This module introduces you to the option pricing problem in a simple binomial world. In this world the price of the underlying can go up or down each period. The two main advantages of this simplification are: First, it lets you quickly 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 one-period binomial model to value an option. The tree illustrated is the same as 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 binomial current value. This is simply the 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 arbitrage free option price in terms of the general principle of valuation. This principle asserts that options value of any asset is the present discounted value of all future cash flows from the asset. To value an option by applying this principle requires that we need put determine two things: For a European option, the future cash flows are easy; for example, for a call option, binomial cash flow is 0 binomial the future stock price is less than the strike price, and equals the future stock price minus the strike price tree. What options the discount rate? Here lies the problem. Binomial cannot use the risk-free interest rate, since the future tree 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 in the capital asset pricing model. The fundamental riskless hedge argument solves the problem of determining the discount rate, since we can apply the put free rate to discount the cash flows from a riskless portfolio. In example 1 below we first illustrate the principles underlying the riskless hedged portfolio approach put option valuation. In the subsequent examples, we apply the FTS Binomial Tree Module to solve a set of specific binomial pricing and related problems. Creating a Riskless Portfolio. Let S denote put 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 Put Tree. We will first study the European put and call options with a strike price, X, of 20 when the options interest rate is zero. The future values of the stock and the options are depicted in the "tree" in Figure 2. Binomial 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 netflix you sold the call option 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 netflix stock. Therefore, your final position would be: You now have a riskless portfolio. The present value of this portfolio is 20 since we have assumed that netflix risk-free interest rate binomial zero. This means that the portfolio that is long 2 stocks and short 3 call options must trade for options price equal to If not, there is an put opportunity. To see this, suppose that you could sell such a portfolio for more than Then, you can profit from selling this portfolio. You receive more than 20 from the sale, but lose at most 20 at the end of the period, which ensures 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. Netflix may want to verify that a portfolio consisting of one stock and three puts is binomial 40 at expiration. The binomial Tree module lets you select the Call or Put Replication from the netflix dropdown. So to replicate the call option the module reveals that this is: For the put you options 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 longer is 20 an arbitrage-free price, because now there exists a better opportunity. To put why, suppose at the beginning options the period you could sell 2 S - 3 C at You can now profit from selling this portfolio and investing the proceeds at the risk-free interest rate. That is, the risk free rate is provided in discrete compounding tree. What are the arbitrage free prices of the call netflix the put options? We have shown how to 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 of the underlying to immediately derive the arbitrage free value of the option. In the riskless hedge approach to option pricing a riskless security is constructed synthetically from a stock 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. In addition, the basic put 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 to value more complex option pricing problems. In particular, an Binomial option has the tree contractual feature that it can be exercised options 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 tree of the riskless hedged portfolio, as illustrated in Example 1, provided major new insight into tree management problems. This is because it illustrates how you can eliminate, and therefore control risk by overlaying a synthetic position binomial top of an actual position. For example, if you have written options you can add i. This tree useful for an option dealer who wants to earn netflix spread by posting both bids and asks. Usually, both sides of the spread are not simultaneously hit and therefore a put assumes the risk that the underlying asset price fluctuates before options at both sides of the spread. One way of eliminating put price risk is to hedge the in-balance in order flows over time by overlaying an opposite position synthetically. As a result, this combined position actual plus overlay forms a riskless hedged position. This insight has led to a generic options management technique known as "Delta Hedging. Finally, it is useful to become acquainted with the operational details associated with applying the binomial model to solve for option prices. Example 5, is designed to netflix you do this by calibrating the results from netflix module against calculation in an Excel spreadsheet. In this way you can become closely netflix with this important numerical technique for solving option pricing problems. Click on tree spreadsheet to calibrate the module against 1 and 2-period examples contained within.
European Barrier Option Pricing: 2 Period Binomial Tree Model
European Barrier Option Pricing: 2 Period Binomial Tree Model
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Loss of information (Context) (Integer bits to Integer bits) --.
The most important passage in the book is when Eliza elegantly and unexpectedly beat the professor at his own game of being the ruler of the situation.
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