## Book solution "Fundamentals of Futures and Options Markets", Hull John - ch 3

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Global Financial Markets - Update A guide to the workings of the world's currency, money and capital, commodities and derivatives markets. It also contains information on developments and research in the international financial markets that might be of interest to students and professionals.

Main Contents Page Currency futures and options markets chapter 8 to the top of this section. Evidence from Metals Markets" by A. It is often difficult, especially in the context of a corporation's complex international transactions, to separate hedging from speculative transactions in the foreign exchange markets.

The action is speculative--it is precipitated by an individual's forecast of the yen's direction, not purely by the company's natural business. In short, the company, like most, is using its foreign-currency receivables as an excuse to take a view on a currency in the hope of profiting. These were accumulated by the firm's treasury department, apparently without authorization.

The contracts, taken out in and subsequently rolled over, bought the dollar forward at an average exchange rate of Y, to which level the yen had briefly weakened that year. At the end ofthe yen was trading at Y per dollar. In Showa Shell's case, because the losses were "unrealized," i.

Banks in Japan routinely allowed their counterparties to defer settlement of lossmaking contracts by rolling them over until they were advised by the ministry of finance to desist. A fractal approach is used to analyze financial time series, applying different degrees of time resolution, and the results are interrelated. Some fractal properties of foreign exchange FX data are found.

In particular, the mean size of absolute values of price changes follows a "fractal" scaling law a power law as a function of the analysis time interval ranging from a few minutes up to a year. In an autocorrelation study of intraday data, the absolute values of price changes are seen to behave like the fractional noise of Mandelbrot and Van Ness rather than those of a GARCH process. Intraday FX data exhibit strong seasonal and autoregressive heteroskedasticity.

This can be modeled with the help of new time scales, one of which is termed intrinsic time. These time scales are successfully applied **currency futures and options markets chapter 8** a forecasting model with a fractal structure for both the FX and interbank interest rates, which present similar market structures as the Foreign Exchange. The goal of this paper is to demonstrate how the analysis of high-frequency data and currency futures and options markets chapter 8 finding of fractal properties lead to the hypothesis of a heterogeneous market, where different market participants analyze past events and news with different time horizons.

This hypothesis is further supported by the success of trading models with different dealing frequencies and risk profiles. Intrinsic time is proposed for modeling the frame of reference of each component of a heterogeneous market.

In mid the Philadelphia Stock Exchange, the self-styled "world's largest organized market for currency options," announced that it would henceforth offer market participants tailor-made currency futures and options markets chapter 8 and hedging tools, but with all the safeguards of a regulated exchange.

In its first phase, the market offered customizable strike prices, a choice of matching any existing currency pairs, and inverse European-quoted contracts. This is an example of how futures and options exchanges are attempting to market themselves more currency futures and options markets chapter 8 as regulators cast a negative light on the over-the-counter derivatives market. Quanto options, mentioned on page in Chapter 8, sound exotic. They give the right to buy or sell an unknown quantity of foreign currency, where the quantity depends upon some market variable such as equity prices or interest rates.

In fact, they have some very down-to-earth applications. Quanto options can be useful to international equity portfolio managers who wish to take a view on share prices in a foreign market, but not on the exchange rate.

Such a portfolio manager might invest in a foreign market, purchasing shares valued at X units of the foreign currency. One way to hedge this is by selling forward X units of the currency. To solve this, he can buy a quanto option whose payoff depends on both the exchange rate and Y. A quanto option could be useful to a corporation that plans to buy a fixed amount of a commodity in a foreign market at a certain date in the future.

For example, a Japanese utility may anticipate purchasing 20 million barrels of oil for the winter months. Instead of hedging by buying a fixed dollar amount forward, the utility could employ a quanto option where the number of dollars purchased at a fixed exchange rate depends on the price of oil.

Quanto options are needed to hedge the risks entailed in writing a differential swap. Banks that write diff swaps guarantee to exchange the interest rate in one currency for that in another, but the exchange is all done in one currency. For example, NatWest may agree to pay or receivein dollars, the difference between US dollar and sterling Libor every six months for two years.

Hedging this requires NatWest to engage in separate interest rate swaps in the dollar and sterling markets, and to use a quanto option to hedge the dollar-sterling exchange rate risk. The quanto would assure NatWest of the dollar-sterling rate at which sterling Libor could be exchanged, whatever the level of sterling Libor.

Some readers have pointed out that the section on page entitled "Hedging Options with Futures Versus Hedging Currency futures and options markets chapter 8 with Options" is a little hard to follow. Even the title is a mouthful. I thought I'd amplify the ideas with an example and three diagrams. Suppose we are FX options traders at a bank.

In response to a customer's request, we sell at-the-money call options on sterling. We must now hedge the position. From the slope of the options price line see Fig. Hence to hedge the option position with futures contracts, we buy 50 sterling futures. We are now delta-hedged. As the diagram below shows, hedging options with futures entails hedging with a number of futures determined by the slope of, or tangent to, the options price line. But this slope changes whenever the underlying currency's value changes.

In our example, when the price of sterling moves up, currency futures and options markets chapter 8 options writer is underhedged the slope has increasedso we lose more on our short option position than we gain on our futures position.

When sterling falls, the options writer is overhedged, so we gain less on our options position than we lose on the futures. The reason for our problem is simple. We are using an instrument whose price line is linear to hedge one whose price line is curved. The degree currency futures and options markets chapter 8 curvature is measured by the option's gammawhich is the second derivative of the price with respect to the price of the underlying instrument.

The solution is to use options to currency futures and options markets chapter 8 options: The proportion of one option we use to hedge another option is determined by the relation between the deltas, the hedge ratios. Even so, the hedging-gap cannot be zero unless the curvature or gamma of the options we've bought, and those that we've sold, are identical.

Being delta-neutral is not enough. Normally the gamma of the one we've bought is greater than the ones we've sold we are "long gamma"or vice-versa "short gamma". Gamma is a desirable property for options one owns, and undesirable for options one has written. For options one owns, the more the curvature the more losses are cushioned and gains accelerated.

As the diagram below shows, being short gamma but delta neutral gives losses for large upside and downside moves. It's a little like selling a straddle. The valuation of many types of financial contracts and contingent claim agreements is complicated by the possibility that one party will default on their contractual obligations. This paper develops a general model that prices Black-Scholes options subject to intertemporal default risk.

The explicit closed-form solution is obtained by generalizing the reflection principle to k-space to determine the appropriate transition density function. The European analytical valuation formula has a straightforward economic interpretation and preserves much of the intuitive appeal of the traditional Black-Scholes model.

The hedging properties of this model are compared and contrasted with the default-free model. The model is extended to include partial recoveries. In one situation, the option holder is assumed to recover alpha a constant percent of the value of the writer's assets at the time of default. This version of the partial recovery option leads to an analytical valuation formula for a first passage option - an option with a random payoff at a random time.

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