The Binomial Options Pricing Model
Essentially, all models are wrong, but, some models are useful.
— George E. P. Box
Option is a kind of financial instrument, where, a buyer buys the right to exchange some given commodity or asset in future at a price which is fixed in the present. Note the word “right”, since it is solely on the buyer’s discretion to exercise the option. The seller on the other hand is bound(or obligated) to fulfil the option if the buyer exercises it. But, what is the seller’s incentive then to enter into such an agreement in the first place?
The seller enters into the agreement only after he is paid some premium for selling such a “right” to the buyer. Of course, the buyer pays the premium to first buy the exercising “rights”.
Still confused? Have a look at this article by CFI to understand better. It is of immense importance that you know what are options before proceeding because our arguments further will be based on the assumption that the reader knows about options and related jargon!
Now that you know how options work, let us consider options based on a peculiar underlying stock. What is the peculiarity? The underlying stock’s price can only move to two possible prices on the next day, either an up price or down price. Moreover both of these movement happens with some fixed factors which is common knowledge in the market. Mathematically,
S (price of stock today) can become :
u*S (where u is the up price factor) with probability p and
d*S (where d is the up price factor) with probability (1-p)
Then, the expected price tomorrow (denoted by Pt) :
Pt = p*(u*S) + (1-p)*(d*S)
And the present value of the “expected price tomorrow” is
(Pt / r) where r is (1+ the risk free interest rate)
A little note on present value : 1000 $ today will not be of same buying power as of 1000 $ ten years later because of inflation. In fact, if we deposit 1000 $ in some central bank today and assume rate of interest to be 5% per annum, the 1000 $ will be worth 1000*(1+0.05)¹⁰ = 1630 $ ten years later considering compounded growth.
In a similar fashion we can find the present worth of some future amount by discounting it by the interest rate. For example, the present value of (1000 $ to be received 10 years in future) will be 1000/(1+0.05)¹⁰ = 614 $
An interesting fact here is that r must lie between d and u, or,
d < r < u
Why?
If r>u, then even in the case of up price movement, price on the next period will be (u*S)/r which is less than S because u/r < 1. Hence, a rational trader will short the stock today and buy bonds from the money received by shorting. The next period he can sell the bonds and buy back the stock to settle his liabilities. Net gains :
On Day1 : S - S =0 (i.e. sold stock of value S and bought bonds of value S)
On Day2 : S - (u*S)/r > 0 (i.e. sold bonds of value S and bought stock of value (u/r)*S )
For calculating gains on Day2, we are doing calculations on present value
Hence, we see that the trader makes profits by riskless arbitraging. But, since, the up and down factors are common knowledge in the market such an arbitraging opportunity will never arise and so r>u is infeasible.
Similarly we can derive that r<d is also infeasible. I shall the urge the readers to try proving this for better understanding!
Binomial Options Pricing Model(or BOPM from now on) tries to predict the correct premium a buyer must pay at any time to buy the rights of an option which is based on the underlying stock that we discussed above. Let us write the assumptions that the model makes during prediction of premiums at any given time:
- There are only two possible prices for the underlying asset on the next day, either an up movement or down movement. Moreover both of these movement happens with some fixed factors. Price does not remain constant!!!!
- The stock is arbitrage-free. If the underlying stock is not arbitrage-free, all rational traders will make riskless profit arbitraging the stock bringing the stock price back to its correct arbitrage-free price.
- No dividends are paid out during the option’s life. Why? Because the author wants it that way.
- The risk-free rate is constant throughout the life of the option. Borrowing and lending are both done at the same risk free rate
- Trading is frictionless. There are no transactional costs, operational costs or taxes on profits for that matter xD.
- Investors are risk neutral i.e. if there is a scope of profiting from a certain trade the investor will go on to trade it irrespective of risk to reward ratios.
- We can long or short any fraction of underlying stock.
An important point that I should stress here is that option premium can take any value and the constraints of up and down price movements apply only on the underlying stock!
Let us consider the call option with one day to expiry. The premium that should be paid tomorrow(i.e. the expiry day)to buy the call option will be max(0, u*S-K) or max(0, d*S-K) if the price moved up or down respectively (here, K is the strike price of the call option)
In the above arguments B can be negative as well (for example when Cd=0 and Cu>0) but Δ will always be positive because Cu>Cd and u>d.
Now, since in our hedging portfolio the final price is same as that of the option, any discrepancy in their initial prices will lead us to a riskless arbitrage opportunity. Considering this fact, the initial prices of the hedging portfolio and the call option should be equal and we as investors will arbitrage if the prices differ. This is the crux of BOPM.
Once we have understood how the hedging portfolio helps us in arbitraging and pocketing the mispricing between the call premium and the hedging portfolio, we can extrapolate the case to a multi period call option. The fundamental idea still remains the same : assuming that the final prices of the option and hedging portfolio are the same we will perform backward induction to find the correct value of the call premium at any given period.
For example, let us look at the two period BOPM :
Here, we should focus on the point that the model requires dynamic hedging that is after every period we should adjust our portfolio of the underlying stocks and bonds so as to remain hedged for the next period and next period only.
The significance of the bold text above is to delineate the fact that hedging portfolio protects us just for one period and to remain hedged successfully until the expiry of the call option we should hedge our portfolio every day according to the formulae shown above.
Now, the above case is easily generalizable to an N-period call option, if you notice the occurrence of binomial coefficients for pricing the premium of the call option. The formula that we will achieve thus :
Any value of call premium that differs from this and hurrah we can book some profit in the long run with riskless arbitraging.
Advantages of BOPM :
- Simple. Why? Try Black-Scholes and come back to agree with me.
- Recombinant i.e. price after a combination of some k ups and m downs will be same. For example price after two ups and 1 down = after 1 up then 1 down then 1 up = after 1 down and two ups or (u*u*d*S)=(u*d*u*S)=(d*u*u*S). This helps to bring time complexity of complete simulation to O(n²), where n is the number of time periods.
- Many applications. BOPM works optimally for European, American and Bermudan options although it is used most frequently with American options.
What next? Curious readers can find how BOPM also leads to Black-Scholes option pricing formula in the limiting case by going through these :
