Binomial models in finance pdf

Underlying Price Input current underlying price. Strike price Strike price at time to maturity 7. Maturity It is time to maturity in unit years. Number of Steps Choose number of steps of binomial tree 9.

Start date Input date start date of calculation. End date Input maturity date Black schools smoothing This cell shows the result of calculation using Black scholes smoothing model.

Richardson extrapolation This cell shows the result of calculation using Richardson extrapolation method with Black scholes smoothing model Tian This cell shows the result of calculation using the Tian model Call option price This cell shows calculation option price for call option using Black scholes model. It returns 0 when using with an American option. Put option price This cell shows calculation option price for put option using Black scholes model. We can also build binomial tree by this application to see the value of option and underlying price at each nodes.

To compare the accuracy of each method, we use convergence function in the application. When we click at convergence button, it will calculate the option price with different methods of calculation and number of steps.

Then, the result is shown in graph as the picture below Option price Figure 2: The convergence of each method using data from the example From the picture, It implies that Richardson extrapolation method purple line will convert to Black scholes value faster than any other method. The convergence of this method is close to Black scholes smoothing because we apply Richardson extrapolation with Black scholes smoothing.

Tian model has oscillation very huge but it converts faster than the Cox-Ross- Rubenstein model. The Cox-Ross-Rubenstein model converts slowly and has too much oscillation. In this example, we need at least steps of calculation to get the result from the Cox-Ross-Rubenstein model close to Black scholes value.

Lecture notes in Analytical Finance I. Wikipedia, Richardson Extrapolation. This is just the same as the European Call Option except that the right to buy is replace by the right to sell. Every time EXP issues bills for the export goods, it can take up to 3 months to receive payment e.

Again we have long and short contract, depending on whether one is buying or selling. Proof For the one-step binomial asset pricing model. Using equation 3.

Remark 3. Formula 3. For example, consult Anthony [2] and Jacque [40]. If this did not happen, then a U. The covered interest rate parity formula shows a relationship between exchange rates and interest rates in the two countries.

This is an arbitrage opportunity in contradiction to the axiom. This is again an arbitrage opportunity in contradiction to the axiom. Later we shall treat exchange rate futures contracts. For this we will need to introduce the concept of a margin account. Then you 3. This is the view from Canada. From the U. In addition from the U.

From United States We need to develop the U. Now all is denominated in USD, which is regarded as the domestic currency. This is a put. F USD. The quantity P t, T will be usually unknown until time t. It is the value at time t of a zero-coupon bond expiring at time T with face value 1 USD. This observation implies restrictions on the possible interest rate models in this framework. We now describe some choices. Some references for this model are Panjer et al.

References for this model include Panjer [59] and Black, Derman and Toy [7]. In Choice 1, the Ho and Lee model, k is a measure of volatility spread of the interest rates.

Here equation 3. If this contract is entered by no payment of premium, then the present value of this contract is zero. Proof of equation 3. This shows that 3. Recall that there is an inverse relationship between bond prices and interest rates.

Here are some of the details. The reverse inequality in 3. Now 0. We shall return to interest rate concepts after we have discussed multi-period models. Suppose we have a market with two tradeable assets as in the basic model of the previous chapter. Show the following: 1. This leads to an alternative calibration of the the binomial asset pricing model. Exercise 3. Study Examples 3. The two currency problems below are modelled in a one-step binomial asset pricing model.

JPY is the abbreviation for the Japanese yen. Write Rd resp. The day annual interest rates for the various currencies are 5. Suppose X 0 and J 0 were 1. Let P 0 resp. Ra Rd 3. When you have done this, verify the formula 3. Use 1. By this she hopes to transfer cash from Japanese operations to the U.

The latter will be used when we discuss path-dependent options. For now we shall consider only recombining trees, though the general methodology will be the same for both. Here we have generalized the one-step model notation. An interpretation of node n, j is the following. With recombining trees, there are many ways to reach node n, j. This requires that 0 4. Again, restrictions must apply so that 0 4. We can now use the backward induction formula 4.

R This recursion has an explicit solution, 4. RN 69 4. There are various proofs of 4. One can use the principle of mathematical induction to establish the general result 4. The result now follows from 4. It is provided by the following lemma. Lemma 4. We shall prove 4. The formula 4. Now assume that 4. In summary, 4. Corollary 4. We can simply apply Corollary 4. Therefore, a exists.

Toss a biased coin n times. Theorem 4. Remark 4. Formula 4. Recall the formulae 4. These can be compared with the formulae 3. The proof is exactly the same as in the exchange rate example. For completeness we write out some details. Then the stock In the one-step binomial model let B S S becomes the riskless asset, and the riskless asset becomes risky. The Black and Scholes formula, or perhaps better, the Black, Scholes and Merton formula, was presented in Black and Scholes [8] and Merton [53] using continuous time stochastic calculus methods.

Black died in , but M. Scholes and R. Merton received the Nobel Prize for Economics in for this work. In the s, economists were not conversant with the mathematical tools used, so Cox, Ross and Rubinstein, wrote the paper [18] in which they rederived the Black and Scholes formula as a limit from the binomial model. Show how formulae 4. Exercises 4. Exercise 4. This is also called the As You Like It option.

The chooser option is discussed in Hull, [37] pages —2. Then compute W 0, 0 by usual backwardization. You may wish to have either the option to sell your stock at T for K if events are bad, or buy at T for K if events are good. Zhang [79] has the whole of Chapter 23 on the chooser option for those who wish to read more. Let us study the call option on a call option, although there are obviously many other combinations. We again need three dates as in 4. Now compute W 0, 0 , by the usual backwardization.

Hint: Think of a way of doing both 4. A good application of compound options is described in Hull [37] on page Such products are also engineered so that their present values can be estimated and importantly the product can be hedged see Chapter 5. Some would say that hedging is even more important. Hedging is a general strategy, independent of any model. However, in this book we discuss hedging in our binomial framework. We now show how to hedge, or replicate, a claim.

We start with an amount of cash. In each time period see below we divide our wealth between an investment in a bank, with interest, and the purchase of S.

This portfolio can be adjusted at each time, without the addition of extra cash or the removal of any cash, but at expiry T, it must have value equal to the claim. We are then said to have hedged, or replicated, the claim. The initial cost of the hedge must be the present value of the claim. We now provide some details, and give some examples. Suppose that the general contingent claim that we are trying to hedge is denoted by W , and that W n, j is its value at n, j.

In fact the values of W n, j can be obtained from the values of W N,. We now write down generalizations of H0 and H1 which we introduced in Section 2. We then have from 5. The quantity H1 n, j is called the hedge ratio; formula 5.

Alternately, we can regard H1 n, j as indicating the level of exposure of W with respect to S at n, j. Also note that 5. This implies 5. The conditions 5. Example 5. There are four rows in this table. We show how to use the calculations. Increase the stockholding to 0. Decrease stockholding to 0. This is the same as the value of the call in 3, 2. As the call is short held, we may now cash settle this call.

We now have no unfunded liabilities at expiry. Thus, we have an arbitrage. In a multiperiod model we need to trade periodically as we have illustrated. Remark 5. If you write a call, then you collect W 0, 0 at time 0 and initiate the hedging procedure just described.

This is called hedging. All this works under the proviso that the model is correct. The book by Natenburg [57] is particularly good in describing similar hedging methods to lock in arbitrage opportunities. This book is popular with practitioners, particularly with market makers. Table 5. Produce the spreadsheet for Example 5. This should have a value at 0, 0 of 0 for a fair value of F 0, 0. We could be more dramatic. Jones has agreed to buy 50 hogs from Farmer Bill on a certain Friday, but Jones does not turn up.

What is Farmer Bill to do? Futures contracts are standardized by futures exchanges e. Margin Accounts Each exchange has its own rules. Each of the two counterparties opens a margin account with the exchange. To some extent a margin account is like any other bank account that earns interest, but it must contain a minimum amount.

Further, the clearing house of the exchange must have access to it, in the sense that it can add and remove amounts from it on a daily basis, as we shall explain. When the amount in the margin account gets too low, a margin call is put out, asking that the margin account be topped up. If the call is not answered, the contract is closed out. For the meantime we shall assume that no margin calls are necessary which is equivalent to all margin calls being answered.

Let us consider the long side initiated by company XYZ. Suppose XYZ opens a margin account with initial amount M 0, 0.

The exchange will specify the minimum amount that should be placed in this margin account. This could depend on the size of the contract. We now wish to determine G n, j for each n 6. Further, if the futures price G rises, then the increase is added to M and removed from L, and vice versa if the futures price drops.

The clearing house does the adding and removal of amounts. This process of adjusting the margin accounts in this way is called marking to market. The procedure of collecting and pairing variations in margin accounts is called resettlement. Remark 6. In general futures prices are not equal to forward prices, as the following example shows. Example 6. Equations 6. Let us note that the value of M 3, 1 depends on the path that led to 3,1. Analysis of a Default Suppose that the long side defaults at n, j.

Then XYZ would have received a margin call. Let us assume that XYZ ignored this call. In that case the futures contract is closed out. This is the same value in the margin account as would have been there had no default occurred.

We saw in Example 6. In this example R n, j and r n, j were state-dependent, which means that interest rates were not deterministic. Theorem 6. We say that interest rates are deterministic if R n, j and r n, j for each n do not depend on j. In that case we will write R n rather than R n, j , and r n in place of r n, j. We only need to show that when interest rates are deterministic F and G satisfy the same backwardization equation.

From 6. Show that the values obtained for F 0, 0 and G 0, 0 in Example 6. Produce a spreadsheet calculation.

Exercise 6. Recall that an American style option is one that can be exercised at any time up to and including the expiry date. These are the kind of option most frequently traded on stock exchanges. Because of the results in Section 2. When a stock pays dividends, it is often optimal to exercise the American call option early. We defer discussion of this situation until we have discussed the payment of dividends. At any time prior to expiry, an American style option can be a exercised; b sold; c held.

Example 7. Let us compute the American put option with data as in Example 5. It can also be noted that since the American put is worth more than the European put, there must have been times and corresponding states where it was optimal to exercise the American put option early.

In fact these occur at 1,0 , 2,0 and 2,1. We again use a CRR model, as in Section 4. The reader should carry out these calculations on a spreadsheet and determine the early exercise nodes. One should 7. There are basically two types of barrier options: 1. Knock-out options These options cease to exist when a barrier is struck by the underlying price.

There are four examples: down-and-out calls and puts, up-and-out calls and puts. Knock-in options These options come to existence when a barrier is struck by the underlying price. There are four examples: down-and-in calls and puts, up-and-in calls and puts. While barrier options may once have been regarded as exotic options, they are now rather commonplace. They are not traded on exchanges but in the over the counter OTC market. Barrier options often involve currencies and are issued by banks who have the technology to price and hedge them.

Barrier options were introduced because plain vanilla options are often too expensive and features of the vanilla options may not match client requirements. This will be apparent in the examples below. In fact most exotic options and various investment products are introduced because there is a demand for such products.

You will recall our earlier discussion on the reason why there is a market for call options. There is no point introducing a brown bear option if there are no buyers or sellers. We are talking here of the exchange rate that is usually quoted on the news—the indirect quote. This means that the CAD value of the import will rise. What is ABC to do? There are various solutions. However, the premium for this call option may be too large. ABC thus looks for a cheaper way to obtain a similar protection.

This corresponds to the market quote of 74 cents. If the CAD rose to 74 cents, then the option ceases to exist. This is the same as saying in direct terms: When the exchange rate goes down to 1.

After the knockout, the option no longer exists, and so there is no longer any protection against a falling dollar in market terms.

ABC will have to take the chance that if the CAD rose to 74 cents then it will not fall below 72 cents afterwards. Let us discuss some terminology. For example, you wish to forgo insurance when you do not think it will be needed.

Barrier Monitoring. This is a new feature of barrier options. The question to be answered is this: How does one determine that the barrier has been crossed? End-of-day monitoring would mean that the barrier is deemed to have been crossed if it is crossed at the end of a trading day.

However, we leave this subject to the interested reader. In our binomial models, we shall monitor at nodes in the tree. For a barrier option we need to specify which of the eight types it is, the strike price, the barrier, the expiry date and, if needed, the face value. You need to be good at computer programming to do this. This example uses the CRR model for a stock price. When the stock price reaches this level, the call option ceases to exist.

Let V denote the value of this option. The value 0 is given to such barrier nodes. So you can see the savings involved. These are formulae used in the spreadsheet, say to compute the barrier option prices. That is, you are bullish. We have already talked about leverage in an earlier lecture.

It is however cheaper to buy a knock-in call option, an up-and-in call option. It goes without saying that the premium for this barrier option will be less than that of the vanilla option. Cheaper options can provide greater leverage! Let us price this option using the same data as in Example 7. Again let V denote the value of this option. At these nodes the call option is calculated as it is for the vanilla call option. Read through these examples. Do not be too concerned if the various exchange rates are not the current rates.

Consider a U. The company is concerned that the Euro may appreciate against the U. To insure against this risk, the company could purchase an ordinary foreign currency European call option, allowing it to purchase Euros at a stated price. Over the next 51 days, it is possible that the Euro depreciates against the dollar. If this happens the company may feel it no longer wants the insurance provided by an ordinary call option. What should the company consider instead? Discuss how it works.

You are concerned that the Euro may appreciate against the USD. If it went up to 1. There are various ways of dealing with future uncertainty.

Do nothing and pay the million Euros in USD at the going exchange rate. Enter a forward contract to pay the million Euros in USD at the forward exchange rate. You will then know what you must pay in 51 days. Purchase a European call option on million Euros with strike price 1. You therefore have the right to purchase the million Euros for 1.

However the premium may be too large for you to consider this possibility. This option will be cheaper than the one in 3. If you set the barrier at 1. You will have to make a choice of where to set it. This will depend on the premium you are prepared to pay and the risks you are prepared to take. The gamble you take with this barrier option is this: If the exchange rate falls to 1.

You will have to decide based on advice whether this is a reasonable gamble. Consider now a company that expects to receive a payment of million Euros in 47 days time. The company is concerned that the Euro may depreciate against the dollar. One way of insurance would be for the company to purchase a put option on the Euro. Suppose the 7 American and Exotic Option Pricing Euro has been appreciating against the dollar and the company expects this general trend to continue.

Can the company do better than the vanilla put? Discussion In this case you are to receive million Euros in 47 days time. This will be about million USD. In this case you do not mind if the Euro rises. If it rose to 1. However, you are now concerned that the Euro will fall in value. If it fell to say 1. We could repeat the discussion of Example 7. This means that the put will not exist until the exchange rate actually falls to 1.

You are supposing this is not going to happen, but you do not want to take chances in case it does. The lower you set the barrier, the cheaper the put option is going to be. So if the exchange rate falls to 1. If it does not reach the barrier you may get only a little more than million USD.

The company faces the risk that the yen will depreciate against the dollar. Discuss this product. Discussion The domestic market is now U.

If the exchange rate goes down, the USD company will lose value. For example if the exchange rate falls to 0. Buying an up-and out put with barrier at 0. If this put option has strike rate 0. By taking this slight gamble, the U. The company is concerned that the Euro may appreciate against the dollar.

However the company has observed that the Euro has been depreciating 7. Why could the company solve its problem with the purchase of an up-and-in call option? Discussion At the moment 1 USD is about 1. If this is the case then the directly quoted rate for the Euro would be about 0. Thus 50 million Euros is about 46 million USD. The company could consider buying an ATM European call strike rate 0.

However, if the exchange rate were to rise above 0. However, if the barrier is not reached, you will pay at most That is the risk that is being taken by purchasing the cheaper option. The additional feature to be considered here is this. What should be the value of X?

Discussion Please refer to the Example 7. Let V n, j be the value of this option in state n, j. So far, these were the details provided in Section 7. Let W n, j also satisfy 7. We can modify the analysis. Verify the results in Example 7. Exercise 7.

This needs a ten-step binomial tree. Show the following results. Then use the backwardization formula for American put option valuation to obtain these inequalities for the smaller values of n. Study Section 7.

The booster option has two barriers, called L and H for Low and High. The booster option pays the holder an amount that is proportional to the time that the stock price say stays between the barriers. The same would apply to the lower barrier L. These are the most common examples. These examples need to be analyzed using non-recombining binomial trees. This causes problems as the number of states is growing exponentially. There are special techniques to help with this problem which, a good practitioner must know about, but a discussion of them is beyond the scope of this book.

Those who wish to explore this important issue further should look at J. Hull and A. White [36]. You may see now what you could do with trinomial trees. What is nice about this notation is that with 2, we can work out the past history: Start at 0, then up 1 , then down 0.

With this notion all our earlier formulas do not change much. This average will depend on the path that was followed from 0, 0 to N, j. The exchange rates are X 0.

MML may be happy with that. If however the direct exchange rate rose, then the bill may become unacceptable. Suppose that MML will accept an average exchange rate of 1.

It can be shown that the sum of the value of these four calls is greater than the cost of the Asian or average rate option. Forward and Futures Contracts. American and Exotic Option Pricing. Path-Dependent Options. The Greeks. Implied Volatility Trees. Implied Binomial Trees. Interest Rate Models. Real Options. The Binomial Distribution. An Application of Linear Programming. Volatility Estimation. Existence of a Solution. Some Generalizations. Page 1 Navigate to page number of 2.

About this book Introduction This book deals with many topics in modern financial mathematics in a way that does not use advanced mathematical tools and shows how these models can be numerically implemented in a practical way.