This module introduces you to the option pricing problem in a option binomial world. In this world the price of the underlying can go up or down each period. The two main advantages of example simplification put First, it lets you example become acquainted with the economic intuition that underlies the Black Scholes tree pricing model. Second, the binomial tree has resulted in numerical solutions to more general option pricing problems than what is covered by the Black Scholes model. For example, the American put option can be valued this way. In the following example, we show that you can use the option binomial model to value an option. The technique illustrated is the same american applied in the original Black Scholes option pricing paper. This technique identifies the arbitrage free price of an option by creating a riskless portfolio synthetically, using the stock and an option. Here, riskless means that the portfolio has a known value at the end of the period, no matter what happens to the stock price. If the portfolio is riskless, then we know its current value. This is simply proposal future value discounted by the risk-free interest rate. Then, since we know the portfolio value and the stock price, we can immediately identify the arbitrage free option price from the difference. From this approach we can interpret the option free option price in terms of the general principle of valuation. This principle asserts that the value of any asset is the present discounted value of all future cash flows from the asset. Binomial value an option by applying this principle requires that we need to determine two things: For a European option, the future cash flows are easy; for example, for a call option, the cash flow is 0 if the future stock price is less than the strike price, and option the future stock price minus the strike price otherwise. What about the discount rate? Here lies the problem. We cannot use the risk-free interest rate, since the future cash flows are not risk-free; they depend on the unknown future stock price. If people are risk-averse, then they will hold risky securities only if they can get a return greater than the risk-free interest rate. In fact, much of the problem in valuing risky securities is determining the appropriate discount rate, as option the capital asset pricing model. The fundamental riskless hedge argument solves the problem of determining the discount rate, since we can apply the risk free put to discount the cash flows from a riskless portfolio. In example 1 below we first illustrate the principles underlying the riskless hedged portfolio approach to option valuation. In the subsequent examples, we apply the FTS Binomial Tree Module to solve a set of specific option pricing and related problems. Creating a Riskless Portfolio. Let S denote the current stock price and assume that at the end of one period the stock value is either 10 or In the display window set Asset Price to equal 20, strike price equal to 20, uptick to 2 and downtick to 0. From the top drop down above the button Draw Tree select Asset Process, the second drop down select European Option and then click on Draw Tree. We will first study proposal European put and call options with a strike price, X, of 20 when the risk-free interest rate is zero. The american values of the stock and the options are depicted in the "tree" in Figure 2. You can see that the tree values are the same as in Figure 2. Now suppose you form a portfolio: Sell 3 call options. Consider what happens at the end of the period if a downtick occurs. Since the stock is worth 10, the call option finishes out-of-the-money and the trader you sold the call tree to would not exercise it. The final payoff from your portfolio is 20, which is the value of the two stocks. Suppose an uptick occurs. Each stock is worth 40, but the calls would now be exercised against you. You would be required to give three stocks to the person who bought the options, and you would receive 20 for each stock. Therefore, your final position would be: You now have a riskless portfolio. The present value of this portfolio is 20 since we have binomial that the risk-free interest rate is zero. This tree that the portfolio that is long 2 stocks and short 3 call options must trade for a price equal example If not, there is an arbitrage opportunity. To see this, suppose that you could example such tree portfolio for more than Then, you can profit from selling this portfolio. You receive more than 20 from the sale, example lose at most 20 at the end of the period, which option you of a profit. Each of these situations presents an arbitrage opportunity i. Puts can be priced in the same manner by considering portfolios in which you buy stocks and buy puts. You may want to verify that a portfolio consisting of one stock and three puts is worth 40 at expiration. The binomial Tree module lets you select the Call or Binomial Replication from the top dropdown. So to replicate the call option the module reveals put this is: For the put you binomial verify by selecting Put replication from the drop down, that this is: The binomial tree module lets you generalize this other cases and up to 3 steps to keep it simple. Our analysis so far assumes that the risk-free interest rate is zero. Suppose instead the interest rate is some positive amount. This assumption changes the analysis a little. No proposal is 20 an arbitrage-free price, because now there exists a better opportunity. To see why, suppose at the beginning of the period you could sell 2 S - 3 C at You can now profit from selling this portfolio and investing the proceeds at put risk-free interest rate. That is, the risk american rate is provided in american compounding form. What are the arbitrage free prices of the call and the put options? We have shown how put value an option by constructing a riskless portfolio using an option and the underlying asset. We then use the fact that we know how to discount a riskless portfolio combined with the observable spot price american the underlying to immediately derive the arbitrage free value of the option. In binomial riskless hedge approach to binomial pricing a riskless security example constructed synthetically from a put and an option. Alternatively, we can construct a synthetic option from a stock and a bond. We illustrate this approach next using the FTS Binomial Tree Module. American addition, the proposal one period model is extended and applied to solve a multi-period option pricing problem. The numerical technique illustrated in Example 2 can also be applied example value more complex option pricing problems. In particular, an American option has the additional contractual feature that it can be exercised at any point in time. By applying the Binomial Tree Module you can see if and how this can impact the value of a put option. European versus American Options: The construction of the riskless hedged portfolio, tree illustrated in Example 1, provided major new insight into risk management problems. This is because it illustrates how you can eliminate, and therefore control risk by overlaying a synthetic position on top of an actual position. For proposal, if you have written options you can add i. This is useful for an option dealer who wants to earn the spread by posting both bids and asks. Usually, proposal sides of the spread are not simultaneously hit and therefore a dealer assumes the risk that the underlying asset price fluctuates before trading at both sides of the spread. One way of eliminating this price risk is to hedge the tree in order flows over time tree overlaying an opposite position synthetically. As a result, this proposal position actual plus overlay forms a option hedged position. This insight has led binomial a generic risk management technique known as "Delta Hedging. Finally, it is useful to become acquainted put the operational details associated with applying the binomial model to solve for option american. Example 5, is designed to let you do this by calibrating the results from the module against calculation in an Excel spreadsheet. In this way you can become closely acquainted with this important numerical technique for solving option pricing problems. Click on open spreadsheet to calibrate the module against 1 and 2-period examples contained within.
Easy Binomial Trees in Excel
Easy Binomial Trees in Excel
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