In the previous chapter we showed how central to a borrowing activity (particularly in the context of development banking) is the simultaneous trading of a bond and a swap where one leg of the swap mimics the bond. We have seen that the remaining leg of the swap is a vanilla leg (floating rate plus asset swap spread), which means that as far as the complex structure in the bond and the swap is concerned we have two opposite positions, short in the bond and long in the swap. Because of this we are protected against the risks originating from the complexity of the structure. This is a form of hedging in which we replicate, by copying it exactly in the swap leg, the bond's structure. The fact that we hedge our position is actually very important when it comes to assessing its value, and to understand this better we are going to give a brief overview of the concept of replication using as an example, for their simplicity, equity options.

In finance an assumption that applies to the near totality of situations is that when we assess the value of a financial instrument we do so within a risk-neutral framework. Risk neutrality, as the name implies, means that, when assessing the value of a financial instrument, we are indifferent to risk.

Although it is not always obvious to say whether risk neutrality is at the source of hedging or vice versa, one can say that there are some fundamental concepts that support both. Crucial is the absence of arbitrage. We have already mentioned this in Section 2.2.4, but the absence of arbitrage means the impossibility of making a riskless profit. This in turn hinges on the assumption that information travels freely in the marketplace (everyone has the same level of knowledge) and on the assumption that everyone has the same tools, including speed, with which to act upon this information. Knowledge and tools mean that, should there be the possibility of making a riskless profit, everyone would know about it and acting on it would immediately erase this possibility.^{[1]}

Absence of arbitrage is a requirement, replication is what makes risk neutrality what it is and one cannot discuss replication without mentioning hedging. Hedging can be described as taking opposite positions in two similar financial instruments or assets: this definition is left vague on purpose and much depends on the meaning of similar.

In its simplest form hedging is a form of protection, a known curtailing of one's profit for the sake of protecting oneself against potential losses. Cases of it have been known to take place in ancient times. As hedging became more sophisticated, also thanks to advances in the divulgation of financial information, Black and Scholes [14] built a theory out of it, a theory based on replication. After Black and Scholes' rigorous construction, hedging became even more precise and frequent.

The biggest difference between classical physics and finance is that in the latter the action of the user has a direct impact on the variables and the model itself. To put it bluntly, a financial model is only valid if people use it. This means that the activity of hedging gave birth to a theory that prompted even more precise hedging, which in turn gave further backing to the theory itself. Tet us be more specific.

Central to Black and Scholes' theory is that if we sell an option with value V that depends on an underlying S, by building a portfolio

(7.1)

in which next to the selling of V we purchase an amount of S as dictated by the quantity A, we are indifferent to risk and this enables us to find the risk-neutral price of V. Since these are the foundations of finance, volumes have been written on it (to this day one of the clearest descriptions is the one given by Baxter and Rennie [10]). It is probably useful, however, to remind the reader that the act of replication exemplified by Equation 7.1 has a profound impact on the probabilities governing what, after all, are events driven by stochastic variables (to this end in Appendix F we remind briefly how replication leads to probability values). These probabilities are risk- neutral probabilities as opposed to real-world probabilities, which govern nonfinancial events.

Making a very general and approximate statement, the value V of an instrument offering a payout C in case some events X take place, can be written as

(7.2)

where P(X) is the probability that the event takes place. The loss of a suitcase is a stochastic event as much as the behavior of a Ford Motor share.

The probability of the former event, however, which is behind the premium charged by an insurance company, is very much different in nature from the one driving the latter and as a consequence the value of an option written on the share.

An insurance premium is driven by probabilities belonging to the real- world measure, an option price is assessed using probabilities from the risk- neutral measure.

While we can buy and sell shares in Ford Motor according to the Δ in Equation 7.1, we cannot do the same with suitcases: this fact gives rise^{[2]} to two different types of probabilities. This means that the value of an instrument, such as an option, offering protection against an event, although it can roughly be calculated with Equation 7.2, will be driven by very different P (X) depending on whether we can purchase the underlying or not.

The assumption of risk neutrality is so general that it often is applied to situations that, under closer scrutiny, should instead be exempted. Let us observe a few examples.

Let us imagine that bank ABC trades a vanilla option written on Ford Motor stock with bank XYZ. Both will use Black and Scholes and they will more or less agree on the value of the option. Let us imagine that ABC also sells to retail clients (the Italian old ladies or the Mrs. Watanabe of retail banking lore) a structured product with an embedded option that is identical in structure to the one just traded with XYZ. A structured product with this embedded option can be thought of as a bond, such as the one discussed in Section 6.2.1, with Fj a vanilla equity option on Ford Motor.

The value of the option by itself within the bond (i.e., the value of Fj) will be priced and sold in an identical way to the one traded with XYZ, and the retail client will accept the price. Is this correct? Both ABC and XYZ are able to trade the underlying whereas, no matter how much financial prowess one attributes to Mrs. Watanabe, the retail client cannot. Since we have said that it is the act of replication that drives the probabilities, which in turn drive the option price, the investment bank and the retail client should envisage two different prices so the answer to the question is no. This in practice does not matter since the retail client is ready to purchase the structured product anyway and to view essentially this investment equal to the purchase of a lottery ticket.^{[3]}

A similar concern surrounds options written on underlyings difficult to purchase: should they be priced within a risk-neutral framework? This is very important in the context of the use of historical versus actively traded (and therefore implied) data.

Let us imagine that we need to price an option, exotic or vanilla, on the Nikkei index. The most important input, the volatility of the index, is readily available since options on the Nikkei are actively traded and the Nikkei futures used to hedge them are also actively traded. In this case we would be more than justified in pricing the option within a risk-neutral framework.

Let us imagine now that we are going to price an option not on the Nikkei but on some proprietary index. A proprietary index is an index, that is, a variable tracking the performance of some assets, published usually by investment banks at their own discretion whose value is obtained through a proprietary algorithm. Neither futures nor options are traded on it, the only available data would be past fixings of the index. Using past fixings we can of course calculate historical volatility and pretend we can simply plug this into, say, Black and Scholes formula. Would it be correct though?

The essence of risk neutrality is that when trading an option we can at the same time purchase the underlying. In this case we cannot, so we should not, technically, be pricing this option the same way we would price the option on the Nikkei. In practice, however, many market participants would do it anyway. The danger is that, should something dramatic happen to the components of the index (let us imagine that they are stocks) affecting in turn the index and the option written on it, we would not be able to hedge ourselves since futures on the index are not traded and the composition of the index itself, which could enable us to trade on the stocks directly, is not openly known being the index composition proprietary.

A counterargument has been made by Derman and Taleb [30] and by Taleb [78]. To the extent that pure dynamic replication within the requirements of Black and Scholes is most of the time an illusion,^{[4]} by making a virtue out of a necessity and following market practice, one should price options written on non-easily tradable underlyings in the same way one would price them when written on liquid ones. The fact that, as we have said before, in finance only the model that is actually used is considered valid, which gives some support to this view.

The final issue we are going to consider has to do with leverage as we anticipated in the previous section. In Section 3.1.2 we introduced how the danger of derivatives consists of breaking the nexus between option and underlying. We specifically described the situation where one would trade the protection offered by a CDS without ever holding the bond the CDS was supposed to render riskless.

When we trade an underlying there are physical limits attached to it: for bonds it is the total outstanding principal of debt, for shares it is the market capitalization of the company. If a company has a market capitalization of $1M, it is impossible to purchase more than that amount (assuming all of it is traded). However, is it possible to write on the same underlying an option with a principal of $5M? Certainly. When we use Black and Scholes to price it and we are told, from Equation 7.1, that we need to buy, say, $2.5M of Δ, what are we going to do?

This is another side effect of leverage, the one of making the fundamental assumptions of financial modeling dangerously inapplicable. However, this is not as bizarre as it might seem. Although an extremely large principal would make a trader think twice about trading a particular option, a large but reasonable principal would not push a trader to actually make sure that there is enough stock to buy in order to hedge.^{[5]}

We have outlined in Equation 7.1 (and in Appendix F) the basic principles of replication. We are now going from simple replication to full hedging.

[1] Of course some players try to build powerful means (often nonhuman) that enable them to act faster than everybody else and hence exploit the possibility to make a riskless profit: the presence of these players, however, is not strong enough to change the assumption of risk neutrality.

[2] As mentioned previously, much has been written already about the theory behind this, but the reader is directed, as a reminder, toward a rigorous mathematical backing given, for example, by Baxter and Rennie [10] or by Etheridge [37].

[3] The probabilities of a winning lottery are drawn from a real-world probability distribution.

[4] The argument goes roughly along these lines: the assumptions of Black and Scholes are so strict theoretically that in practice, particularly as far as the idea of continuous and costless hedging is concerned, they can never be satisfied and therefore the use, sometimes, of historical data and/or nontradable data is not as serious as one would think.

[5] Of course the issue is not necessarily with the amount of stock available at inception: we need to remember that the more in the money an option goes, the greater the Δ becomes and therefore the greater the need for the underlying becomes.

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