# FD/U3 Topic 5 Binomial Option Pricing Model, Trading with Option

**Binomial Option Pricing Model**

The binomial option pricing model is an options valuation method developed in 1979. The binomial option pricing model uses an iterative procedure, allowing for the specification of nodes, or points in time, during the time span between the valuation date and the option’s expiration date.

The model reduces possibilities of price changes and removes the possibility for arbitrage.

### Basics of the Binomial Option Pricing Model

With binomial option price models, the assumptions are that there are two possible outcomes, hence the binomial part of the model. With a pricing model, the two outcomes are a move up, or a move down.

The major advantage to a binomial option pricing model is that they’re mathematically simple. Yet these models can become complex in a multi-period model.

*Fast Facts*

*The binomial option pricing model values options using an iterative approach.**WIth the model, there are two possible outcomes with each iteration.**The model is athematically simple.*

### Real World Example of Binomial Option Pricing Model

A simplified example of a binomial tree has only one step. Assume there is a stock that is priced at $100 per share. In one month, the price of this stock will go up by $10 or go down by $10, creating this situation:

**Stock price**= $100**Stock price in one month (up state)**= $110**Stock price in one month (down state)**= $90

Next, assume there is a call option available on this stock that expires in one month and has a strike price of $100. In the up state, this call option is worth $10, and in the down state, it is worth $0. The binomial model can calculate what the price of the call option should be today.

For simplification purposes, assume that an investor purchases one-half share of stock and writes or sells one call option. The total investment today is the price of half a share less the price of the option, and the possible payoffs at the end of the month are:

**Cost today**= $50 – option price**Portfolio value**(up state) = $55 – max ($110 – $100, 0) = $45**Portfolio value**(down state) = $45 – max($90 – $100, 0) = $45

The portfolio payoff is equal no matter how the stock price moves. Given this outcome, assuming no arbitrage opportunities, an investor should earn the risk-free rate over the course of the month. The cost today must be equal to the payoff discounted at the risk-free rate for one month. The equation to solve is thus:

**Option price**= $50 – $45 x e ^ (-risk-free rate x T), where e is the mathematical constant 2.7183.

Assuming the risk-free rate is 3 percent per year, and T equals 0.0833 (one divided by 12), then the price of the call option today is $5.11.

Due to its simple and iterative structure, the binomial option pricing model presents certain unique advantages. For example, since it provides a stream of valuations for a derivative for each node in a span of time, it is useful for valuing derivatives such as American options – which can be executed anytime between the purchase date and expiration date. It is also much simpler than other pricing models such as the Black-Scholes model.

**Trading with Option**

**Leverage –** Option trading is very attractive for small pocket traders. By taking position in options, one can reduce their cost significantly. Let’s take an example, suppose you wish to buy 12,000 shares of ABC Ltd., which is currently trading at Rs 130 level. Rs 130 call option of the same company is quoting Rs 5. Lot size for the company is 6000. In case you decide to buy stocks, your investment would be (130*12,000) = Rs 15,60,000. But if you choose to go options way, to take the same exposure, you will have to buy two lots for which your investment would come to (5*2*6000) = Rs 60,000. This means your cost of investment in options trading is just 3% to 4% of the investment required in stock trading.

**Limits risk –** Another benefit of options buying is that the risk is limited to the investment you make. Suppose, in context to the above example, you buy shares of ABC Ltd. and on the next day, the company comes up with the news of closing one of its subsidiaries and aftermath the stock opens 15% below your entry price. Now, the stock price falls to Rs 110.50, while the 130 call becomes zero. In case of stock, you would incur a loss of Rs 2,34,000; while in options, you would lose Rs 60,000 – the entire investment amount, which is far less than the one incurred in stock trading.

**Higher potential returns –** By trading in options, one will experience higher percentage returns compared to stocks. Lets assume, the delta of ABC Ltd. is 0.80, which suggests that options price will rise by 80% of stock price. If stock moves up to Rs 13, you will earn 10% return. While your option position will gain Rs 10.40 on the investment of Rs 5. Here, the return on investment is nearly 208%, which is much better than the return on stock. On the flip side, if you are on the wrong side of trade, one may end up losing the entire investment amount.

If you have been a seller of an option and the stock moves in the opposite direction than you thought, your take home losses accordingly.

**Works in different market scenarios –** One of the key advantages of options trading is alter strategies as per different market conditions. There are various strategies for all kind of markets, whether bullish, bearish or sideways – Long call, Bull call spread, Long Put, Bear put spread, long straddle, short straddle, etc. One can switch his strategies as per market condition.

**Hedging –** Every individual trading in stock market is exposed to a certain risk. In the event of any adverse market movements, hedging simply protects your trading positions from incurring loss. Suppose, you picked a stock of ABC Ltd. for Rs 100 six months back and now it is trading at Rs 125; here, you are earning a return of 25% on your investment. Now due to result season, you realize that the markets may soon enter a turbulent phase, which may also result in losing the money you earned during this time frame. In such scenario, you can hedge you position by simply buying ATM put option for same quantity, which will limit your downside during adverse market condition.

**Income from existing portfolio –** Any long term investors, who would like to earn some return or who want to lower the cost of their existing portfolios, can opt for covered call writing. In this case, one can write call option of the stocks he holds, which may give him some income on investment. Suppose, you have 7000 shares of XYZ Ltd., which you bought for Rs 130. Now, this stock is giving you handsome returns and therefore, you wish to hold it further. Later, you witness that this stock has consolidated and foresee the 135 level as a hurdle for the stock. Hence, you write 135 call option of same series for Rs 4 due to which you will get Rs. 28,000 (4*7000), if stock closes anywhere below Rs135. Thus, the same money that you have blocked in your portfolio fetches you some income.

Above are some of the advantages of options trading, but at the same time, one should not ignore the important factor called ‘time value decay’ while trading into options segment. Readers can refer to the previous article for more clarity on behaviour of time value in options.

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