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However, in the present chapter we shall be concerned with a portfolio of several securities held simultaneously over a single time step. Even though an investment in either stock separately involves risk, we have reduced the overall risk to nil by splitting the investment between the two stocks. In addition to the description of a portfolio in terms of the number of shares of each security held developed in Section 4.

This means that wk is the percentage of the initial value of the portfolio invested in security number k. If the stock prices change as in Example 5. However, a small investor may have to face some restrictions on short selling.

We can see in Example 5. This indicates that the return on the portfolio should depend only on the weights w1 , w2 and the returns K1 , K2 on each of the two securities. Proposition 5. Proof Suppose that the portfolio consists of x1 shares of security 1 and x2 shares of security 2. V 0 Exercise 5. On the other hand, as will be seen below, formula 5. Theorem 5. Portfolio Management Exercise 5. Then 5. The last equality is equivalent to 5.

Figure 5. The bold segments correspond to portfolios without short selling. It follows from the proof of Proposition 5. These parametric equations describe the line in Figure 5. Portfolio Management and 5. Solving for s gives the above s0. This is illustrated in Figure 5. The bold parts of the curve correspond to portfolios with no short selling. The parameter s can be any real number whenever there are no restrictions on short selling.

Portfolios without short selling are indicated by the bold line segments. Corollary 5. The above corollary is important because it shows when it is possible to construct a portfolio with risk lower than that of any of its components. In case 1 this is possible without short selling. In case 3 this is also possible, but only if short selling is allowed. In case 2 it is impossible to construct such a portfolio.

Does this portfolio involve short selling? We conclude this section with a brief discussion of portfolios in which one of the securities is risk-free. The variance of the risky security a stock is positive, whereas that of the risk-free component a bond is zero. As usual, the bold line segment corresponds to portfolios without short selling.

The at- tainable set consists of all portfolios with weights w satisfying 5. Suppose that the returns on the securities are K1 ,. T Exercise 5. We shall solve the following two problems: 1. It will be called the minimum variance portfolio. Portfolio Management 2. Since the variance is a continuous function of the weights, bounded below by 0, the minimum clearly exists in both cases.

To this end we can use the method of Lagrange multipliers. Substituting this into con- straint 5. The asserted formula follows by substituting the solution into the expression for w. Portfolio Management From Proposition 5. The minimum variance line can be computed using Proposition 5.

Substituting these into the formula for w in Proposition 5. What are the weights in this portfolio? Also compute the expected return and standard deviation of this portfolio. Compute the weights and the standard deviation of this portfolio. One is presented in Figure 5.

Here two of the three weights, namely w2 and w3 , Figure 5. Of course any other two weights can also be used as parameters. The vertices of the triangle represent the portfolios consisting of only one of the three securities. The lines through the vertices correspond to portfolios consisting of two securities only. For example, the line through 1, 0 and 0, 1 corresponds to portfolios containing securities 2 and 3 only.

Points inside the triangle, including the boundaries, correspond to portfolios without short selling. Points outside the triangle correspond to portfolios with one or two of the three securities shorted. The minimum variance line is a straight line because of the linear dependence of the weights on the expected return. It is represented by the bold line in Figure 5. This is sometimes called the risk—expected return graph. The three points indicated in this picture correspond to portfolios consisting of only one of the three securities.

For instance, the portfolio with all funds invested in security 2 is represented by the point 0. The lines passing through a pair of these three points correspond to portfolios consisting of just two securities. These are the two-security lines studied in detail in Section 5. Portfolio Management lios containing securities 2 and 3 only lie on the line through 0. The three points and the lines passing through them correspond to the vertices of the triangle and the straight lines passing through them in Figure 5.

The shaded area both dark and light , including the boundary, represents portfolios that can be constructed from the three securities, that is, all attainable portfolios. The boundary, shown as a bold line, is the minimum variance line. The shape of it is known as the Markowitz bullet. The darker part of the shaded area corresponds to the interior of the triangle in Figure 5. Namely, the w2 , w3 plane is folded along the minimum variance line, be- ing simultaneously warped and stretched to attain the shape of the Markowitz bullet.

In other words, each point inside the shaded area in Figure 5. However, each point on the minimum variance line corresponds to a single portfolio. The minimum variance line without short selling is shown as a bold line in both plots. For comparison, the minimum variance line with short selling is shown as a broken line. Also sketch the set of all attainable portfolios with and without short selling. Portfolio Management use.

Employing the techniques of Section 5. Proof By Proposition 5. This proposition is important. It means that the minimum variance line has the same shape as the set of portfolios constructed from two securities, studied in great detail in Section 5. A necessary condition for a constrained max- imum is that the partial derivatives of F with respect to the weights should be zero. Accurate estimation of C may pose considerable problems in practice. Within the CAPM it is assumed that every investor uses the same values of expected returns, standard deviation and correlations for all securities, making investment decisions based only on these values.

The return on the risk-free security will be denoted by rF. The standard deviation is of course zero for the risk-free security. By Proposition 5. According to the assumptions of the CAPM, every rational investor will select his or her portfolio on this half-line, called the capital market line. This argument works as long as the risk-free return rF is not too high, so the upper half-line is tangent to the bullet.

If rF is too high, then the upper half-line will no longer be tangent to the bullet. Portfolio Management Figure 5. Because of this property it is called the market portfolio. In practice the market portfolio is approximated by a suitable stock exchange index. This additional return above the risk-free level provides compensation for exposure to risk. From the proof of Proposition 5.

Substituting the numerical values from Example 5. Compute the weights in the market portfolio constructed from the three securities in Exercise 5. Also compute the expected return and standard deviation of the market portfolio. In Figure 5. In what follows we discuss another interpretation of the beta factor. The market portfolio involves only this kind of risk. This interpretation of the beta factor is of crucial importance.

The weights in the market portfolio will be denoted by wM. Thus, by Proposition 5. We have proved the following remarkable property. This is shown in Figure 5. However, 5. What are the coordinates of this point? Everyone is holding a portfolio of risky securities with the same weights as the market portfolio. As a result, the demand and supply of all securities will be balanced.

This will remain so as long as the estimates of expected returns and beta factors satisfy 5. The new estimated values may no longer satisfy 5. Demand will exceed supply, the price of the security will begin to rise and the expected return will decline. In this case supply will exceed demand, the price will fall and the expected return will increase.

This will continue until the prices and with them the expected returns of all securities settle at a new level, restoring equilibrium. The above inequalities are important in practice. They send a clear signal to investors whether any particular security is underpriced or, respectively, overpriced, that is, whether it should be bought or sold. The party to the contract who agrees to sell the asset is said to be taking a short forward position. The other party, obliged to buy the asset at delivery, is said to have a long forward position.

The principal reason for entering into a forward contract is to become independent of the unknown future price of a risky asset. A forward contract is a direct agreement between two parties. It is typically settled by physical delivery of the asset on the agreed date. As an alternative, settlement may sometimes be in cash. Let us denote the time when the forward contract is exchanged by 0, the delivery time by T, and the forward price by F 0, T.

The time t market price of the underlying asset will be denoted by S t. No payment is made by either party at time 0, when the forward contract is exchanged. Figure 6. We begin with the simplest case. Stock Paying No Dividends. A typical example is a stock paying no dividends. We shall denote by r the risk-free rate under continuous compounding and assume that it is constant throughout the period in question.

An alternative to taking a long forward position in stock with delivery at time T and forward price F 0, T is to borrow S 0 dollars to buy the stock at time 0 and keep it until time T. The amount S 0 erT to be paid to settle the loan with interest at time T is a natural candidate for the forward price F 0, T. The following theorem makes this intuitive argument formal. Theorem 6. Forward and Futures Contracts Proof We shall prove formula 6. In this case we construct the opposite strategy to the one above.

The proof of 6. Exercise 6. Is there an arbitrage opportunity? Find the highest rate d for which there is no arbitrage opportunity. Remark 6. The last formula is in fact more general, requiring no assumption about con- stant interest rates. Including Dividends.

We shall generalise the formula for the forward price to cover assets that generate income during the lifetime of the forward contract. The income may be in the form of dividends or a convenience yield. We shall also cover the case when the asset involves some costs called the cost of carry , such as storage or insurance, by treating the costs as negative income. Suppose that the stock is to pay a dividend div at an intermediate time t between initiating the forward contract and delivery.

At time t the stock price will drop by the amount of the dividend paid. We shall construct an arbitrage strategy. Does either of the strategies in the proof of Proposition 6. Dividend Yield. Another example is foreign currency, attracting interest at the corre- sponding rate.

If the dividends are reinvested in the stock, then an investment in one share held at time 0 will increase to become erdiv T shares at time T. The situation is similar to continuous compounding. This observation is used in the arbitrage proof below.

Between time 0 and T collect the dividends paid continuously, reinvesting them in the stock. At time T you will have 1 share, as explained above. Between time 0 and T you will need to pay dividends to the stock owner, raising cash by shorting the stock. Your short position in stock will thus increase to 1 share at time T. What is the forward price of euros expressed in dollars that is, the forward exchange rate? As time goes by, the price of the underlying asset may change.

Along with it, the value of the forward contract will vary and will no longer be zero, in general. We shall derive formulae to capture the changes in the value of a forward contract. This is good news for an investor with a long forward position initiated at time 0. This discounted amount would be received or paid, if negative by the investor with a long position should the forward contract initiated at time 0 be closed out at time t, that is, prior to delivery T.

This intuitive argument needs to be supported by a rigorous arbitrage proof. For a stock paying no dividends formula 6. Growth above the risk-free rate results in a gain for the holder of a long forward position. The value of this contract at time t will be given by 6. Futures contracts are designed to eliminate such risk.

These prices are unknown at time 0, except for f 0, T , and we shall treat them as random variables. As in the case of a forward contract, it costs nothing to initiate a futures position. Forward and Futures Contracts if positive, or will have to pay it if negative. The opposite payments apply for a short futures position.

The following two conditions are imposed: 1. The second condition means that, in particular, it costs nothing to close, open or alter a futures position at any time step between 0 and T. Each investor entering into a futures contract has to pay a deposit, called the initial margin, which is kept by the clearing house as collateral.

The opposite amount is added or subtracted for a short futures position. Any excess that builds up above the initial margin can be withdrawn by the investor. On the other hand, if the deposit drops below a certain level, called the maintenance margin, the clearing house will issue a margin call , requesting the investor to make a payment and restore the deposit to the level of the initial margin.

A futures position can be closed at any time, in which case the deposit will be returned to the investor. In particular, the futures position will be closed immediately by the clearing house if the investor fails to respond to a margin call. As a result, the risk of default is eliminated. Example 6.

The table below shows a scenario with futures prices f n, T. At the end of the day the investor decides to close the position, collecting the balance of the deposit. This is possible due to standardisation and the presence of a clearing house. Only futures contracts with particular delivery dates are traded. Moreover, futures contracts on com- modities such as gold or timber specify standardised delivery arrangements as well as standardised physical properties of the assets.

The clearing house acts as an intermediary, matching the total of a large number of short and long futures positions of various sizes. The clearing house also maintains the margin deposit for each investor to eliminate the risk of default. This is in contrast to forward contracts negotiated directly between two parties. Let r be the risk-free rate under continuous compounding. The argument below can readily be extended to cover more frequent marking to market.

At time T close the risk-free investment, col- lecting the amount F 0, T , purchase one share for F 0, T using the forward contract, and sell the share for the market price S T. This construction cannot be performed if the interest rate changes unpre- dictably. The futures prices are random, but this is caused entirely by the randomness of the prices of the underlying asset. One relatively simple way to hedge an exposure to stock price variations is to enter a forward contract.

However, a contract of this kind may not be readily available, not to mention the risk of default. Another possibility is to hedge using the futures market, which is well-developed, liquid and protected from the risk of default. Suppose that we wish to sell the stock after 3 months. To hedge the exposure to stock price variations we enter into one short futures contract on the stock with delivery in 3 months.

The payments resulting from marking to market are invested or borrowed , attracting interest at the risk-free rate. Note that if the marking to market payments were not invested at the risk-free rate, then the realized sum would be In reality the calculations in Example 6. Some limitations come from the standardisation of futures contracts. If we want to close out our investment at the end of April, we will need to hedge with futures contracts with delivery date beyond the end of April, for example, in June.

To protect ourselves against a decrease in the asset price, at time 0 we can short a futures contract with futures price f 0, T. The price f 0, T is known at time 0, so the risk involved in the hedging position will be related to the unknown level of the basis. This uncertainty is mainly concerned with unknown future interest rates. If the goal of a hedger is to minimise risk, it may be best to use a certain optimal hedge ratio, that is to enter into N futures contracts, with N not necessarily equal to the number of shares of the underlying asset held.

Futures on Stock Index. A stock exchange index is a weighted average of a selection of stock prices with weights proportional to the market capitalisation of stocks. An index of this kind will be approximately proportional to the value of the market portfolio see Chapter 5 if the chosen set of stocks is large enough.

For the purposes of futures markets the index can be treated as a security. The futures prices f n, T , expressed in index points, are assumed to satisfy the same conditions as before. If the number of stocks included in the index is large, it is possible and convenient to assume that the index is an asset with dividends paid continuously.

Find the dividend yield. Our goal in this section is to study applications of index futures for hedging based on the Capital Asset Pricing Model introduced in Chapter 5. As we know, see 5. By V n we shall denote the value of the portfolio at the nth time step. Here we use discrete time and ordinary returns together with periodic compounding in the spirit of Portfolio Theory.

We can form a new portfolio with value V n by supplementing the original portfolio with N short futures contracts on the index with delivery time T. The initial value V 0 of the new portfolio is the same as the value V 0 of the original portfolio, since it costs nothing to initiate a futures contract. Forward and Futures Contracts Proposition 6. Corollary 6. For example, suppose that an investor is able to design a portfolio with superior average performance to that of the market.

By entering into a futures position such that the beta of the resulting portfolio is zero, the investor will be hedged against adverse movements of the market. On the other hand, should the market show some growth, the expected return on the hedged portfolio will be reduced by comparison because the futures position will result in a loss.

It needs to be emphasized that this type of hedging with futures works only on average. Let us conclude this chapter with a surprising application of index futures. However, index futures were traded. Due to the fact that one of the indices WIG20 was composed of 20 stocks only, it was possible to manufacture a short sale of any stock among those 20 by entering into a short futures position on the index, combined with purchasing a suitable portfolio of the remaining 19 stocks. With a larger number of stocks comprising the index the transaction costs would have been too high to make such a construction practicable.

Here we shall establish some fundamental properties of options, looking at them from a wider perspective and using con- tinuous time. Nevertheless, many conclusions will also be valid in discrete time. Chapter 8 will be devoted to pricing and hedging options. A European put option gives the right to sell the underlying asset for the strike price X at the exercise time T. In other words, an American option can be exercised at any time up to and including expiry.

Apart from typical assets such as stocks, commodities or foreign currency, there are options on stock indices, interest rates, or even on the snow level at a ski resort. Some underlying assets may be impossible to buy or sell. The option is then cleared in cash in a fashion which resembles settling a bet. No payment will be due if the index turns out to be lower than on the exercise date.

This explains why options are often referred to as contingent claims. This would be contrary to the No-Arbitrage Principle. The premium is the market price of the option. Establishing bounds and some general properties for option prices is the primary goal of the present chapter. The next chapter will be devoted to de- tailed techniques of computing these prices. We assume that options are freely traded, that is, can readily be bought and sold at the market price.

The same constant interest rate r will apply for lending and borrowing money without risk, and continuous compounding will be used. Example 7. Options: General Properties Exercise 7. These gains are illustrated in Figure 7. Note that the potential loss for a buyer of a call or put is always limited to the premium paid. For a writer of an option the loss can be much higher, even unbounded in the case of a call option.

Figure 7. Consider a portfolio constructed by and writing and selling one put and buying one call option, both with the same strike price X and exercise date T. This motivates the theorem below. The balance of these transactions is 0. Exercise 7. What is the price of a European put with the same strike price and exercise date? Find an arbitrage opportunity. Remark 7. Because of this, one could argue that put and call prices should be independent of the expected return on stock.

We shall see that this is indeed the case once the Black—Scholes formula is derived for call and put options in Chapter 8. Formula 7. For American options put-call parity gives only an estimate, rather than a strict equality involving put and call prices. Theorem 7. In this case we can write and sell a put, buy a call and sell short one share, investing the balance on the money market. The theorem, therefore, holds by the No-Arbitrage Principle.

They hold because an American option gives at least the same rights as the corresponding European option. Nevertheless, it does not necessarily follow that the inequalities in 7. Options: General Properties Figure 7. Similar inequalities are of course valid for the more valuable American options. In what follows we are going to discuss some further simple bounds for the prices of European and American options.

The advantage of such bounds is that they are universal. They are independent of any particular model of stock prices and follow from the No-Arbitrage Principle alone. On the exercise date T we could then sell the stock for min S T , X , settling the call option. These results are summarised in the following proposition and illustrated in Figure 7. The shaded areas correspond to option prices that satisfy the bounds. The choice of 0 as the starting time is of course arbitrary.

This means that the American option will in fact never be exercised prior to expiry. This being so, it should be equivalent to the European option. In particular, their prices should be equal, leading to the following theorem. Then, at time T you can use the European call to buy a share for X and close your short position in stock.

It might, therefore, appear that the American call option should be more valuable than the European one. Nevertheless, there is no contradiction. Example 8. On the other hand, it often happens that an American put should be ex- ercised prematurely even if the underlying stock pays no dividends, as in the following example.

The value of a put option cannot possibly exceed the strike price, see 7. This gives a sharper lower bound than that for a European put. However, the upper bound has to be relaxed as compared to a European put. These results can be summarised as follows. Proposition 7. Next we consider options on dividend-paying stock. These can be variables characterising the option, such as the strike price X or expiry time T , variables describing the underlying asset, for example, the current price S 0 or dividend rate rdiv , variables connected with the market as a whole such as the risk-free rate r, and of course the running time t.

We shall analyse option prices as functions of one of the variables, keeping the remaining variables constant. These inequalities are obvious. The right to buy at a lower price is more valuable than the right to buy at a higher price. Similarly, it is better to sell an asset at a higher price than at a lower one. Proof By put-call parity 7. Since, by Proposition 7.

This is illustrated in Figure 7. Alternatively, an arbitrage argument can be given along similar lines as for call options. Geometrically, this means that if two points on the graph of the function are joined with a straight line, then the graph of the function between the two points will lie below the line. The current price S 0 of the underlying asset is given by the market and cannot be altered. We shall study the dependence of option prices on S.

Options: General Properties Remark 7. The inequality for puts follows by a similar arbitrage argument. Both terms on the left-hand side are non-negative by the previous theorem, so each is strictly less than the right-hand side. In other words, the call and Figure 7.

This means that the call and put prices are convex functions of S. To cover this liability we can exercise the other options. The inequality for put options can be proved by a similar arbitrage argument or using put-call parity. In addition, we have to take into account the possi- bility of early exercise. Dependence on the Strike Price.

The following proposition is obvious for the same reasons as for European options: Higher strike price makes the right to buy less valuable and the right to sell more valuable. The proof is similar for put options. Once again, we shall con- sider options on a portfolio of x shares. As a result, this strategy will provide an arbitrage opportunity.

Proof By the inequalities in Theorem 7. Dependence on the Expiry Time. For American options we can also for- mulate a general result on the dependence of their prices on the expiry time T. The argument is the same for puts. A European option in the money is no more than a promising asset. We can see that the intrinsic value is zero for options out of the money or at the money. Options in the money have positive intrinsic value.

The price of an American option prior to expiry may be greater than the intrinsic value because of the possibility of future gains. The price of a European option prior to the exercise time may be greater or smaller than the intrinsic value. The option price must be at least equal to the intrinsic value, since the option may be exercised immediately.

Typically, the price will be higher than the intrinsic value because of the possibility of future gains. The positive price of the put is entirely due to the possibility of future gains. Similar relationships for other strike prices can be seen in the table. The time value of a European call as a function of S is shown in Figure 7. The market value of a European put may be lower than its intrinsic value, that is, the time value may be negative, see Figure 7.

The risk that the stock price will rise above X in the meantime may be considerable, which reduces the value of the option. For an American put a typical graph of the time value is shown in Figure 7. Proof We shall present an argument for European calls. The proof for other options is similar. We have already learnt the basic method of pricing options in the one-step model see Section 1.

Not surprisingly, this idea extends to a general binomial tree model constructed out of such one- step two-state building blocks. Developing this extension will be our primary task in this chapter. Theorem 8. Replication also solves the problem of hedging the position of the option writer. If the cash received for the option is invested in the replicating strategy, then all the risk involved in writing the option will be eliminated.

In this chapter we shall gradually develop such pricing methods for options, starting with a comprehensive analysis of the one-step binomial model, which will then be extended to a multi-step model. Finally, the Black—Scholes formula in continuous time will be introduced. Here we shall reiterate the ideas in a more general setting: We shall be pricing general derivative securities and not just call or put options.

This will enable us to extend the approach to the multi-step model. By Theorem 8. Analyse the impact of a change of d on the option price. Exercise 8. Draw a conclusion about the relationship between the variance of the return on stock and that on the option. No transaction costs apply when the stock is bought. Compare this value with the case free of such costs.

Find the values of the replicating portfolios for a put and a call. Is the answer consistent with the put and call prices following from Theorem 8. Option Pricing Figure 8. This means that the one-step procedure can be applied once again to the single subtree at the root of the tree. This proves the following result. Find the option price and the replicating strategy. The emerging pattern is this: Each term in the square bracket is characterised by the number k of upward stock price movements.

For example, there are three paths through the 3-step tree leading to the node udd. The result can be summarised as follows. Remark 8. Option Pricing Theorem 8. At which step will the delta of a European call become 1? Of course, it can be exercised only once. The price of an American option at time n will be denoted by DA n. The stock values are n 0 1 2 The option holder will choose the higher value exercising the option in the down state at time 1.

This gives the time 1 values of the American put, n 0 1 2 0. Taking the higher of the two completes the tree of option prices, n 0 1 2 0. Suppose that a dividend of 14 dollars is paid at time 2. Option Pricing Exercise 8. The ex-dividend stock prices are n 0 1 2 As a result, the price of the American call is higher than that of the European call.

In place of this, we shall exploit an analogy with the discrete time case. As a starting point we take the continuous time model of stock prices de- veloped in Chapter 3 as a limit of suitably scaled binomial models with time steps going to zero. This means, in particular, that S t has the log normal distribution. As in the discrete-time case, see Theorem 8. Here we shall accept this formula without proof, by analogy with the discrete time result.

The proof is based on an arbitrage argument combined with a bit of Stochastic Calculus, the latter beyond the scope of this book. Let us compute this expectation using the real market probability P. So far we have dealt with conditional expectation where the condition was given in terms of a discrete random variable, see Section 3. Here, however, the condition is expressed in terms of S u , a random variable with continuous distribution.

In this case the precise mathematical meaning of 8. Now we shall con- sider a European call option on the stock with strike price X to be exercised at time T. The general formula 8. What we have just derived is the celebrated Black—Scholes formula for Eu- ropean call options. The choice of time 0 to compute the price of the option is arbitrary. It is a property anal- ogous to that in Remark 8.

It is interesting to compare the Black—Scholes formula for the price of a European call with the Cox—Ross—Rubinstein formula. There is close analogy between the terms. The precise relationship comes from a version of the Central Limit Theorem: It can be shown that the option price given by the Cox—Ross— Rubinstein formula tends to that in the Black—Scholes formula in the continuous time limit described in Chapter 3.

Figure 8. Though m is irrelevant for the Black—Scholes formula, it still features in the discrete time approximation. Even with as few as 10 steps there is remarkably good agreement between the discrete and continuous time formulae. The presentation will be by means of exam- ples and mini case studies. First, we shall present methods for eliminating or reducing the risk involved in writing options.

Next, we shall analyse methods of reducing undesirable risk stemming from certain business activities. Our case studies will be concerned with foreign ex- change risk. It is possible to deal in a similar way with the risk resulting from un- expected future changes of various market variables such as commodity prices, interest rates or stock prices.

We shall introduce a measure of risk called Value at Risk VaR , which has recently become very popular. Derivative securities will be used to design portfolios with a view to reducing this kind of risk. Finally, we shall consider an application of options to manufacturing a lev- ered investment, for which increased risk will be accompanied by high expected return.

Theoretically, the loss to the writer may be unlimited. We shall see how to eliminate or at least reduce this risk over a short time horizon by taking a suitable position in the underlying asset and, if necessary, also in other derivative securities written on the same asset. In practice it is impossible to hedge in a perfect way by designing a single portfolio to be held for the whole period up to the exercise time T. Nevertheless, here we shall only discuss hedging over a single short time interval, neglecting transaction costs.

This can be seen in a slightly broader context. Its dependence on S can be measured d by the derivative dS V S , called the delta of the portfolio. This can be achieved by ensuring that the delta of the portfolio is equal to zero. Such a portfolio is called delta neutral. Proposition 9. Exercise 9. For the remainder of this section we shall consider a European call option within the Black—Scholes model.

By Proposition 9. It is convenient to choose y so that the initial value of the portfolio is equal to zero. To construct the hedge we buy We shall analyse the value of the portfolio after one day by considering some possible scenarios. The time to expiry will then be 89 days. Suppose that the stock volatility and the risk-free rate do not vary, and consider the following three scenarios of stock price movements: 1 1.

Our debt on the money market is increased by the interest due. The position in stock is worth the same as initially. The balance on day one is stock 34, On the other hand, for a holder of a delta neutral portfolio the loss on the options is almost completely balanced out by the increase in the stock value: stock 35, Financial Engineering 1 3. The value of the stock held decreases too. The portfolio brings a small loss: stock 34, As we shall see later in Exercise 9. Going back to our example, let us collect the values V of the delta neutral portfolio for various stock prices after one day as compared to the values U of the unhedged position: S V U If we do not hedge, at least we have a gamble with a positive outcome whenever the stock price goes down.

Meanwhile, no matter whether the stock price goes up or down, the delta neutral portfolio may bring losses, though considerably smaller than the naked position. Let us see what can happen if some other variables, in addition to the stock price, change after one day: 1. Some loss will result from an increase in the option value. The values of the hedging portfolio are given in the second column in the table below. The option price will increase considerably, which is not compensated by the stock position even if the stock price goes up.

Financial Engineering As we can see, in some circumstances delta hedging may be far from satis- factory. In what follows, after introducing some theoretical tools we shall return again to the current example. For instance, to hedge against volatility movements we should construct a vega neu- tral portfolio, with vega equal to zero.

A delta-gamma neutral portfolio will be immune against larger changes of the stock price. Examples of such hedging portfolios will be exam- ined below. The Black—Scholes formula allows us to compute the derivatives explicitly for a single option. Remark 9. Financial Engineering 9. Delta-Gamma Hedging. The construction is based on making both delta and gamma zero.

This allows us to make the delta of the portfolio zero. To make the gamma also equal to zero an additional degree of freedom is needed. That is, we take long positions in stock and the additional option, and a short cash position.

The reverse happens if the stock price declines. The values of the portfolio are given below for comparison we also recall the values of the delta neutral portfolio : 1 S delta-gamma delta Delta-Vega Hedging. Next we shall hedge against an increase in volatility, while retaining cover against small changes in the stock price. This will be achieved by constructing a delta-vega neutral portfolio containing, as before, an additional option.

The examples above illustrate the variety of possible hedging strategies. The choice between them depends on individual aims and preferences. We have not touched upon questions related to transaction costs or long term hedging. Nor have we discussed the optimality of the choice of an additional derivative instru- ment.

Portfolios based on three Greek parameters would require yet another derivative security as a component. They could provide comprehensive cover, though their performance might deteriorate if the variables remain unchanged.

In addition, they might prove expensive if transaction costs were included. The selling price S 1 is random. Example 9. Here N x is the normal distribution function 8. Financial Engineering Exercise 9. The methods will be illustrated by a simple example of business activity. Case 9. The investment to start production is 5 million pounds. The sales are predicted to generate 8 million dollars at the end of the year. The manufacturing costs are 3 million pounds per year. The current rate of exchange is 1.

We assume that the other values will be as predicted. To begin with, suppose that no action is taken to manage the risk. Unhedged Position. If the exchange rate d turns out to be 1. However, if the exchange rate d becomes 2 dollars to a pound, the com- pany will end up with a loss of 0.

In an optimistic scenario in which the pound weakens, for example, down to 1. The question is how to manage this risk exposure. Forward Contract. The forward rate is 1. Full Hedge with Options. However, this may be costly because of the premium paid for options. The company can buy call options on the exchange rate. A European call to buy one pound with strike price 1. The interest is tax deductible, making the loan less costly.

If the exchange rate drops to 1. This strategy leads to a better result than the hedge involving a forward contract only if the rate of exchange drops below 1. Partial Hedge with Options. To reduce the cost of options the company can hedge partially by buying call options to cover only a fraction of the dollar amount from sales. Suppose that the company buys 2, , units of the same call option as above, paying a half of the previous premium.

A half of the revenue is then exposed to risk. Financial Engineering 5. Combination of Options and Forward Contracts. Finally, let us inves- tigate what happens if the company hedges with both kinds of derivatives. Half of their position will be hedged with options. A book like this isn't something you would want to curl up on the couch and read for enjoyment.

This is a reference guide, and anything that can help it be less boring is a plus. Cross- references in the front of this book make it easy to look up formulas. Along with a listing of the chapters of the book, there is another table of contents that lists the pages to find a specific formula.

Most people who buy this book will use it as a reference for looking up formulas, so this cross- reference in the front is helpful. One thing that will probably make many people hesitate to purchase this book is the high price tag. The book is only pages in length, so I am not sure why the cost is so much.

With a price this high, many potential buyers might want to forgo buying it new and seek out a used copy instead. You can save yourself a lot of money by purchasing this book used. Computing rates of return, investment growth, the effects of taxes and inflation, and other financial measurements takes some time and effort. With this book, a novice can gain a better understanding of how different financial computations are made. It's not a perfect book, and I did find a few typos as I thumbed through the pages.

But it does provide a decent starting point to understanding investment computations. This book has many errors, misconceptions, and shows a lack of understanding of the time value of money principles. Thomsett uses the erroneous sinking fund formula for annuity due problems. He thinks the internal rate of return IRR is a method that is inexact and requires many assumptions when in fact it is a staple of the financial community that is precise, there are no assumptions, and it is a valuable tool.

He simply does not understand the internal rate of return method. His method of calculating the rate of return of mutual funds is a erroneous cook book method, and his example of how to calculating the rate of return of a mutual fund is a factor of 10 in error. Anyone reading this book will not learn the subject, and should be totally confused by the erroneous approach and examples provided by the book.

It is amazing that any publisher would publish this book. Great reference book. See all reviews. Customers who bought this item also bought. James Altucher. Only 2 left in stock - order soon. There's a problem loading this menu right now. Learn more about Amazon Prime. Get free delivery with Amazon Prime. Back to top. Get to Know Us. Amazon Payment Products.

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Louis This book is an investigation of the interrelationships between mathematics and music, reviewing the needed concepts in each subject as they are encountered. Along the way, readers will augment their understanding of both mathematics and music. Bass Lecture notes on mathematical finance - figuring out the price of options and derivatives. The text civers elementary probability, the binomial asset pricing model, advanced probability, the continuous model, and term structure models.

Powers - University of Notre Dame Multidimensional calculus, linear analysis, linear operators, vector algebra, ordinary differential equations. Directed at first year graduate students in engineering and undergraduates who wish to become better prepared for graduate studies. Uploaded by ia-mario on December 7, Search icon An illustration of a magnifying glass. User icon An illustration of a person's head and chest. Sign up Log in. Web icon An illustration of a computer application window Wayback Machine Texts icon An illustration of an open book.

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Download or read it online for free here: Download link multiple formats. Louis This book is an investigation of the interrelationships between mathematics and music, reviewing the needed concepts in each subject as they are encountered. Along the way, readers will augment their understanding of both mathematics and music.

Bass Lecture notes on mathematical finance - figuring out the price of options and derivatives. The text civers elementary probability, the binomial asset pricing model, advanced probability, the continuous model, and term structure models. Powers - University of Notre Dame Multidimensional calculus, linear analysis, linear operators, vector algebra, ordinary differential equations. See what's new with book lending at the Internet Archive. Uploaded by ia-mario on December 7, Search icon An illustration of a magnifying glass.

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He has published dozens of books and written hundreds of articles about options, stock investing, candlestick charting, and technical analysis. His popular blogs reach millions of readers daily. Product details Item Weight : 1.

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Verified Purchase. Mathematics of Investing is a reference guide intended to assist people in understanding how various financial calculations are made. The guide has chapters on such topics as time value of money, taxes and inflation, options trading, mutual funds, and others. Graphs, tables, and other illustrations accompany a good number of the pages in this guide.

This is helpful, I think, because it makes some of the terminology easier to understand and it makes the book a little more interesting than it would otherwise be. A book like this isn't something you would want to curl up on the couch and read for enjoyment. This is a reference guide, and anything that can help it be less boring is a plus. Cross- references in the front of this book make it easy to look up formulas. Along with a listing of the chapters of the book, there is another table of contents that lists the pages to find a specific formula.

Most people who buy this book will use it as a reference for looking up formulas, so this cross- reference in the front is helpful. One thing that will probably make many people hesitate to purchase this book is the high price tag. The book is only pages in length, so I am not sure why the cost is so much. With a price this high, many potential buyers might want to forgo buying it new and seek out a used copy instead.

You can save yourself a lot of money by purchasing this book used. Computing rates of return, investment growth, the effects of taxes and inflation, and other financial measurements takes some time and effort. With this book, a novice can gain a better understanding of how different financial computations are made. It's not a perfect book, and I did find a few typos as I thumbed through the pages. But it does provide a decent starting point to understanding investment computations.

This book has many errors, misconceptions, and shows a lack of understanding of the time value of money principles. Thomsett uses the erroneous sinking fund formula for annuity due problems. He thinks the internal rate of return IRR is a method that is inexact and requires many assumptions when in fact it is a staple of the financial community that is precise, there are no assumptions, and it is a valuable tool.

He simply does not understand the internal rate of return method. His method of calculating the rate of return of mutual funds is a erroneous cook book method, and his example of how to calculating the rate of return of a mutual fund is a factor of 10 in error. Anyone reading this book will not learn the subject, and should be totally confused by the erroneous approach and examples provided by the book. It is amazing that any publisher would publish this book. Great reference book.

See all reviews. Customers who bought this item also bought. James Altucher. Only 2 left in stock - order soon. There's a problem loading this menu right now. Learn more about Amazon Prime. Get free delivery with Amazon Prime.

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It emphasizes underlying mathematical motivation, applied perspective of the engineer. Reviewer: iliana1 - favorite favorite favorite favorite favorite - *1e12 mathematics of investment* the mathematics of investment nvjkhfhufcvkhg. PARAGRAPHImages Donate icon An illustration of a heart shape Donate Ellipses icon An illustration of text ellipses. The Mathematics of Investment Item. Reviewer: bukaw - favorite favorite - June 25, Subject: df. Reviewer: mong - favorite favorite. Reviewer: xar mei - favorite book deals with applied mathematical. Mathematical results are derived from without full mathematical rigor. Advanced embedding details, examples, and help. Download or read it online favorite - November 6, Subject: 30, Subject: : so helpful.