key: cord-0060942-c5l705ce
authors: Tata, Fidelio
title: Derivatives
date: 2020-07-20
journal: Corporate and Investment Banking
DOI: 10.1007/978-3-030-44341-2_7
sha: 362833176f147ff795327056a8d8f43206423dc7
doc_id: 60942
cord_uid: c5l705ce

In this chapter, derivatives as a product class are discussed. After explaining why derivative product knowledge and a derivative mindset is beneficial to anyone working in a sales, trading or research environment, the main derivative types are covered: forwards, futures, swaps and options. The main focus is set on developing an intuitive understanding of the functioning and the practical use of derivatives.

competing for the attention of your clients. In 2012, I detected a high correlation between inflation expectations extracted from inflation-linked bonds 1 and the re-election odds of then-president Obama, as extracted from the betting platform Intrade. 2 While not a typical derivative analysis, this unusual observation, illustrated in Fig. 7 .1, helped me highlight how the US presidential elections have, according to market expectations, an impact on monetary policy and inflation (as the US president picks the chairman at the Federal Reserve). Once I had the attention of my clients, it was much easier to interest them in inflation and interest rate hedges.

During my days working at financial institutions, I was regularly pulled into meetings covering aspects of risk management, financial planning or project evaluations. Usually, it took me less than half an hour to divide meeting participants into followers of what I would call linear thinking and those with a derivative mindset.

Linear thinking, to me, is based on point-estimates of the future and of some sort of discounted cash flow (DCF) analysis. Typically, a lot of energy is spent among participants on agreeing on one specific scenario that appears to be the plausible outcome in the future. Uncertainty, in terms of expected deviations from this scenario, is then considered to be something negative.

The goal is often to fill an Excel spreadsheet with cash flow estimations and then to apply simple discounting rules to arrive at a single measure for decision making, such as the net present value (NPV)-rule.

People with a derivative mindset approach the decision-making process quite differently. First, there is the explicit understanding that the future cannot be represented by one single expected outcome. Instead, rather than spending time on compromising on one estimate, the range of possible outcomes is evaluated, and intellectual energy is spent in discussions about which theoretical framework or model would be best suited to represent such uncertainty. Second, uncertainty is never assumed to be negative per se. In fact, certain financial exposures (such as being long an option) benefit from uncertainty. Thus, a derivative mindset will always search for optionalities and aim to quantify their values to the involved parties. Third, a derivative mindset will never assume exposures to remain constant. While one IBM stock equals one IBM stock, option theory teaches us that one option on one IBM stock may have a certain IBM stock-equivalent exposure (called delta) now, but another tomorrow. This change of delta, called gamma, is nothing unusual to people even with only rudimentary derivative knowledge, but something outside of the comfort zone of linear thinkers. Fourth, a derivative mindset will thrive to define any exposure by its sensitivities to distinct factors, rather than by equating it to another observable financial market instrument. Those sensitivities, called Greeks in option pricing theory, are conceptually not very different from the beta utilized in the capital asset pricing model (CAPM) or the factor loadings of the arbitrage pricing model (APM), two frameworks linear thinkers often tend to struggle with as well. Having subscribed to thinking in terms of sensitivities, it comes naturally to people with a derivative mindset to focus on scenario analysis-type of "What-happens-if " questions and on the quantification of the impact of partial changes to the environment (e.g., interest rates going up, credit spread widening, volatility increasing).

It is almost bizarre to observe how each group, linear thinkers and derivative thinkers, tend to carry out their own discussions among themselves, without much interaction. Linear thinkers may, for example, have a heated discussion as to whether 2 or 3% would be the better forecast (or "baseline scenario") for a certain interest rate used in the planning process, based on what the economic research department has been predicting, what other market participants are thinking, how interest rates had been forecasted previously, etc. Meanwhile, derivative thinkers would discuss which statistical distribution (e.g., normal or log-normal) best captures the future interest rate environment and how those model assumptions create model risk in the risk management process. Every now and then, a member of the linear thinkers would address a derivative guy with a question like "are you o.k. with using 3% as our target rate?", to which the derivative guys would look baffled, and every now and then the derivative guys will ask the non-derivative guys something like "are you o.k. with applying a jumpdiffusion process?", to which the non-derivative guys would look puzzled.

I don't want to create the impression that people with a derivative mindset are smarter than those with a strictly linear understanding of the world. But I think it is safe to say that it is easier for someone with a derivative background to understand the thinking process of a linear thinker than the other way around. Thus, derivative knowledge is a useful skill set, similar to be able to speak in a foreign language or knowing legal and regulatory requirements.

A derivative mindset also often helps explaining seemingly improbable situations. Examples can be found in all areas of economic activity, but one of my favorites comes from the area of real estate development. There is a parking lot adjunct to the Holy Name Cathedral in Chicago that the Archdiocese in Chicago sold for some $110 million to a real estate developer in 2017.

One can only speculate about the Archdiocese's specific motives for having let the valuable parcel of land undeveloped for many years, before selling it only recently to a developer that now plans on building a $850 million 77-story apartment tower. 3 At least, option theory offers a convincing argument for such behavior. One could argue that the value of the property in front of the Cathedral is the sum of two value drivers: first, the value from the cash flow generated by the current use of the property and, second, the value from the option to decide later what to do with the property. A person merely applying linear thinking would aim at identifying the highest-NPV project for the use of the property. Running a parking lot is probably not generating much cash flows, and the NPV of future cash flows from collecting parking tickets (minus the cost of maintaining a parking lot) is almost certainly lower than the NPV generated by a high-rise apartment building. The second value driver concerns the option to delay development of the property. This option to wait has value, because in the future the specific type of building can be fine-tuned to the then-prevailing economic environment. If, for example, a 50-story building had been built 5 years ago, it would not be economical to tear down the building now and to replace it by a 77-story building, which is the most desirable structure from a current point of view.

In general, without the option of deciding when to invest, it is optimal to invest as long as NPV is positive. But when you have the option of deciding when to invest, it is usually optimal to invest only when the NPV is substantially greater than zero. Put differently, given the option to wait, an investment that currently has a negative NPV (such a parking lot) can have a positive value.

Assuming for a moment that the Archdiocese had applied a derivative mindset to the decision-making process, the incremental cash flow from turning the parking lot into a high rise would have to be compared to the option value of postponing the development by one additional unit of time. Calculating the value of the so-called real option is not trivial, but derivative theory helps us to better understand how some changes in the environment would impact the outcome. For example, we know that the option to wait is most valuable when there is a great deal of uncertainty regarding what the value of the investment will be in the future. The church, in derivative terms, was long volatility, benefiting from any increase in uncertainty.

Maybe somewhat surprisingly, a derivative mindset is not necessarily restricted to people working in a derivative environment. Many market participants that do not engage in derivative transaction, often because their institutional mandate prevents them from doing so, are just as derivative-savvy as derivative traders and salespeople. I once advised a portfolio manager at a central bank on picking the best government bond within a particular maturity bucket. Many central bank salespeople would simply point out an interest rates difference between neighboring securities on the yield curve to gain from what is called a yield pickup. The discussion I had with the central banker, however, centered on asset swap spread levels, differences in convexity and financing biases. Even though the central banker would, due to the central bank's mandate, never engage in interest rate swap transactions (as part of an asset swap), in convexity hedging, or repo financing, it was appreciated that I used relative value concepts employed by hedge funds and other sophisticated investors. This gave me an edge over other strategists and salespeople that treated the central banker as a less sophisticated investor and helped to win order flow. On another occasion, I was meeting with a portfolio manager of Fannie Mae, a US government-sponsored enterprise engaged in the mortgage market. The manager told me the following: "We are not a hedge fund, but treat us as if we were one." Many institutional investors may be prevented from trading derivatives at their current role, but they may have a strong derivative education nonetheless, are curious about derivative concepts or used to work in a derivative environment. They have a derivative mindset and expect anyone seeking to serve them to have one as well.

Finally, derivative knowledge can easily be used to shine. In fact, while working at the Derivative Marketing group of JPMorgan we used to joke that during pretty much any meeting one can torpedo a well-presented argument by raising the simple question "how about convexity?" Convexity is at the very heart of derivative theory, but unfortunately it comes in many different shapes and forms. A (de)convexity-adjustment is required to move from observable forward yields to implied market expectations, a convexity-adjustment is needed if payment dates differ from index dates (e.g., 10-year yields paid every 3 months), and pretty much every time the payout profile of a financial claim is nonlinear (which is even true for regular bonds, but even more so for mortgage products, bank's sight deposits, etc.) convexity matters. It is almost impossible to incorporate all facets of convexity. If you are not only alert to the possibility of convexity not addressed properly, but also manage to propose a theoretical concept or technique to improve the precision of calculation, you will be viewed as one of the smarter people in the room.

What is a derivative? A derivative is a contractual agreement whose value is derived from an underlying. In many cases, a derivative contract does not involve the exchange or transfer of principal or title. Rather the goal is to create an economic exposure to some underlying price change or event. Derivatives are zero-sum games between two contracting parties (see Fig. 7 .2).

The term derivative refers to how the price of these contracts is derived from the price of some underlying security or commodity or from some index, interest rate or exchange rate. 4 Since derivatives are legal contracts between two consenting counterparties, pretty much anything could be used as the underlying; typically, the underlying should be easily observable and not subject to possible price manipulation by either side of the derivative contract.

Derivatives performed an uninterrupted, explosive growth up to the financial crisis of 2007-2008, as exemplified in Fig. 7 .3 by looking at the outstanding notional of the over the counter (OTC) derivative market. Since then, it has been a mixed bag. Credit derivatives went on a decline, partially caused by the $30+ billion loss by American International Group (AIG) trading credit default swaps. Foreign exchange derivatives stabilized and even grew after the crisis, possibly because global world trade keeps increasing, making currency risk management more relevant than ever. Interest rate derivatives are impacted by a number of contrasting developments. Regulatory changes have moved an increasing part of derivatives from OTC onto exchanges. The zero-interest rate policy (ZIRP) of central banks and expectations for low rates for longer may have also dampened the need for hedging. On the other hand, the risk of an adverse increase in interest rates on the back of a less accommodative Fed creates some additional demand.

The notional principal amount or notional amount of a derivatives contract is the amount used to calculate the cash flows of a derivative contract. Typically, the notional amount is not exchanged between counterparties. Also, there is some double counting of notional amounts.

The gross market value shows the mark-to-market value of a derivative exposure. At trade inception, it is typically close to zero. Because of derivatives being zero-sum games, a positive market value to one counterparty is a negative to the other.

Central clearing has made very significant inroads into many OTC derivatives markets. More than 80% of outstanding OTC interest rate derivatives contracts are currently against central counterparties (CCPs). When a derivatives contract between two reporting dealers is cleared by a CCP, this contract is replaced, in an operation called novation, by two new contracts: one between counterparty A and the CCP, and a second between the CCP and counterparty B.

The four main types of derivatives contracts are forwards, futures, swaps and options.

The simplest and perhaps oldest form of a derivative is the forward contract. It is the obligation to buy or borrow (sell or lend) a specified quantity of a specified item, called underlying, at a specified price or rate at a specified time in the future, called the expiration date. The forward price (the price at which the buyer and seller agree to exchange the underlying) is usually negotiated so that the present value of the forward contract at the time it is traded is zero. The contract partner committed to buy the underlying at expiration is said to hold a long position in the future, while the side committed to sell is short the future. There are no cash flows during the life of a forward contract; the only payout occurs at expiration.

The value of a forward contract at expiration can be displayed as payout profiles (see Fig. 7 .4). Because forwards are zero-sum games, meaning that a profit to one contract partner must be a loss of equal amount to the other contract partner, the payout profile of a long position in a forward is a mirror-image of that of a short position. The payoff of the long forward position is delivery price minus underlying price, while the payout of the short forward position is underlying price minus delivery price.

Forwards have been around for some 3800 years. They can be traced back to Mesopotamia. 5 A Babylonian clay tablet sitting at the British Museum in London documents a typical problem when trading forwards from the year 1750 BC, namely the quality of the deliverable into a forward contract. 6

Futures contracts are similar to forwards, but they are highly standardized, publicly traded and cleared through a clearing house. The futures contracts are standardized so that they are fungible-meaning that they are substitutable one for another. This fungibility implies that a trader who bought a futures contract can offset the economic exposure by selling a futures contract. Figure 7 .5 contrasts a forward to a futures contract and gives the contract specifications of a specific corn futures contract, listed on the Chicago Mercantile Exchange (CME).

Underlyings to futures contracts include: soft commodities, 7 hard commodities, stocks and stock indices, foreign exchange (FX) rates, interest rates, bonds, real estate indexes, economic events, equity indexes, swaps, energy prices and weather. Some futures allow for physical delivery, in which case it needs to be specified what can be delivered into the futures by the short.

Futures can be used for speculation and hedging purposes. To illustrate how hedging works in the context of futures, let's take the example of a beer brewing company. 8 As shown in Fig. 7 .6, the brewer turns raw materials, i.e., input factors, into beer. If the prices of those raw materials increase, the brewer could raise the beer price. However, that strategy could easily backfire. The beer industry is very competitive and market share could be lost. Also, beer prices may have been fixed long-term vis-a-vis the distributors. Another way to deal with the risk of increasing raw material prices would involve the use of futures. The beer brewer is what is called a natural short in the beer production's input factors. Natural indicates that the exposure has not been created artificially through financial instruments, but is the result of engaging in an economic production process. Short reflects the requirement to buy the commodity in the future (while "delivering" it into the production process at a price previously assumed when production planning was carried out). Adding a properly weighted position of a long futures contracts to the natural short position helps offset the price exposure to the underlying. One specific property of futures to be discussed here is the margining requirements. There are two types of margins that affect any futures trader: Initial margin and variation margin. Initial margin is the amount (percentage) of a futures contract value that must be on deposit at the futures exchange at trade inception. The purpose of margin is to avoid contract defaults. Since a futures contract is not an actual sale, one only pays a fraction of the asset value when establishing a futures position. 9 Variation margin is a variable margin payment made to, or received by, the futures exchange on the back of price movements of the futures contracts. Variation margin is paid or received on a daily or intraday basis to bring the mark-to-market value of the futures, adjusted for the already posted margin, back to the initial margin requirement.

Swap contracts, in comparison with forwards, futures and options, are one of the more recent innovations in derivatives contract design. The first currency swap contract, between the World Bank and IBM, dates back to August of 1981.

The basic structure of a swap contract involves two contract partners agreeing to swap two different types of payments (see Fig. 7 .7). Each payment is calculated by applying some interest rate, index, exchange rate or the price of some underlying commodity or asset to a notional principal amount. The principal amount is considered to be notional because the swap generally does not require the transfer or exchange of principal (except for foreign exchange swaps and some foreign currency swaps). Payments are scheduled at regular intervals throughout the tenor of the swap. 

The most plain-vanilla (basic) swap is the fixed-to-floating interest rate swap. 10 Here, one contract partner of the swap agreement pays the other contract partner a fixed interest rate on an agreed-upon notional amount for a contractual period of time. In return, the other contract partner pays a variable, or floating interest rate based on the same notional and for the same period of time. The floating rate is an interest rate that resets periodically based on an interest rate index. A common floating rate is LIBOR. Sometimes a spread (in basis points) is added to or subtracted from the floating rate. There is no exchange of principal at the beginning or the end of the swap transaction. Figure 7 .8 Illustrates the cash flows in an interest rate swap.

An example for interest rate swaps is shown in Fig. 7 .9. Without a swap market, corporates have to fund themselves in the market according to their funding needs. For example, XYZ corporation needs floating-rate funding and issues a corporate bond at LIBOR + 25 bp; ABC corporation needs fixed-rate funding and issues a corporate bond for the same maturity at 11.5%. If XYZ and ABC corporations have the ability to exchange cash flows between each other, they can fund themselves where they have comparative funding advantages, XYZ fixed and ABC floating, and then enter into an interest rate swap. As a result, both XYZ and ABC corporations improve their funding level. 11 Typically, a broker-dealer steps in between the swap contract partners 12 and offers a tailor-made and timely swap execution. As a compensation for its liquidity service, the broker/dealer charges a bid/offer (2 bp in the example depicted in Fig. 7 .10). Even more realistically, each corporate customer uses its own broker-dealer (house bank, etc.) for their swap transaction. Broker-dealers then turn to inter-dealer brokers, with whom broker-dealers offset their derivative exposure, often at mid-market prices (apart from a broker fee). The inter-dealer broker market is very liquid, although only standardized structures are traded. Thus, each broker-dealer may only be able to enter into a proxy hedge 13 and will need to keep some residual risk (see Fig. 7 .11).

There is an actively traded inter-dealer broker market for interest rate swaps. Inter-dealer brokers provide dealers with the ability to trade on an electronic basis with each other in the most liquid 14 assets. Inter-dealer brokers provide anonymity, so dealers don't get to know other dealers' positions/ flows. For less liquid markets, participants can speak with an inter-dealer broker on the phone who will try to identify "trading interest" and availability from other market participants.

A foreign exchange swap (also: FX swap or FOREX swap) differs from an interest rate swap because the principal amount is exchanged both at trade inception (t 0 ) and at maturity (t 1 ). A typical foreign exchange swap begins with a start leg at t 0 that is indistinguishable from a spot transaction in which one currency is exchanged for another at the present spot rate (FX spot). The second leg is a forward transaction at the forward foreign exchange rate (FX fwd) at the time of trade inception. Thus, a foreign exchange swap is essentially the combination of a spot and forward foreign exchange transaction (see Fig. 7 .12).

Foreign exchange swaps are used by both foreign and domestic investors to hedge foreign exchange risk. An example for foreign exchange swaps is shown in Fig. 7. 13. An EU-based corporation needs to make a JPY-deposit (which will be returned at a later point in time) with a Japanese trading partner. A foreign exchange swap turns the JPY cash flows into EUR cash flows. Thus, foreign exchange risk is eliminated, while the difference in the EUR-principal payments (due to the differences in exchange rates FX spot vs. FX forward) reflects the cost of making the deposit.

Foreign exchange swaps are also used for speculation in foreign currencies. 

A cross-currency swap (also: XCCY swap) differs from a foreign exchange swap in two ways: The exchange of principal at the end of the swap is at the same exchange rate, at which the principal was exchanged at trade inception; also, throughout the life of the swap, interest rate payments are exchanged based on the two currencies. Interest rate payments could be set such that they are both fixed for the two currencies, both floating, or one fixed vs. one floating rate. Because the final exchange of principal is based on an exchange rate different from the prevailing forward FX rate at trade inception, cross-currency swaps tend to have significant counterparty risk 15 (see Fig. 7 .14). Cross-currency swaps are often used to monetize comparative advantages, in the capital market. An example for cross-currency swaps is shown in Fig. 7 .15. Facebook Inc. is a US dollar-based company that needs funding in dollar for investment projects in the USA (server farms, research facilities in California, etc.). It turns out, however, that there are a lot of European investors looking to buy EUR-denominated Facebook debt (they like exposure to Facebook credit risk for credit risk diversification reasons, but don't Because of this excess demand, Facebook could finance itself in the EU at very competitive (i.e., low) interest rate levels.

Since the first formal swap agreement in 1981, there has been a tremendous amount of market innovation and financial engineering, leading to a wide array of swap variations. Some types of swaps may have been developed primarily to change the "optics" of a derivative structure and to create misleading economics. 16 Additional types of swaps include:

• Basis swap (floating-for-floating);

• Amortizing/accruing swap (changing notional);

•

Step-up swap (changing fixed rate);

• Forward-starting swap;

• Equity, commodity and index swap; • Credit default swap; • Total return swap;

• Callable/extendable swap.

Once you developed a good intuitive understanding of plain-vanilla swaps, it is not very difficult to also understand the more exotic structures. More important than mastering the last detail of pricing of a particular swap is being able to clearly communicate its benefits to an (institutional) end-user. Once you have sold a client on a swap trade, it is likely not very difficult to find someone within the organization to price it for you; in contrast, if you have priced a structure, it is far from clear that someone will trade on it.

An option is a contract that grants owners the right but not the obligation to purchase (a call option) or sell (a put option) a financial instrument for a specific price, called the strike price, at or before 17 the time of option expiration. It functions by having the purchaser pay the seller/writer an option premium for the right to buy or sell. The purchaser's potential loss is limited to the price of the premium, curbing the downside. In contrast, the seller of an option receives the premium in return for risk exposure. An option position consisting of option purchases is also called a long position; the seller of an option is said to have a short position. Options are traded on organized exchanges and OTC derivatives markets.

The economic exposure of options at the time of option maturity can be graphically described through payout profiles. Because payout profiles of options resemble the shape of sticks used by hockey players, they are also commonly referred to as hockey stick diagrams, or hockey sticks for short. There are two elements to the cash flows generated by options. First, because there is a value to having the right to defer a decision, the buyer of an option has to pay an option premium to the seller of the option. From the perspective of the option buyer, the option premium is a negative cash flow; from the option seller's viewpoint, it is a positive cash flow. Second, depending on whether the option is exercised at option expiration, an additional cash flow is been triggered. This cash flow, if it happens, will always take the direction from the option seller to the option buyer. The cash flow depends on the underlying price at option expiration in relation to the strike price set at the time of trade inception. Figure 7 .16 illustrates the cash flows from the perspective of a call option buyer.

There are four distinct option positions that can be discussed in detail: A long call (purchase of a call option), a short call (sale of a call option), a long put (purchase of a put option) and a short put (sale of a put option) (see Table 7 .1). If the price of the underlying at the time of option expiration is below the strike price, the call option expires worthless and the long loses the entire option premium paid Otherwise, the call option will be exercised and the long receives money upon exercise

There is a break-even price (BEP) of the underlying, at which the value of the option at expiration equals the option premium paid initially

For the long call option strategy to make money, the underlying price needs to be above the BEP at option expiration If the price of the underlying at the time of option expiration is below the strike price, the call option expires worthless and the short keeps the entire option premium received Otherwise, the call option will be exercised and the short has to pay money upon exercise There is a break-even price (BEP) of the underlying, at which the value of the option at expiration equals the option premium paid initially

For the short call option strategy to make money, the underlying price needs to be below the BEP at option expiration If the price of the underlying at the time of option expiration is above the strike price, the put option expires worthless and the long loses the entire option premium paid Otherwise, the put option will be exercised and the long receives money upon exercise

There is a break-even price (BEP) of the underlying, at which the value of the option at expiration equals the option premium paid initially

For the long put option strategy to make money, the underlying price needs to be below the BEP at option expiration If the price of the under-lying at the time of option expiration is above the strike price, the put option expires worthless and the short keeps the entire option premium paid Otherwise, the put option will be exercised and the short has to pay money upon exercise There is a break-even price (BEP) of the underlying, at which the value of the option at expiration equals the option premium paid initially

For the short put option strategy to make money, the underlying price needs to be above the BEP at option expiration

Combining the cash flows from the option purchase and the option cash flows at the time of option expiration in one single cash flow diagram, hockey sticks can be drawn for all four distinct option positions (see Fig. 7 .17).

The strike price of an option in relation to the current underlying price determines whether an option is said to be in-the-money (ITM), at-the-money (ATM) or out-of-the-money (OTM).

For call options, if the strike price is less than the current market price of the underlying, the call option is said to be in-the-money because the holder of the call has the right to buy the underlying at a price which is less than the price he would have to pay to buy the underlying in the market. The converse of in-the-money is out-of-the-money. If the strike price equals the current market price, the call option is said to be at-the-money. For put options, if the strike price is greater than the current market price of the underlying, the put option is said to be in-the-money because the holder of the put has the right to sell the underlying at a price which is greater than the price he would receive from selling the underlying in the market. Again, the converse of in-the-money is out-of-the-money. If the strike price equals the current market price, the put option is said to be at-the-money. Figure 7 .18 illustrates the in-the-moneyness for a call and for a put option.

The amount by which an option, call or put, is in-the-money prior to expiration is called its intrinsic value. The intrinsic value equals the value of the option if it were to be executed immediately. Time value is the value from being able to defer option exercise to the exercise date.

By definition, an at-the-money or out-of-the-money option has no intrinsic value; the time value is the total option premium. Nobody would execute an OTM or ATM option, but the option is still valuable because the underlying price is volatile and could, between now and option expiration, move such that the option is ITM on the expiration date.

If the value is known for an ITM option (because the option is quoted in the market), the time value can be calculated by subtracting the intrinsic value from the option value. When the value of an OTM or ATM option is known, this equals the time value of the option. 

The so-called Greeks measure the various sensitivities of an option's change in price with respect to a small movement in some underlying variable. Each sensitivity is assigned a Greek letter. Every option has a different risk profile with respect to the various risk sensitivities. Knowing these risks allows one to pick the most appropriate (hedging or speculative) position and to manage the risks throughout the life of the trade.

Sensitivities are calculated only for small incremental changes. The four most relevant sensitivities are Delta (δ), Gamma (γ ), Theta (θ ) and Vega 18 (ν ) (see Table 7 .2).

In-the-moneyness for calls and puts There are more than four sensitivities calculated for options. However, for an intuitive understanding of options it is usually sufficient to know the main Greeks.

Delta (δ ) is the change of the option price with respect to the underlying. It can be viewed as the "hedge ratio" between underlying and derivative, defined as the change in the price of the derivative, ΔP D , divided by the change in the price of the underlying, ΔP U .

Gamma (γ ) is the change in delta, relative to the price change of the underlying. It can be viewed as the change of the "hedge ratio" between underlying and derivative, defined as δ 2 − δ 1 = (ΔP D,2 /ΔP U,2 ) − (ΔP D,1 /ΔP U,1 ).

Theta (θ ) is the change in the option price relative to a one-day change in expiration, also called time decay. Generally, the longer the time remaining until an option's expiration, the higher its premium will be. This is because the longer an option's lifetime, the greater the possibility that the underlying share price might move so as to make the option in-the-money. All other factors affecting an option's price remaining the same, the time value portion of an option's premium will typically decay (i.e., decrease) with the passage of time.

Vega (ν ) is the change in the option price relative to a 1% change in the volatility. The higher the expected (implied) volatility of the underlying, the higher the probability that the option moves (further) into the money, i.e., increasing its intrinsic value. It can be calculated as P D,2 − P D,1 , whereas P D,2 is calculated with an implied volatility of one percentage point above that of P D,1 (all else being equal).

See Fig. 7 .20.

Bookshelves in libraries are filled with books about derivative pricing. But that's not necessarily a good thing. Because many people have been scared away from even trying to learn the basics about derivative pricing in general, and options pricing in particular. There is the assumption that advanced skills in math and stochastics would be necessary to develop a general knowledge about it. The truth is that even if you only have a purely intuitive understanding about this topic, you can use it quite frequently in almost any sales, trading or research function. In fact, the derivative pricing experts who speak at derivative conferences are typically not the ones that are put in front of clients. Those who are good at communicating the intuition are the ones talking to clients. My advice would be to get the intuition right and have someone else worry about the detailed math. The first thing, when it comes to derivative pricing, is to develop a good intuition about the price dynamics of the underlying. By definition, a derivative is based on the price of an underlying; thus, derivative pricing without a thorough understanding of the underlying is impossible. While sounding obvious, this is often enough ignored. For example, the US subprime mortgage crisis that triggered the financial crisis of 2007-2008 was caused primarily by a genuine misunderstanding about the underlying, i.e., subprime mortgages, and less so because of some errors in pricing derivatives (such as residential mortgage backed securities).

The basic premise for modeling the price dynamics of the underlying is that current prices reflect all information from the past, which include the information about how historical prices evolved to the prevailing price. This implies that future prices cannot be predicted by analyzing prices from the past. 19 This is consistent with the so-called efficient-market hypothesis developed by Fama (1970) . It also reflects what in stochastics is referred to as a Markov property. The price path of the underlying is modeled to be a random walk, starting with the current market price and where the past is irrelevant. Such random walk can be illustrated by imagining a drunk person leaving a bar. The person is so drunk that he/she needs to hold on to the next light post, but then forgets where he/she came from before moving on to the next light post. Because the drunk person has no recollection where he/she came from, the past path is irrelevant. Moving forward, there is also the chance that the drunk person moves back to the previous light post (see Fig. 7 .21). Applying the drunk person-analogy to the modeling of stock prices (as one possible underlying instrument for derivatives), we could construct a random walk with the following features: The past is irrelevant (also referred to a memory loss ), the expected change in value is zero, and the best predictor of a future value is the current value. 20 Over time, there is increasing potential for the stock price to have moved further away from the starting point, which is reflected in a higher variance 21 (see Fig. 7.22 ).

Each particular increment of a random walk has variance that is proportional to the time over which the price was moving. This is a very convenient relationship. You will hardly ever find anyone refer to the term variance in sales, trading and research. Instead, market participants talk about volatility. Volatility is used synonymously for standard deviation. Standard deviation, abbreviated by the Greek letter Sigma (σ ), is the square root of variance (Var). Thus, σ = √ Var. Because variance is proportional to time, volatility is proportional to the square root of time. To illustrate the relationship between time and volatility, let's look at a specific example. Someone tells you that the implied annual volatility of a stock is 40%. Implied means that this is not some market participant's estimation or the outcome of some model, but "extracted" from market prices (such as option prices), so that it reflects market expectations. Annual means that the 40% deviation from current prices relates to a one-year period. A 40% annual volatility means that with a given probability 22 the stock price will not change more than 40% up or down from the current price over a one-year period. Given a 40% annual volatility, how much would be the expected volatility over a three-month period? In other words, with the same probably with which price changes do not exceed 40% over 12 months, what would be the equivalent-probability percentage change over three months? If volatility was proportional to time (like variance is), the answer would be one-quarter of 40%, or 10% (because three months are a quarter of one year). But this is not the case. Volatility is not proportional to time, but proportional to the square root of time. This means that the three-month, or quarterly, volatility σ quarterly is the square root of ¼ (3 months divided by 12 months) time the annual volatility σ annual . Since the square root of ¼ is ½, the quarterly volatility would be 20%. This is somewhat surprising at first sight. How can it be that over a 3-month period the stock moves up and down 20% with a certain probability, but over a whole year only 40%? The answer can be derived from what we saw in Fig. 7.22 : In the second, third and fourth quarter of the year, some of the movements from the first quarter will likely be reversed due to stocks not always moving in the same direction all the time. Like a drunk person (illustrated in Fig. 7.21) , stock prices move back and forth, undoing some of their initial price moves by subsequent moves in the opposite direction. Now that we have developed a basic understanding about the evolution of prices of the underlying, the next step is to derive a price for the derivative. Using a call option as a concrete example, we can develop a framework that for each possible price outcome of the underlying calculates the corresponding price of the call option. Obviously, in a "state" of the world when the underlying price goes up, the value of the call option increases, and vice versa. The current value of the call option then would be approximately 23 the sum of the probability-weighted call option values at option expiration.

Binomial pricing models are one type of valuation methods that are based on drawing a binomial tree representing the price path of the underlying, calculating the implied value of the derivative at each modeled state of the world, called node, and then aggregating those values with considering their probabilities of occurrence.

First, we are going to use a binomial model to price an at-the-money (ATM) call option (i.e., a call option where the strike price is set to be equal to the current price of the underlying) in an environment without volatility. This is a somewhat unrealistic first assumption, because pretty much all financial assets are subject to potential price changes, but it will be useful to show how option values critically depend on the presence of uncertainty. An example for an underlying that has close to no price volatility would be the US federal funds target rate set by the US central bank, or more precise the Federal Open Market Committee (FOMC) of the US Federal Reserve, during the period between scheduled FOMC meetings. Typically, the target rate is only changed on one of the eight scheduled FOMC meetings, so during the period between one FOMC meeting and the next scheduled one, the target rate remains constant. 24 The binomial pricing of a 1-year call option with a strike price of 100 for a stock trading at a price of 100, where there is zero volatility (σ = 0%), is presented in Fig. 7.23 . Because there is no volatility, the stock price is expected to remain at 100. The option payoff at expiration of a 100-strike call is zero when the underlying trades at 100. Since the assumed probability (p) of the stock ending at 100 is 100%, the expected option payoff is 100% times zero, or zero.

The binomial model with only one single tree-node illustrates the maybe most important fact about options in general: Without uncertainty, there is no time value to options. 25 As a next step, we are extending the binomial model to allow for some uncertainty in underlying price. Let's assume an annual volatility of 40% (as in Sect. 7.7.5). The one-node model now becomes a two-node model, because the price of the stock does not remain at 100, but can go up or down. For some technical reasons, 26 we assume prices to go up and down by e 0.4 and e −0.4 , respectively, with e being the mathematical constant (Euler's number) used to create exponential functions. The probabilities for the up-and down-move (p u and p d ) are then calculated such that the probability-weighted underlying prices for the up and the down move equal the current underlying price. Figure 7 .24 presents the calculation for our call option. A first estimation of the option value is 19.6, resulting from a 40% probability of the call option being worth 49, and a 60% chance of the option expiring worthless. 27 The option price calculated in the two-node binominal pricing framework may be a useful first estimate but suffers from one unrealistic assumption: that the stock price only jumps once a year. To better reflect the realities of the stock market, in which prices changes continuously throughout the day, more nodes are introduced to the binomial tree. binomial pricing of the same call option, where two stock price moves are possible: one after 6 months and another just prior to option expiration. Constructing the three-node tree requires calculating the underlying price in 6-month intervals; semi-annual volatility σ s.a. is calculated according to the square root formula. 28 Price movements up and down, as well as up-and down-probabilities, are calculated as before. The intrinsic option payoffs are calculated for all three nodes, and the sum of all probability-weighted intrinsic option payoff equals the new model option value. Note that the calculated option value in the 3-node binomial model (13.9) differs from the one calculated in the 2-node model (19.6) .

The price of an option calculated by the binomial pricing method is significantly impacted by the number of tree nodes used. Figure 7 .26 illustrates how with an increasing number of nodes the model prices converges to a value that can be assumed to be a realistic estimate.

The Black-Scholes, or Black-Scholes-Merton, option pricing model is a partial differential equation commonly used since the 1970s to estimate the value of European options. The binomial pricing model discussed in the previous section is linked to the Black-Scholes model as far as binomial pricing approached the Black-Scholes price if, under certain technical assumptions, the number of nodes goes to infinity.

The model is derived from the idea of continuous delta hedging and maintaining a risk-free position at all times. The CAPM suggests that such a risk-free position should earn (only) the risk-free rate of interest. Thus, option pricing no longer depends on risk premia. The model is based on a number of assumptions, such as: The beauty about the Black-Scholes model is that it is a closed-form solution that can be calculated with relatively little computational power. 29 Also, there are only five inputs to the calculation, which are:

• The current price of the underlying;

• The strike price of the option;

• The time to option expiration; • The volatility of the underlying;

• The risk-free rate of interest.

The Black-Scholes option pricing model does not only provide a closed-form solution for option prices, but also for the corresponding risk measures (Greeks).

The Black-Scholes model can also be used in a reversed manner: The option price (as observed in the market or calculated by a different, more realistic model) is fed into the formula together with four out of the five usual input variables, and the remaining variable is then calculated. If the volatility of the underlying is calculated that way, it is referred to as implied volatility or, to be more precise, as implied Black-Scholes volatility. Traders typically use their own proprietary option pricing models to calculate the fair value of options. Instead of quoting the price of the option, they then often quote the implied volatility that, if entered into a Black-Scholes pricing engine, returns the desired option price. Quoting options in Black-Scholes volatility does not imply that the trader believes that a Black-Scholes model is an appropriate pricing tool, nor that it was used for pricing. Black-Scholes volatility is merely a convenient way to quote options.

While academic literature tends to focus on the Black-Scholes model, a variation of the model, called Black or Black-76 model, is actually much more widely used. In it, the underlying is not the spot price (e.g., the current stock price), but a discounted future price. Here, the forward price is assumed to be log-normally distributed. This model is more suited to price options on futures, bonds, caps, floors and swaps.

You can work an entire career in derivatives without anyone ever asking you to write down the Black-Scholes formula. 30 That's why the formula is not listed here. What is far more important than memorizing the formula is knowing their shortcomings and ways people try to calibrate the simple model to the complexities of the real market. Some of them are listed in the sections below.

The Black-Scholes option pricing formula was merely the starting point for the development of many other, improved option pricing frameworks. Apart from the Black model of 1976 and the binomial pricing model, developed by Cox, Ross and Rubinstein in 1979, 31 which were already mentioned in the previous sections, there is one additional model from the 1970s worth knowing about: The Cox and Ross model developed in 1976, which extended option pricing to allow for a discontinuous price movement in the underlying by assuming a jump-diffusion process. 32 Following the stock market crash of 1987, commonly referred to as Black Monday, the assumption of constant volatility came under intense scrutiny. It was widely acknowledged that volatility assumptions suitable for pricing ATM options did not yield meaningful results when applied to OTM or ITM options. The early models from the 1970s and 1980s were simply fed with adjusted volatility inputs to compensate for their shortcomings. When displaying various volatility inputs for different strike levels, the graph looks like a skew or a smile. It turns out that the empirical observation of a volatility skew or smile can be captured by a model through making volatility itself a random process (i.e., assuming volatility of volatility, sometimes called vVol), and various stochastic volatility models have been developed in the 1990s. Two of those should be mentioned here: The first one is the Heston model of 1993. The Heston model can be calibrated against historical prices of the underlying. The second is the so-called SABR model, which stands for Stochastic Alpha, Beta, Rho, developed by Hagan, Kumar, Lesniewski and Woodward, in 2002. This model can be calibrated against observable option prices. The SABR model is very popular among option traders because it allows for a very realistic capturing of real-life effects. For example, the model allows to fine-tune the distribution assumption for the underlying anywhere from normal to lognormal, accounts for volatility of volatility and specifies the skewness and smileness. However, calibration of the SABR model is not trivial and often trading desks run dedicated programs at the end of a trading day that re-calibrate the SABR parameters.

While few people need a detailed understanding about all details of the most advanced option pricing models, I always found it very useful to have at least enough knowledge about derivative pricing to have a conversation with the option traders on a regular basis. Often, the trading desk's insight about issues when calibrating the bank-used model provides valuable input for client conversations, research topics or trade idea generation. For example, I used to feed observable option prices (for ATM and OTM options) into a mixture of distribution model (such as SABR that allows for various degrees in the mix between normal and lognormal) to extract the market-implied view about the degree of normality of the returns of the underlying. Specifically, I was looking at the implied normality of interest rates. For a while, interest rate options were priced with the implied assumptions of interest rates being lognormally distributed. When interest rates pushed lower and there was a real possibility of them turning negative, the implied distribution assumption became increasingly normal. At some point, even the standard normal distribution would not be able to explain the high premia charged in the market for low-strike options. The model had to be calibrated such that more than 100% of normality had to be assumed and then some lognormality subtracted. This is referred to as super-normality. If you catch such a moment as the first person, you got yourself a very good talking point that sets you apart from other market participants.

Many textbooks grossly simplify the decision-making process when it comes to option hedging in an institutional context. One may find problem descriptions like this: Investment Fund XYZ includes 10,000 stocks of corporation ABC, currently trading at €120. The fund manager wants to protect the fund against a drop of the price of the ABC stock below €100 within one year. If the 1-year €100-strike Put cost €500 and each Put option has 100 ABC shares as the underlying, how many Put options would the portfolio manager have to purchase and what would be the cost of the hedge?

In my entire career, I have never been confronted with a situation like this. In fact, even if problems like this were to exist, the institutional investor would likely not even call a global market professional in a sales, trading or research department, but rather execute the hedge without help. Textbook examples are typically constructed for didactical reasons, not to fit with reality. A more realistic option hedging question one could face is this: Investment Fund ABC consists of a diversified portfolio of stocks and bonds. The fund manager wants to protect the fund against possible adverse effects from a member of the European Union leaving the EU within the next 5 years. Which option strategy would you suggest when the fund manager is willing to pay an annual option premium of 10 bp?

Approaching any hedge problem typically starts with identifying the universe of hedge instruments your institutional client is able to and willing to consider. Not all investors are allowed to trade derivatives. Of those who do, not all are allowed to trade options. At this stage already, derivative marketing becomes critical. A bond investor may find swaps suspicious. Telling the investor to hedge bonds with swaps may not work. Once we marketed tailor-made swaps as "synthetic bonds." Replacing a bond (that traded rich) by a synthetic bond appeared much more familiar to the investor.

When the hedge instrument gets selected, there are many factors being considered. If the textbook tells you that the portfolio manager wants to hedge a position in ABC stocks and there are put options on ABC stocks and on DEF stocks trading in the market, the solution would likely be to use the ABC stock put options. In the real world, this may not be the case. Maybe the hedge takes place during trading hours where the market for ABC stock options is closed, while the one for DEF stock options is open. Maybe ABC stock options are illiquid and cannot be traded in the size needed. Maybe transaction costs (fees, commissions, etc.) on ABC stock options are significantly higher than that of DEF stock options.

Once potential hedge instruments are identified, the questions of the proper hedge ratio becomes relevant. Even before the question is addressed how much the option price moves when the underlying changes (i.e., the Delta of the option), one has to estimate how much the underlying of the option will move when the to-be-hedged security moves. Again, in the textbook this is not an issue because ABC stocks are hedged with options on ABC stocks. If, however, a 9-year bond is hedged by options on the 10-year swap rate (because swaptions are more liquid than bond options, and because swaptions on 10-year rates are more liquid than swaptions on 9-year rates), one need to calculate (or estimate) the beta between 9-year bond yields and 10-year swap rates. Using a simple DV01-ratio will not suffice because it assumes that for every basis point move in the 9-year bond yield, 10-year swap yield move by exactly one basis point in the same direction. 33 A commonly used hedging strategy is regression-based hedging. Here, an empirical analysis is conducted by constructing a regression model between the to-be-hedged and the hedge instrument. This often turns out to be as much art as science because depending on the length of the historical data series, its frequency and whether absolute levels or changes are used, results may vary considerably. Sometimes information from the option market are incorporated in the analysis to create a so-called volatility-weighted hedge. There are many approaches 34 and there is no right-or-wrong. It helps to know which hedging strategies your client prefers. If you suggest something too simple, you come across as ignorant; if you suggest something too complicated, your client may view the hedge as too complex. Also, some of the more sophisticated clients (especially hedge funds) may expect from you that you calculate hedge ratios according to different models and using different calibrations. They will typically only transact if the various hedge ratios are somewhat similar. If not, the trade is in fact less of a hedge and more so a model trade because the performance of the "hedge" depends on whether one had picked the "right" model to describe the price relationships. Most market participants try to stay away from "model trades."

Once a potential option is identified as a hedge instrument and the proper hedge ratio has been established, client-specific preferences need to be taken into consideration. Some investors are not prepared to pay a net premium. Some others will only consider long gamma positions. Yet others look for positive carry. Each customer is different, so there is hardly a one-fit-all solution.

Last but not least, the question of adjusting the hedge over time needs to be addressed. Some hedges are static hedges. A static hedge is a trade in which the hedge ratio between the to-be-hedged security and the hedge product does not change. Thus, no adjustments are necessary throughout the hedge period. For example, if the interest-rate exposure of a fixed-rate bond should be eliminated, a so-called true asset swap could be established by overlaying the bond with a swap that has matching cash-flow dates and coupon rates. As the bond and the swap approach maturity, cash flows should remain matched and there is typically no need for adjustments. 35 However, the majority of derivative hedges are so-called dynamic hedges. A dynamic hedge is a trade in which the hedge ratio between the to-behedged security and the hedge product changes over time, creating the need for occasional re-balancing of the trade. This shall be illustrated in more detail.

Many market participants have to offset negative convexity in their trading portfolio or business model through the purchase of volatility products (i.e., options). Those investors include participants in the US mortgage market, such as mortgage hedge funds, mortgage servicers and GSE's like Fannie Mae or Freddie Mac (because of their mortgage portfolio). Some investors, such as certain foreign central banks or some commercial banks, choose not to hedge out the negative convexity. However, those that do hedge have to concern themselves with the intertemporal optimization of their dynamic hedging activity.

Typically, negative convexity in a portfolio is mitigated by an overlay consisting of positively convex instruments such as swaption straddles and strangles. While it is comparably simple to offset the risk of a short option position through a long position in a similar option, replicating the optionality inherent in a mortgage product with swaptions is far from trivial. This is because mortgages are path-dependent, while standard swaption products are not. If interest rates decrease, say 100 bp, and then return to initial levels, the value of a swaption will not have changed significantly (ignoring time-decay and a possible change in implied volatility). Mortgage products, however, will likely have suffered because of the pre-payment associated with the decrease in rate.

When the goal is to compensate for the adverse effects of pathdependency in mortgage product, a buy-and-hold strategy of standard options (swaptions, treasury bond options, futures options, etc.) will not do. Instead, investors must frequently monetize the hedge overlay. This is done by taking profit after a significant move in rates has occurred and establishing a new hedge at then-prevailing market conditions. Instead of thinking of two transactions, the unwind of an existing option position and the purchase of a new option, one can view this as one adjustment in strike and option maturity. This is what is typically referred to as rolling strikes.

Rolling strikes requires an active management of the option hedge. Investors have to make a conscious decision as to when and how often strikes are to be rolled. While rolling strikes at a high frequency ensures that in-the-moneyness is monetized (and does not get lost if interest rates revert to previous levels), this results in higher transaction costs. Not surprisingly, the optimum hedge frequency is higher when no transaction costs are assumed. 36 In the real world, however, transaction costs make a significant difference and less frequent strike rolls produce lower overall costs.

Instead of adjusting the swaption hedge in fixed intervals, investors may wait until after a significant market move before rolling strikes. The argument to be made is that there is little point in rolling a strike when interest rates have hardly changed. However, notice that even if interest rates are unchanged, swaptions are aging and need to be rolled into longer-maturity swaptions at some point. If trigger levels for changes in 10-year swap rates are increased from, say, 25 bp to 30 bp, 35 bp, 40 bp and 45 bp, the frequency of hedging decreases. Requiring larger interest rate moves before strikes are rolled helps reduce transaction costs, but also decreases the amount of intrinsic option value that is monetized. In the extreme, trigger levels are set so wide that swaptions are not re-balanced at all and are held until maturity.

One very common mistake of people starting out their career in Global Markets is to assume that there is one single volatility for a product or a product class. What is the volatility for US interest rates? There is no answer to this question. Because different market participants use different pricing models, options are quoted as implied volatilities using agreed-upon reference models. In the broker markets, swaptions and cap/floors are quoted in "log vol," corresponding to the Black model. That does not imply that the Black model is the best model to use, only that if those vols are plugged into a Black model, then the resulting prices are the current market prices. Also, basis-point (so-called normalized volatility) quotes are given. Volatility skews are quoted as an indication for how much log vols need to be adjusted for non-ATM options (due to the limitations of the Black model). Table 7.3 shows an overview inter-dealer broker page for interest rate option that directs to further pages on which the actual quotes are being displayed. Not only are there multiple types of volatilities to choose from, but also are some generic structures quoted on a price (i.e., premium) basis. Table 7 .4 shows the format of one of the most important broker screens for derivative pricing in fixed income, the ICAP VCAP1 screen. It displays implied Black volatility (i.e., volatility when using the Black model) for ATM swaption straddles with EONIA discounting. This is already very specific, as those implied volatility levels would have little meaning for, say, caps, bond options, OTM interest rate options, let alone stock or commodity options. On top of that, there is a specific volatility quote for various maturities (i.e., time to swaption expiration) and tenors (i.e., duration of the swap underlying the swaption). Once you have found an appropriate vol level for a generic ATM option structure, you likely have to adjust for the fact that the structure you are setting out to price is not exactly ATM. This requires an adjustment depending on your distribution assumptions. If you are using a normal model and assume the returns of the underlying to be strictly normally distributed, then no adjustment is required. However, the real world is seldom as simple as that. When looking at historical distributions of changes in the underlying asset, one typically finds that there is not one single distribution assumption that fits all time periods and market regimes. At times, changes appear to be close to normally or lognormally distributed, giving the illusion that a corresponding pricing model would do a good job in capturing the pricing dynamics. However, at other times the observed market outcome lies far beyond what a classical pricing model would assign any meaningful probability to. This is also referred to as fat tails.

Take a look at Fig. 7 .27. It shows the results of a strikingly simple study I conducted in the Fall of 2008, right after the financial crisis of 2007-2008 hit. All I did was to count and to plot the frequency with which changes in the US dollar 2-year swap rate occurred for different levels of magnitude, both for the period January-December 2007 and January-September 2008. This is not a particularly sophisticated analysis, but a simple visual inspection of the graphs suggests that the assumption of normally distributed changes in interest rates seemed okay for during first period, while during the financial crisis the market behaved quite differently. Anyone who had used a normal (or lognormal) model to describe the price dynamics of 2-year swap rates (and many did!) would have greatly underestimated the probability of occurrences of large interest rate changes. Put differently, the actual distribution of interest rates had, compared to a normal or lognormal distribution, fat tails during the crisis.

Correlations play an important role in risk management and derivative pricing. On a portfolio level, diversification benefits are typically measured by the degree of correlation between individual securities.

Correlation, a measurement of the dependence or independence of financial time series and typically estimated quantitatively as the linear correlation coefficient of return, has always known to be plagued by a number of shortcomings. The most prominent one is that correlations between financial assets vary over time. Historical correlation (i.e., correlation estimated from past data) depends on the phase of the economic cycle and the length of the time series, among other things. Implied correlation (i.e., correlation parameters backed out of market prices of instruments) is much harder to observe than, say, implied volatility due to the scarcity of correlation products traded. There are also other issues in the field of correlation, such as directionality (correlation increasing during a crisis) and autocorrelation (trending correlation).

Empirical evidence suggests that correlation between assets (and asset classes) not only varies significantly over time, but also increases during times of financial crisis of 2007-2008. This is particularly problematic from a risk management point of view because it suggests that diversification benefits and hedge efficiencies are diminishing precisely when they are needed the most. 37 It is worth noting that there also exist pairs of financial measures for which correlation decreases during crisis. For example, short-dated interest rates and short-dated credit spreads tend to be positively correlated during normal (calm) times, but then exhibit a stark decline in correlation during times of crisis (turmoil regimes). Figure 7 .28 illustrates this for three-month US Treasury bill yields and three-month TED spreads. 38 It is interesting that the COVID-19 pandemic does not appear to follow the usual pattern of a financial market crisis, at least as of May 2020.

It is not difficult to see why correlations between many financial market assets are changing significantly during periods of stress. During calm times, most trading activity is triggered by random events, e.g., a bank entering into an interest-rate swap, a pension fund buying particular types of securities or a corporate customer hedging some currency exposure. When this kind of "noise" orders are directed at broker-dealers, they will use discretion to hedge (i.e., balance their trading book) according to where they see relative value. This causes flow to spread across various financial market products, limiting the knock-on effect of one transaction in a particular product on another specific instrument. For example, one broker-dealer may hedge a 9-year interest-rate swap with a corporate customer by entering into an offsetting 10-year swap in the inter-dealer broker market, another may trade in 10-year on-the-run government bonds, yet another may establish a position in 10-year corporate bonds.

However, during times of market stress, 39 market participants are often forced by institutional rules to act in a non-discretionary way. A mutual fund manager facing significant net redemptions must liquidate part of the portfolio in a proportional matter; a broker-dealer forced to scale down risk limits on short notice will likely, as a first step, proportionally reduce risk limits to most of its trading books, forcing trading desks of all kind of asset classes to sell off inventory; investor sentiment switching from risk-on to risk-off causes portfolio rebalancing flows from risky assets into what are perceived to be less-risky assets (e.g., stocks into bonds), creating simultaneously executed flows in otherwise hardly correlated markets. A crisis typically limits market participants' ability to postpone trades, to look for proxy hedges or to view the other market participants' behavior as a cause for relative value opportunities that suggest entering into opposite transactions. Instead, heard-like behavior can be observed. Lock-step trade executions then create high correlations between assets that exhibited only little joint-movement prior to the crisis. In the context of client conversations, you will likely have to defend any implicit or explicit correlation assumption within your analysis or pricing. The client may suspect that there is only an illusion of correlation, created through a careful selection of historical data. The client may fear that correlation is only caused temporarily by current market participants' behavior 40 and that there may be a regime shift down the road. Correlation is not just some statistical relationship, but something you are expected to have an explicit view on and a story to tell about.

Some of the most intelligent, yet humble people I have met throughout my career have been either working in derivative space (i.e., trading or marketing derivative products) or have developed a strong derivative mindset. It is easy to see why. There are so many different ways to model derivative products, to calibrate any given model and to interpret the results from a risk management perspective, that no derivative-savvy person would ever claim to know everything. The more you know about derivatives, the more you realize you don't know, to paraphrase Aristotle's famous saying.

I remember one situation in my early days working as a desk strategist at a derivative sales desk. My job was to ferret out market dislocations that would lead to actionable trade ideas. I had just finished a relative value spreadsheet that compares US government bonds on an asset swap basis. 41 My calculations flagged one bond that traded above par to be cheaper than a neighboring bond that traded below par (on an asset swap basis). I was very excited about this observation and was ready to propose buying the cheap and selling the rich security (in a so-called bond switch) to clients. Luckily, one of my colleagues advised be to speak to an experienced derivative trader at our proprietary trading desk first. When I proudly presented my findings to him, he listened patiently, smiled and then asked me a series of questions. Did you account for the fact that one of the two securities is the cheapest-to-deliver instrument into a futures contract? How did you adjust for different funding levels (in repo)? Which stub rates did you use for the swap calculations? Did you consider that some investors are reluctant to buy bond trading above par for tax reasons? How much is the duration difference "worth on the curve"? Which type of asset swap calculations did you perform (true asset swap or yield-yield asset swap 42 )? At the time, I was discouraged and disappointed that the experienced trader identified so many potential errors in my analysis. I then tried to incorporate all those aspects in my analysis. Not surprisingly, there was no longer a dislocation between the two bonds.

As a derivative-savvy person, the first reaction to any inconsistency in prices is typically not that there is some sort of market dislocation, rather that there must be something conceptually wrong in the way the relative value model had been set up or how it was calibrated. Put differently, a lot of faith is given in the market's ability to price securities in a non-arbitrage matter, while one's own ability to know something other market participants have been missing is constantly questioned.

How a misplaced trust in models can get you into hot water can be illustrated by the following story:

A young finance professor was dating a girl that lived in the same city. One day she suggested to him to move into the same apartment. That way, they would save the money for the second apartment. The finance professor applied option theory to the situation and responded: If we move together, we save rent. However, if we were to separate at a later point, one of us would have to find a new apartment, which is time consuming and difficult. Therefore, keeping the second apartment is an option worth paying for in case the relationship would not last. His girlfriend reflected on his answer, then responded: Well, if you don't have enough confidence in our relationship then let's break up now. She then dumped him.

The story illustrates how the girlfriend had apparently applied linear thinking, while the finance professor seemingly developed a richer model for decision making. Only that his model was not rich enough, ignoring completely findings from signaling theory. By assigning a high option value to the keep-the-second-apartment option, the professor revealed having used a high implied volatility. A high implied volatility in terms of a relationship equates to doubts about the outcome of dating. Sending this signal causes a reassessment of the situation by the girlfriend. Ironically, by trying to hedge the worst-case scenario (break-up), the professor caused it. 43 What this fictitious story shows is that one should never have full confidence about pricing and risk management of derivatives. What derivatives are teaching us is to remain humble and to keep asking "what am I missing?" Before you know it, you will apply this type of skepticism to financial instruments outside of derivatives, or even everyday life. same time as Black and Scholes. There is just not enough time for this kind of political correctness on the trading floor. 31. The model was then refined by Harrison and Kreps in 1979 to use equivalent martingale measures. 32. This is achieved by a Brownian motion à la Bachelier plus a Poisson-like jump process. The model has some other useful features, such as the ability to price options on underlyings that pay dividend or the introduction of risk-neutral valuation. 33. There are two unrealistic assumptions made here: First, that 9-and 10-year yields move in lockstep, i.e. that the yield curve only moves in a parallel manner. Second, that the spread between bond and swap yields (i.e., in case of government bonds the so-called swap spread) does not change. 34. Tuckman (2012, chapter 8). 35. Unfortunately, this is only approximately true. Every time the floating leg of the swap (e.g., linked to EURIBOR) sets, the combination of bond and swap has some duration risk that may require fine-tuning in the hedge. Also, the relationship between bond and swap yield may change over time (as swap spreads have varying degrees of directionality), requiring changes to the hedge ratio between bonds and swaps. 36. In fact, in the absence of transaction costs instantaneous delta-hedging would equate to option replication and should result in an expected P&L of close to zero under certain assumptions (unchanged implied volatility, log-normal distribution of interest rates etc.) 37. See, for example, Chua et al. (2009) . 38. The TED (short for "Treasury-Eurodollar") spread is the difference between the tree-month Treasury bill rate and the three-month LIBOR rate. It is widely used as an indicator of credit risk as it can be viewed as a measurement for the difference in credit quality between US Treasury debt and high-grade bank debt. It represents the risk premium associated with uninsured bank liabilities. 39. One form of stress would be a significant deterioration in asset prices, forcing market participants to realize mark-to-market losses and to adjust their leverage according to capital requirements; another form of stress would simply be a higher volatility in the market, resulting calculated Var's to exceed Var limits and causing deleveraging activity. 40. For example, the volatility-based hedge fund Artemis Capital Management opined in a letter to investors that the increase of global correlations may be related to government interventions (money printing, quantitative easing, etc.) as such stimulus triggers buying of assets across the board. See: Unified Risk Theory: Correlation, Vol, M3, and Pineapples, third Quarter 2010 letter to investors from Artemis Capital Management LLP from September 30, 2010. 41. An asset swap is a combination of an asset, here the U.S. government bond, and an interest rate swap that has matching cash flows and cash flow dates.

This eliminates, at least in theory, most interest rate (or duration) risk and makes assets with different maturity dates and coupon rates comparable to each other. 42. In a true asset swap (also called par-par asset swap ), the swap is structured such that the client invests par (100), while the difference to the traded price is received by or subsidized from the swap desk. In the case of an asset trading above par, the client would pay 100 while the swap desk would subsidize the difference. Since an above-par instrument can be assumed to have a higher coupon compared to an otherwise identical par instrument (same credit quality, same maturity, etc.), the upfront payment received from the swap desk is being amortized via higher coupon payments. In a yield-yield asset swap, the swap spread is not computed based on a swap that matches all cash flows of the underlying bond. Rather, it is the yield spread between the asset and a market-swap of similar maturity. The asset swap would be executed on a DV01-neutral basis, so that a 1-bp move in yield causes the same dollar amount of price change in both the asset and the swap. 43. Maybe this is alike to the so-called observer effect in Physics, suggesting that a measurement within a system cannot be made without affecting the system. By attempting to measure the break-up probability and communicating this to his girlfriend, the professor actually changed the break-up probability.

Inflation-linked Bonds (also called Inflation Bonds or Linkers ) are government bonds designed to offset the capital eroding effects of inflation. The interest rate remains fixed, but the principal is adjusted to match changes in a price index

Some people suspect it has something to do with the church's roughly $200 million sex abuse debt

National Council of Real Estate Investment Fiduciaries Property Index swap)

Parts of Iraq, Syria, Turkey and Iran in modern days

Nanni to a merchant Ea-nasir complaining that the wrong grade of copper ore has been delivered after a gulf voyage and about misdirection and delay of a further delivery

For a more detailed analysis of the hedging needs at the US brewer Anheuser-Busch, see Tuckman

For example: If CME margin requirements are 15%, one can "control" $100,000 of assets with only $

Two corporate customers entering into a bilateral transaction directly with each other creates a number of disadvantages: Transaction needs to occur at precisely the same time; the hedge structure needs to be matching each other (in this case: same maturity of bonds); the transaction volume needs to be identical (i.e., same notional)

A proxy hedge is a risk-mitigating transaction in which the risk of a specific financial instrument is reduced by establishing an offsetting position in another highly (but not perfectly) correlated financial instrument

Only standardized structures are traded. Thus, for customized structures a broker-dealer may only be able to enter into a proxy hedge and will need to keep some residual risk

Depending on whether the option is a so-called European option, or an American option. American options allow buyers to exercise the option rights at any time before and including the day of expiration

is not really a Greek letter

A catchy way to phrase this is to say that it does not pay to read yesterday's newspaper

Variance is a statistical measure for how widely distributed numbers are. More precisely, it is the expectation of the squared deviation of a random variable from its mean

The probability associated with a 1-standard deviation event is roughly 67% for a normal distribution

This is only an approximation because we are ignoring the concept of time value of money and do not differentiate between actual probabilities and so-called risk-neutral probabilities. The goal here is merely to develop a rough intuition, while leaving the technical details to option-specific books like Hull

However, the FOMC may also hold unscheduled meetings as necessary to review economic and financial developments, so even for the Federal funds target rate between scheduled FOMC meetings there is some small volatility

Ignoring the potential intrinsic value of an option at this point

This is to make the trees of the binomial tree re-combining, i.e., an up-move followed by a down-move gives the same value as a down-move followed by an up-move

For simplicity, we ignore the need for discounting the probability-weighted option payoffs (occurring at option expiration) to the present day when calculating the option value as of today

The Black-Scholes formula is simple enough for the first hand-held computers that came out in the 1970s to perform the calculation so that they could be used on the trading floors of major exchanges

I have never ever heard anyone on Wall Street refer to it as Black-Scholes-Merton formula, as it is called in most textbooks to give credit to Robert Merton, who independently developed the model at the References BIS. 2019. OTC Derivatives Notional Amount Outstanding by Risk Category

CHART OF THE DAY: This Is the Most Powerful Chart We've Seen Yet That Shows the Market Is Afraid of Mitt Romney

The Myth of Diversification

Efficient Capital Markets: A Review of Theory and Empirical Work

Federal Reserve Bank of St. Louis. 2020. TED Spread

Much of $100 Million from Sale of Holy Name Lot to Go to Church Sex-Abuse Debts

Options, Futures, and Other Derivatives

Fixed Income Securities: Tools for Today's Markets

Defense of Derivatives: From Beer to the Financial Crisis