Actuarial Science: Binomial Model Option Pricing

You must have heard people saying that ”Options trading is not for the faint of heart”. Have you ever wondered why do they say so?

I’ll tell you the reason behind it.

If you buy the share of the company, then you are simply making an investment which can be held for an indefinite time frame. You make your decisions based on the sound fundamentals of that company and you have the right to decide whether to keep holding it or sell it off. But this is not the case with options.

For a commoner, an option is a financial instrument which gives the right to the buyer but not an obligation to buy (call option) or to sell (put option) the underlying asset at an agreed price on a specific date in future (though it can be exercised before maturity in case of American options). Therefore, as you can read, options are only for a stipulated time frame. If you make money during that time period, great. If not, you’ve lost the money you invested to purchase the option and that’s it.

Hence, traders usually spend hours to work out the price of the option that should be charged so as to avoid arbitrage opportunities. In a competitive market, to avoid arbitrage opportunities, assets with identical payoff structures must have the same price. Valuation of options has been a challenging task. Various models are designed to price these complex instruments. Among all models, binomial model is the easiest and the most popular one for pricing options.

Just before discussing  this model let me ask you a question.

What do we actually mean by the term ‘pricing of an option’?

As mentioned above, an option gives a privilege to a buyer and hence to get such privilege the buyer has to pay a price. This price is nothing but the premium of the option. Thus, the pricing models designed help us to know the appropriate price of the option.

In this article let’s discuss the simplest of all pricing models- Binomial Option Pricing Model.

Meaning:
The name stems from the fact that it calculates two possible values for an option at any given time. The binomial model was developed in 1979 by John Cox (a well respected finance professor), Mark Rubinstein (a financial economist) and Stephen Ross (a finance professor). The aim is to find the value at time t=0 of a derivative that provides a payoff at some future date based on the value of the stock at that future date.

The basic idea is that if we can construct a portfolio that replicates the payoff from the derivative under every possible circumstance, then that portfolio must have the same value as that of the derivative. So by valuing the replicating portfolio we can value the derivative.

Assumptions:
Every theory works on certain assumptions. Here we assume that:

In the binomial model,  it is assumed that:

  • Investors are risk-averse that is when faced with two investments with a similar expected return, they prefer the one with lower risk.
  • Investors prefer more to less.
  • There are no trading or transaction costs or taxes.
  • There are no minimum or maximum units of trading. In other words, unlimited buying and selling are permitted (including short selling).
  • There is complete divisibility of assets and investors are allowed to hold the fraction of an asset.
  • Stock and bonds can only be bought and sold at discrete times 1, 2, …
  • The principle of no arbitrage holds.

As such the model appears to be quite unrealistic. However, it does provide us with good insight into the theory behind more realistic models. Furthermore, it provides us with an effective computational tool for derivatives pricing.

Notations:
The share price process
We will use St to represent the price of a non-dividend-paying stock at discrete time intervals t (t= 0,1,2,.… This means that St is random.

Here “stock” specifically means a share or equity as opposed to a bond. For the time being, we ignore the possibility of dividends, which would otherwise unnecessarily complicate matters.

Over any discrete time interval from t-1 to t, we assume that St either goes up or goes down. We also assume that we cannot predict beforehand which it will be and so future values of St cannot be predicted with certainty. We will, however, be able to attach probabilities to each possibility and we also assume that the sizes of the jumps up or down are known.

Bt- Stands for the cash process. It is the risk-free rate at which accumulation happens.

The above timeline explains the Cash process. As can be seen, the asset has accumulated at a risk free rate and has grown to ert where:

r= Continuously Compounding Risk free rate of interest.

T=Time period.

All the approaches lead to the same price. The only difference is the way price is calculated.

Replicating Portfolio Approach
At time 0, suppose that we hold  ϕ units (phi units) of stock and ψ units (psi units) of cash. So these phi and psi could be any values and this portfolio could consist of an amount of shares and an amount of cash in whatever amounts we choose. The value of this portfolio at time 0 is V0.

There is yet another derivative which pays cu if the price of the underlying stock goes up and cd if the price of the underlying stock goes down. So, the value of the payment made by the derivative, which we can denote by the random variable C, depends on the underlying stock price.

At what price should this derivative trade at time 0?

At time 0 suppose that we hold ϕ units of stock and ψ units of cash. The value of this portfolio at time 0 is V0 .

Eg: If the stock price went up, then the value of one unit of the stock has increased to S0u and if the price went down then the value of the stock has decreased to S0d. Also, the initial unit in the cash account has now increased to er .

Therefore, we have the following two equations:

Solving these two equations we can get the value of phi and psi and then by the principle of no arbitrage, we have:

Risk Neutral Approach
Under this approach, the investor is risk neutral.

We find the probability of an up move (q) and down move (1-q) in such a way that the payoff is replicated.

Also, it should be noted that the asset grows at a risk-free rate. To better understand this, we will illustrate using excel later.

Price Deflator Approach
Under this approach, we find a real-world probability.

This real-world probability is the actual probability by market participants for an up move. This probability will always be greater than the risk-neutral probability.

Here,

Where Au stands for stochastic discounting factor (AU= er (q/p) ).
CU stands for payoff when there is an up move.
P= real world probability measure.

Let us take an example to understand this model better. We will be solving the question using the risk neutral approach which is the most widely used approach of this model.

Step 1:
Plot the binomial lattice according to the information provided in the question.


Step 2:
Plot the payoff of the call option. This is nothing but what the option will get at maturity.

So,  for call option, the payoff at maturity is max( St-X, 0)= max( underlying asset- Strike price, 0).

As can be seen in the figure we have plotted the payoff of the call.


Step 3:
Find the risk-neutral probability. The formula for the same is given below.

We have found the desired probability using the above-mentioned formulas.


Step 4:
Find the expected payoff of the option by using the below formula.

Expected Payoff= Payoff at maturity * Probability


Step 5:
Find the present value factor. This is nothing but exp(-rn). The risk free rate given in the question is continuously compounding therefore we use an exponential function.

 Also “n” stands for the time period. In our question, it is 1.


Step 6:
We get the value of the call option at t=0 by discounting the expected payoff at t=0. Here it is nothing but step 4* step 5.

Hence, we derive the value of the option at t=o. This is a small illustration having just one time period. Complexity increases when there are more than two time periods.

The price so calculated is the theoretical price of the option. This price gives useful insights to the traders. It helps them in taking, decisions on whether to buy/sell an option.

Also, it should be noted that the real price of an option may not always be same as the theoretical price derived from the model. The variations can be attributed to the changing demand/supply situations, the incorrect assumptions of up move or down move etc.

Though this result in a different price of the option but the significance of the theoretical price cannot be denied. It plays a crucial role in creating various positions.

Thus, this is how the option’s price is calculated using the Binomial Option Pricing Method.

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