Math 20 Project
Yon-Sue Choi
Binomial Options Pricing Model and Implementation
In finance, an option denotes a derivative financial instrument that specifies a contract between two parties for a future transaction. Options are widely used to hedge a portfolio and protect against future losses. With its importance in financial markets, knowing how to value an option is critical in daily trading and measuring overall wealth of a portfolio. Though there are a number of complicated models that practitioners use routinely, most, if not all, have their foundations on the binomial options pricing model (BOPM). For this project, I attempted to understand BOPM for both one and multiple periods, program the model through MATLAB, and compared the results from my own program to Black-Scholes calculator that’s widely available online.
Before discussing the model, I will briefly differentiate several types of options. There are two main types of options: call and put. An option that grants the right to buy an asset at a specific price is called a call. An option that conveys the right to sell an asset at a specific price is called a put. When an option is exercised, the owner receives the difference between the pre-specified price and the current asset price. If the difference is negative, the option is worth zero because no logical person will exercise it. Furthermore, a European option only allows the buyer to exercise his right at the maturity date, whereas an American option allows the right to be exerciseany time before the maturity date.
In this paper, I will only deal with European call or put options. Evaluating American options require more complex approaches that are outside the scope of my project. Furthermore, one of my goals is to accurately model the Black-Scholes formula with BOPM, and the original Black-Scholes formula was only intended for the European options. In addition, there are many papers that argue it’s almost never beneficial to exercise American option earlier than its maturity date except under specific conditions, but I will not be providing details on that point in this paper.
To understand BOPM, it’s best to start with a simple one-period model. We assume that right now we have a stock that is valued at S0. After one period, the value of the stock will be S1(H) =uS0, with a probability of p, or S1(T) = dS0, with the probability of q. We also define p + q = 1. A good analogy is that we flip a coin (not necessarily a fair one) and if head shows, the value of our stock goes up by a factor of u. If tailshows, it goes down by a d. Although it doesn’t have to be true, it’s a standard procedure both in textbook examples and in practice to have d = 1/u. The graphicappearing on the right is from Steven Shreve’s Stochastic Calculus for Financeand it provides a good overview of the one-period model.
In additionto p, q, u, and d, we introduce an interest rate of r. This interest rate of r can safely earned by investing in the money market instead of buying stocks. We rule out arbitrage by defining r’s range. We define that 0 < d < 1 + r < u. If this doesn’t hold true, opportunity to make a safe return from zero initial wealth is created by making a short and long portfolio in stock and the money market.
To value an option at time t = 0, we applyarbitrage pricing theory approach by replicating the option through trading in the stock and money markets. Before using the theory, we must make several assumptions: 1) shares of stock can be subdivided for sale or purchase, 2) the interest rate for investing is the same as the interest rate for borrowing, 3) the purchase price of stock is the same as the selling price, and 4) at any time, the stock can take only two possible values in the next period.
Using these assumptions, we construct our portfolio to replicate the option. For example, let’s assume that initial price of a stock is S0 = 4 and the strike price of a European call option is K = 5. Price of the stock can move up by a factor of u = 2, or go down by a factor of d = 1/u = 1/2. We also set our r = 1/4. Then, at t = 1, S1(H) = 8 and S1(T) = 2. Let’s assume our initial wealth X0 = 1.2 and we buy Δ0 = 1/2 shares of thestock. We can easily see that our cash position at t = 0 is (X0–S0Δ0) = -0.8, which means that we borrow 0.8 from the money market to support our position. At t = 1, if we are lucky, we will have the portfolio of X1(H) = (1/2)S1(H) + (1 + r)(X0–S0Δ0) = 3. If not, X1(T) = (1/2)S1(T) + (1 + r)(X0–S0Δ0) = 0. The portfolio value agrees with the potential option values at t = 1, which are either Max [(S1(H) – 5), 0] = 3 or Max [(S1(T) – 5), 0] = 0. Therefore the initial wealth 1.20 required to construct the replicating portfolio is said to be the no-arbitrage price of the option at time zero. The arbitrage pricing theory suggests that if the value of the particular option is anything other than 1.2 at t = 0, one could arbitrage with a mixture of long and short positions and earn a safe return with no risk and no initialwealth.
In general terms, we define V1(H) to be the payoff of an option at time t = 1 when coin toss results in head and V1(T) to be the payoff for tail. Again, our object is to determine V0 at time 0, which should be the price of the option. From the example before, we saw that
X1 = Δ0S1 + (1 + r)(X0–Δ0S0) = (1 + r)X0 + Δ0(S1– (1 + r)S0).
However, we know that there are two possibilities. Therefore, we can derive
and
.
Combining the two, we get,
.
If we choose p so that , then above equation can be simplified to .
By equating S1(H) = puS0 and S1(T) = (1-p)dS0, we also get formulas
.
Furthermore, in the above process, we can also get the delta-hedging formula
.
In conclusion, for binomial model with time period of one, an option that is valued at V1 at time one should be priced at
at time zero.
Now we extend the idea of BOPM to multiple periods. We use the same idea and have two possible paths at each nod. Using the coin analogy again, we toss a coin multiple times. The stock price will move up when each heads and move down for each tails. If we assume that our initial stock price is S0, then S1(H) = uS0 and S1(T) = dS0. After two tosses, the price will be S2(HH) = uS1(H) = u2S0, S2(T) = dS1(T) = d2S0, S2(HT) = dS1(H) = duS0, and S2(TH) = uS1(T) = udS0. Continuing this process, we will have 23coin sequences after three tosses, although not all prices will be distinct. The graphic to the right is also from Steven Shreve’s book, and it provides an excellent overview.
For multiple-period BOPM, it can be shown that if VN is a random variable (value of an option that at time N) that depends on the first N coin tosses w1w2…wN, its price can be defined recursively in time by the formula
Furthermore, the number of shares that should be bought to recreate a replica of the option is given by
.
The above equations imply a three-step method to determine the price of an option at time t = 0. First, one needs to create a binomial price tree until the maturity date of all possible sequences of price going up and down. Second, option value at each end node must be computed. For call option, it would be Max [(SN– K), 0]. For put option, it should be Max [(K – SN), 0]. And the final step is to recursively find the value of an earlier node for every possible sequence of coin tosses. Repeating this process would yield V0 at the end.
However, doing the above three-step process can become overwhelming very easily, which motivated me to write a program in MATLAB (actually, two programs for each call and put option). Although MATLAB implementation uses the same approach, we need to define p, q and rwith the usual market inputs such as market interest rate and volatility. It is a standard practice to define where r is the risk free rate and dt = maturity (in years) divided by the number of desired steps or nodes. This formula is based on the equation of continuously compounded interest rates. It’s also a market standard to define and which satisfy the convention of u*d = 1.
Studying BOPM has been a very interesting experience. Although it conveys a very simple concept, BOPM has enormous implications in the field of quantitative finance. A background research revealed that almost all models of option pricing have BOPM as their starting point. Another fascinating point was that as I increased the number of steps taken by the BOPM program, the calculated price converged with the Black-Scholes calculator with incredible accuracy. After consulting with Mr. Jonathan Bloom of Dartmouth Mathematics Department, I realized that Black-Scholes model in essence is just a continuous version of BOPM, which impressed me even further.