don’t know (stock price at time 1). However we know (stock price at time zero). Any quantity that we don’t know at time zero, we call it random. So stock price at all future time steps is a random quantity. So the stock price at time step one , time step two , time step three , time step four and so on are random quantities that we don’t know at time zero. At each time step according to the Binomial Asset Pricing Model, we toss a coin and the coin toss determines the stock price. The probability of head is and the probability of tail is and stays the same. Here we introduce two numbers the up factor and the down factor. Up factor is just and the down factor is . We assume is greater than and is less than . If it is other way around, we can easily relabel and and it doesn’t effect our Binomial Stock Pricing Model.

At this stage we need to bring in the interest rate into our model. Interest rates are the back bone of modern economies. We assume that the risk free interest rate in the economy is . In US, treasury bills promise the risk free interest rate. So whatever the interest rate is being offered on the treasury bills we can use them as the short term interest rate. Long term interest rates will be those offered by the treasury bonds. US has a very big bond market. Investors invest in these bond when they want to secure a risk free rate. We use these interest rates in our stochastic models of the financial market. Bond market is also known as the money market. If we have interest rate and we invest $ in the treasury bills we get a return of at time one. On the other hand, instead of investing, if we borrow from the money market at interest rate , we will have to pay back at time step one. For our Binomial Stock Pricing Model, we assume that the investing interest and the borrowing interest rate is the same . Watch this documentary on **stock market flash crash**.

Arbitrage happens when we make risk free profits. In modern financial markets, the opportunity of making arbitrage profits is small. In our model, arbitrage takes place when we have a stock trading strategy that can make money for use without taking any risk meaning we have zero chance of losing. Do you believe that? Can you make risk free profits in today’s stock market. Absolutely not. There is always a risk that you will lose. You cannot make risk free profits. Assume we have from the money market and buy stock ABC share. At time step one we have a profit of either or . It doesn’t matter whether the coin toss is head or tail in both cases we have a profit since we have assumed or which is greater than our loan . So we repay the loan and still keep a risk free profit or . Life will be very sweet and simple if this was true.

We will have to assume that if we want to rule out arbitrage. No Arbitrage is the basic assumption in building stochastic models for the financial markets. We just don’t want arbitrage to take place. It is against the common sense. follows from the simple fact that stock price cannot go below zero. We cannot sell a stock at a negative price. Now assume . If this is true we can start with zero wealth. At time zero even if we have no money we can borrow from the money market at the interest rate . At time step 1 we have to repay . Since we have assumed and still end up with a risk free profit of either or as explained in detail above. So cannot be true and we must have in order to avoid risk free arbitrage to be realistic. Learn how to **plot candlestick charts using python**.

Now assume that . If this was true, we could sell the stock ABC short and invest the money in the money market. At time step 1 we will get paid interest and our investment will become . We can use this to replace the stock ABC at time step 1 and still make a risk free profit. This follows from the assumption that . Once again we are making risk free profits which are not possible in real world. So in order to once again avoid arbitrage taking place and making risk free profits we will have to assume that . So we have proved that in order to avoid arbitrage we should have .

You might be wondering what a simple stock pricing model. This is true. Stock prices are much more complicated as compared to our simple Binomial Asset Pricing Model. Binomial Asset Pricing Model is used by quants because it illustrates the concept of arbitrage and how it can provide risk free profits between the stock market and the money market. With many step Binomial Asset Pricing Model can provide a good approximations to the continuous time stochastic models. This model will also help us develop the concept of conditional expectations and the martingales. So you should not take this simple model easily. Master it as it will lead you to master more complicated continuous time stochastic models.

**How To Calculate The Stock Options Premium?**

We want to know what price should we pay for buying an options contract at time zero with expiry at time step one. This is a fundamental question that comes to the mind of every stock options trader. What should he pay for the stock option at time zero when he has no knowledge of what the stock price will be at time step one. How to solve this puzzle? We know the stock price at time zero but we have no knowledge of what the stock price will be at time step one. Can we use the Binomial Asset Pricing Model? Let’s see. Assume we are trading a European Call Option. Options are derivative products which depend on the underlying stocks. Now a days stock options are very popular with traders and investors all over the world as it gives you the right to buy a stock at time step 1 with strike price . But you have no obligation to do so. You pay a small premium to have the right to buy the stock at time step 1. But you can forgo that right and not exercise it. Mathematically we model this by saying . This expression simply means we take the maximum of and 0.

How to model a stock option? We will use the Arbitrage Pricing Theory and replicate the stock options contract by trading the stock market and the money market. Suppose , , and . Then we have and . Assume that the strike price of our stock options contract is . Further assume that at time zero we have and buy shares of stock at time zero. Stock ABC price is and our initial wealth is only , we will need to go to the money market and borrow additional funds of to buy shares of stock ABC. When we borrow from the money market at the interest rate we will have to repay the loan plus interest at time step one. So our wealth at time zero is . At time step one, our wealth position will be . So we will have to repay to the money market at time step one. At time step one, stock ABC will be either priced at or . Learn **how you can download options data from CBOE and Yahoo Finance.**

Let’s assume at time step one we have a head on the coin toss. If get a head one time step one, the value of our portfolio of stocks and bonds will be:

Similarly if we had a tail, the value of our portfolio of stocks and bonds will be:

If we have a head on the coin toss, stock options price will be and in the case if we got a tail the stock options price will be . So we have successfully replicated the stock options contract by simultaneously trading in the stock market and the bond market. Our initial wealth 1.5 is the No Arbitrage Price of the Stock Options at time zero. If the stock options price was say 1.55, there would be arbitrage opportunity. We can easily sell the stock options contract for 1.55 and invest the excess 0.05 in the money market and use the remaining wealth of 1.5 to successfully replicate the stock options. At time step one, we could pay off the options contract regardless of whether it was a head or a tail and keep the excess 0.05 and the interest 0.0001 accrued on it. So once again we have an arbitrage opportunity and risk free profit of 0.0501 whether the coin toss is a head or a tail. It doesn’t matter.

Now take the opposite case. Suppose we can buy the stock options contract for less than 1.5 say we can buy it for 1.45. In this case we could buy the options contract and replicate the stock options trading strategy in reverse. In this case we can sell the stock ABC short. Our initial wealth is 1.5. We use 1.45 to buy the options contract and invest the remaining 0.05 in the money market by buying a bond. So in both cases arbitrage rules out the possibility of stock options contract being higher or lower than the price of 1.5 What this shows is that unless the stock options contract premium is 1.5, there will be a risk free arbitrage opportunity that is simply not possible. Just as a recap, we have made the following assumptions about our Binomial Asset Pricing Model:

1. We can subdivide the shares of stock ABC for buying and selling. In practice this may not be possible as markets are not that efficient.

2. We have the same interest rate for borrowing and investing. This assumption may also not hold in practice.

3. Bid-Ask Spread is zero so we have the same price for selling stock ABC as for buying. Once again this assumption does not hold in reality. We have to pay a Bid-Ask Spread.

4. Stock ABC can take only two values at time step one. Once again this assumption is too simplistic.

The famous Black Scholes Merton Options Pricing Model uses assumptions 1-2-3 in its derivation. It doesn’t use the assumption 4 as it uses a continuous time stochastic model instead of a discrete time stochastic model.In Black Scholes Merton Options Pricing Model we assume that the stock price is a Brownian Geometric Motion. In practice sometimes this assumption works and sometimes it fails miserably. Similarly sometimes we have a low bid-ask spread and our assumption is nearly satisfied while other times we have a big bid-ask spread and the assumptions get violated terribly. Better models have been developed that don’t use the Geometric Brownian Motion assumption and give good results. But in order to understand those models you have to initially master this simple Binomial Asset Pricing Model.

For our one period Binomial Asset Pricing Model, we assume that we have a derivative contract that pays at time step one if we have a coin toss that results in a head and it pays at time step one if we have a coin toss that results in a tail. Now keep this in mind that options is a derivative contract and so are the futures and the forward contracts. Derivative contract prices are based on an underlying asset using some formula. In the case of European Call Options we have and for a European Put Options contract we have .

In order to calculate the stock options price at time zero we replicate it as shown above in the stock and the bond market. Suppose we start with wealth and buy shares of stock ABC at time zero. If we do that our cash position will be . The value of our wealth at time step one will be:

We want to choose and such that we have and at time step one. We already know the stock options two possible values and . So at time zero we know what the two possible values of our stock options contract can be. But we don’t know which one of these two possible values will be realized. Our replication strategy requires that the following two equations hold if there is to be no arbitrage in the market.

Let’s define two variables and . In terms of these two variables the price of the derivative security at time zero is given by:

This is the price that does not allow arbitrage in the market. For any other price, there will be risk free arbitrage which is simply not possible. The two variables and are not actual probabilities but the arbitrage free probabilities that ensure no arbitrage in our one step Binomial Asset Pricing Model. The two risk free probabilities are given by:

**Multiperiod Binomial Asset Pricing Model**

Now we need to discuss the multi period binomial asset pricing model. Just like the one period binomial asset pricing model, at each step we toss a coin and if the coin toss is head stock price goes up and if the coin toss is tail the stock price goes down.In the money market we get interest for buying a bond and we have to pay interest when we sell a bond. We assume that arbitrage is not possible in the market which means there is no trading strategy that can make risk free profits. The risk free probabilities are the same just the that we use for hedging changes. We can increase the period as much as we like we our Multiperiod Binomial Asset Pricing Model!

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