The previous chapters provided a macro view of risk management, including its relationship to risk measurement, the definition of key concepts such as economic capital, and descriptions of the core risk-measurement techniques. Having built that foundation, we can now move our discussion to a micro level—namely, a detailed discussion of specific risk-measurement tools and techniques for each type of risk.
We will first spend several chapters discussing the measurement of market risk for traded instruments. We will then move on to asset/liability management, credit risk, operating risk, and finally the approaches that can be used to pull all the risks together to measure the total risk to the bank.
Market risk arises from the possibility of losses resulting from unfavorable market movements. It is the risk of losing money because the perceived value of an instrument has changed. The difference between credit risk and market risk is that in credit risk there needs to be a default or failure by a counterparty to fulfill an obligation. In market risk, we deal simply with changes in the prices that investors are prepared to pay.
Over the next few chapters, we will describe the types of transactions that cause market risk, several ways of measuring the risk, and how banks organize themselves to manage the risk.
Before launching into the mathematics of risk measurement, it is necessary to have a reasonable understanding of the types of trades and instruments that give rise to the risk. This chapter will detail the main instruments that banks trade and why they trade them—along with an explanation of how to value each of the instruments. Valuation is very important in risk measurement because risk is all about potential changes in value. Typically, a bank trades the following instruments:
• Debt instruments, also known as fixed income or bonds
• Forward rate agreements
• Equities, also known as stocks
• Foreign exchange, also known as FX or currency
• Forwards and futures
• Swaps
• Options
Futures, swaps, and options are different types of derivatives contracts. They are called derivatives because their value is derived from other instruments, such as bonds. We will look at each of these instruments in detail.
In valuing instruments, one of the principal techniques is to decompose a complex instrument into a set of simple instruments with the same payments. The technical term for this is arbitrage pricing theory. An arbitrage is a trade in which a set of securities are bought and sold such that the combination provides a profit but has no risk. Such a riskless profit is attractive, and once the combination is well-known in the market, many investors will buy and sell the components of the arbitrage until their buying causes prices to adjust and the arbitrage opportunity goes away. In describing these equivalent arbitrage portfolios, we will use the terms long and short. Being long roughly corresponds to having an asset, and being short roughly corresponds to having a liability. If a bank is long an instrument, it owns the rights to the instrument. If it is short, it has promised to give the rights to another party. For example, if the bank is short a bond, it means that they have promised to give the bond to another party.
For debt instruments, we will look at their general structure, their valuation, and how changes in their values can be related to changes in interest rates.
Debt instruments are securities that provide interest payments but no ownership claim on the issuer. Debt is a liability or an obligation on a company or government agency to make specified payments. Debt and fixed-income securities are synonymous. Until recently, all debt contracts had fixed-interest payments, and the term “fixed income” continues to be used even though many debt securities now have interest payments that change depending on market conditions. Debt instruments include bonds, notes, commercial paper, syndicated loans, and 144a issues. These are described further in the introduction to the credit-risk section.
From a market-risk manager’s perspective, there are four important features for debt instruments:
• Maturity
• Issuer credit rating
• Payment structure
• Currency
Let’s look at each instrument in more detail.
Maturity is the time left until final payment. Although the time to final payment is a continuous variable, market convention has different names for different levels of maturity. Instruments with a maturity of over five years are called bonds. Those with a maturity of one to five years are called notes, and those with a maturity of less than a year are called bills or money market instruments. The money market is used by corporations and banks for short-term funding. The short-term interest rates are driven by immediate supply and demand, and are relatively insensitive to the long-term economic conditions. Because of their short maturity, the value of a money market instrument is quite insensitive to the prevailing interest rates, and therefore has relatively low market risk.
As discussed earlier, the credit risk of most bonds is rated by a third-party company called a rating agency, such as Standard & Poor’s. These ratings are very important for three reasons:
• The ratings generally correspond to probabilities of default. Bonds with high probabilities of default trade at lower prices than risk-free bonds. As we discuss later in the chapter, a lower price implies a higher interest rate. The difference between the interest rate for a risky bond and the interest rate for a risk-free bond of the same maturity is called the credit spread.
• Many institutional investors, such as pension funds, must adhere to fund restrictions that require them to buy only bonds of “investment grade.” To be investment grade, a bond must be rated BBB or better.
• The Basel Committee on Banking Supervision is prompting bank regulators, such as the Federal Reserve and the Bank of England, to set bank capital requirements according to the credit quality of the bonds they hold. Table 4-1 below shows one of the options for the amount of capital to be set aside for corporate loans.
TABLE 4-1 The Effect of Credit Rating on Regulatory Capital to be Held
The payments of interest on a bond are called coupons and are either fixed or floating. Fixed-rate bonds pay the same percentage every time, and the rate is fixed when the bond is first issued. Floating-rate bonds pay a variable percentage and the interest rate is reset periodically to a prevailing market rate. For example, a five-year floating-rate bond would make payments every six months. At each time of payment, the amount for the next payment will be set according to the prevailing six-month, interbank borrowing rate. A small additional spread would be added to compensate for the relatively illiquid nature of a five-year bond, compared to reinvesting in a series of six-month bonds.
Debt instruments can be denominated in any currency. The probability of default tends to be less if the currency of the bond is the same as the base currency of the entity issuing the bond. This is especially true of governments who always have the option of printing more money to repay bonds denominated in their own currencies. This makes domestic government bonds almost risk free; therefore, bonds of a given currency have their interest rates set relative to the bonds of the government that issues that currency.
Now that we have a better understanding of the various characteristics of debt instruments and how they are classified, it is time to explain how they are actually valued. This is the foundation for measuring the risk of changes in bond values. We will first explain how to value a bond with a single known payment. Then we will discuss how bonds with multiple payments can be valued using a yield curve. The discussion may at first seem a little circular because we use observed prices to get a yield curve, then use the yield curve to estimate prices. However, the utility of the yield curve is that by observing the price of a few bonds, we can create a yield curve and thereby predict the price for all bonds.
It is important to understand how single payments are valued so we can value complex bonds by decomposing them into single payments.
Any bond promising to pay a certain amount at a future point in time has a value. The value is the price that investors are prepared to pay today to own that bond, and therefore own the right to the future cashflow. If we consider a bond that promises to make a single payment, at time t, then the current value of the bond and the future payment amount can be used to define a discount factor (DF):
The discount factor will be less than one, as it is almost always better to have cash now than the same amount promised in the future. The discount factor includes all the effects that cause the value to be less than the promised amount, including the effects of inflation and the possibility of default. If the cash flow is certain (e.g., a fixed-rate government bond), then the discount is called the risk-free discount. If the cash flow is risky, e.g., if there is a possibility of default, the value will be less, and therefore, the discount factor will be smaller. The discount factor for a given maturity, t, can be used to define a discount rate, rt, which is more familiarly known as an interest rate. The usual expression for the discount rate is as follows:
This expression uses discrete compounding. For continuous compounding, the rate is defined by using the exponential function:
DFt = e–tx rt
e is approximated by 2.7. Discrete compounding is more intuitive for most people, but continuous discounting is easier to use in the mathematics of finance, e.g., in option pricing. The value for rt will be slightly different depending on which convention is used.
Zero-coupon bonds are bonds that do not pay interest explicitly. They have only a single “bullet” payment, and are therefore also known as bullet bonds. The bond is sold at a discount to face value, with the difference between the face value and the sale price implicitly being the interest payment. The value of a risk-free zero-coupon bond is simply the amount of the payment multiplied by the risk-free discount factor:
If there were many zero-coupon bonds being traded with different maturities, we could use the above equation to construct a graph with t in the x-axis and rt in the y-axis. Such a graph is called a yield curve and is illustrated in Figure 4-1. The terms yield and discount rate are often used synonymously.
Figure 4-1 shows the average yield curve for the U.S. dollar over the last 10 years. The curve generally, but not always, slopes up because of the additional cost of liquidity for long-dated cashflows, and because long-dated cashflows have a greater risk of their values being eroded through inflation.
Using the current yield curve we can closely estimate the current value of any bond. Coupon-paying bonds have a series of fixed-intermediate-interest payments. The value of a coupon-paying bond is the sum of the value of the individual payments.
FIGURE 4-1 The Average U.S. Yield Curve over the Last 10 Years
Using the rates from the yield curve, the bond can be valued as the discounted sum of the payments or cashflows:
Here, Ct is the cash flow at time t, and the price is the price at which the bond can currently be bought.
It would be relatively easy to value all bonds if there were many traded zero-coupon bonds whose price could be observed. However, there are many zero-coupon bonds traded for short maturities up to about a year, but bonds with longer maturities typically have coupons, and therefore, it is not possible to directly observe the zero-coupon yield for long maturities.
As an alternative to direct observation, traders synthetically create the implied zero-coupon yield curve using “bootstrapping.” In bootstrapping, we first observe the price of a zero-coupon bond and calculate the discount rate for that maturity. We then observe the price of a coupon bond of a slightly longer maturity, and we subtract the value of the short-term coupons using the discount rate that we observed from the zero-coupon bond. The remaining value must be the price that the market is willing to pay for the longer-maturity cashflow, and we can therefore define a discount rate for this longer maturity. We then repeat the process, steadily observing bonds with longer maturities, and subtracting the value of the earlier payments using the discount rates we found as we progressed up the yield curve.
As an example of bootstrapping, consider two simple bonds with the payments and prices shown in Table 4-2. Bond A has 1 cashflow in 6 months. Bond B has a cashflow at 6 months and one at 12 months. We will use these to find the discount rate for payments in 12 months.
TABLE 4-2 Two Bonds Used for the Bootstrapping Example
From the price of Bond A, we calculate that the 6-month discount rate is 4.1%:
The $3 coupon on Bond B at 6 months is therefore worth $2.94:
The remaining cashflow of $100 at one year must therefore be worth $95.06. This allows us to calculate the one-year discount rate to be 5.2%:
In summary, bonds are valued by decomposing them into a series of single payments and discounting each payment according to the yield curve:
The yield curve may be constructed by directly observing the price of zero-coupon bonds or by using bootstrapping to isolate individual cashflows within a bond of multiple payments.
The yield-curve process described above is the one used in risk measurement. But you may also hear of bond values being described in terms of their “yield to maturity.” The yield to maturity (y) is the internal rate of return for a bond, given the current market price and the future cashflows. It is the single value of y that satisfies this equation:
The yield to maturity is continually changing as market prices change. Notice that there is an inverse relationship between price and yield: as the price increases, the yield drops. A coupon-paying bond is said to be at Par if the yield to maturity equals the coupon rate. This is equivalent to saying that the price equals the face value. It is said to be at a Premium if the coupon rate is greater than the yield, and it is at a Discount if the coupon is below yield.
Duration is a measure of the interest rate sensitivity of the value of a bond or loan. This is useful because once we know the duration and the possible movement of interest rates, we have a measure of how much value the bank could lose in its bond portfolio.
As an example, if a bond had a duration of seven, it would mean that that value of the instrument would decline by seven percent if interest rates rose one percent. Duration is calculated based on changes in the present value of the bond. As discussed in the previous section, the price or value (V) is the sum of the discounted cash flows (C):
For convenience, we will define Vt as the present value of the single cash flow that occurs at time t:
The present value for the whole bond is then given by:
There are two types of duration: Macauley duration and modified duration. Macauley duration is easier to calculate, but modified duration is more accurate. Macauley duration is simply the average time for cash flows, weighted by their present values:
Macauley duration was useful when only slide rules were available, but computers allow us to use the more accurate modified duration. Modified duration is based on the derivative of the value with respect to interest rates:
Modified duration is found by dividing this derivative by the initial value and multiplying by negative one. The multiplication by negative one is used to make modified duration compatible with Macauley duration:
If we had not divided by V we would have had the absolute price sensitivity, which is called duration dollars:
The change in the value of a bond is closely approximated by the derivative of value with respect to rates, multiplied by the change in rates:
From here, the link to duration is obvious:
You may have noticed that the unit of duration is time. This may seem strange, but it is because duration is the sensitivity with respect to interest rates, and interest rates are in units of increase per time period. Therefore, since duration is change in percentage value per change in interest rate, the unit of duration is time.
Having come this far, you may wonder why we bother with duration rather than just using the derivative directly. The simple answer is that it is market custom. Many bank financial reports will give “duration,” but few will give “the derivative of value with respect to interest rates.” Therefore, it is necessary to understand the meaning of duration.
Duration has significant limitations as a measure of risk. It is the first derivative of value with respect to rates. As such, it is a linear measure and only describes changes in the value of a bond based on small parallel shifts in the yield curve. It does not describe the value changes that could result from the convexity of bond prices or complex shifts in the yield curve.
Convexity is the nonlinear relationship between the price of a bond and its yield, as illustrated in Figure 4-2. As duration is a linear description of value changes, it becomes increasingly inaccurate as the rate change becomes larger.
Most yield curve movements can be broken down as combinations of shift, twist, and flex, as illustrated in Figure 4-3. The yield curve is said to shift if rates of all maturities move by the same amount. The curve twists if long-term rates move a different amount compared to short-term rates. Flex occurs if long- and short-term rates move in the opposite direction of medium rates.
In later chapters, when we come to measuring the Value at Risk for bonds, we will find that shift, twist, and flex can be accounted for relatively easily by letting rates at different maturities move by different amounts. However, to account for convexity, we need to give up the linear approximation of duration and use full bond valuation.
FIGURE 4-2 Illustration of Bond-Price Convexity
FIGURE 4-3 Illustration of Yield-Curve Shift, Twist, and Flex
The discussion above assumed that the bond had coupons that were fixed. If the coupons are floating, the duration will be much less because if rates move, the coupon payments will move to compensate. Each time the rates are reset, the value of the bond comes back to par, i.e., the value of the bond equals the principle amount. This means that the value is insensitive to changes in the interest rates and the duration is close to zero. In practice, most floating-rate bonds reset the rate for the next payment a few months in advance. You know that on the date of that next reset, you will get the specified cashflow plus you will own a bond that you can sell for the full principal amount. This is equivalent to having a fixed-rate bond that matures on the next reset date. The duration, then, is equal to the time remaining until that next reset, which is typically much less than the time to maturity.
A forward rate agreement (FRA) is a contract to give a loan at a fixed rate starting at some point in the future. For example, the agreement could be that in 2 years’ time the bank will lend $100 to the customer. The customer will then pay 8% per year for 5 years, and in 7 years’ time the customer will pay back the principal of $100. Customers would want such agreements if they had projects for which they expected to start construction in 2 years, and they wanted to fix the borrowing costs now.
Using arbitrage arguments, a forward rate agreement can be decomposed into being long a fixed-rate bond, with the first coupon being paid at some starting point in the future, and short a zero-coupon bond corresponding to the bank’s initial payment to the company. This is illustrated in Table 4-3 for the example above.
Often, the FRA does not entail an actual loan with all of its associated covenants and payments. Instead, at the date on which the loan would have commenced, the parties pay the difference in net present value (NPV) of the cashflows. The calculation of the NPV uses the market rate on the commencement date to discount the cash flow. For the previous example, the calculation of the payment from the bank would be as follows:
TABLE 4-3 Representation of a Forward Rate Agreement as Two Bonds
The discount rates, r1 to r5, are those existing on commencement date. For example, r5 is the 5-year discount rate that will be observed 2 years from now. If rates had risen to be higher than 8%, the term in brackets will be less than $100, and the bank will owe money to the company. The company can then take this money and use it as compensation to offset the increased cost of borrowing that it would experience if it now went into the market and obtained a loan at the prevailing rates.
Up to this point, we have discussed how to value an FRA once the forward lending rate has been set; now we will discuss how that rate is set. When initiating a forward rate agreement, a bank will set the borrowing rate according to the forward rates currently observed in the market.
Forward rates are obtained from the current yield curve using arbitrage arguments. The argument is that if an investor wants to invest money until some distant time, t2, she has two choices:
1. Buy a bond that matures at t2.
2. Buy a bond that matures at an intermediate time, t1, and buy a forward rate agreement that locks in an interest rate from t1 to t2.
Let us use the symbol r1 for the interest rate currently available for bonds maturing at t1, and r2 for those maturing at t2. We will use f1,2 to represent the forward rate that can be locked in from t1 to t2.
If the amount to be invested is $100, the first investment strategy will yield the following result:
Casht2 = $100(1 + r2)t2
The second strategy will yield the following:
Casht2 = $100(1 + r1)t1) (1 + f1,2)t1–t1
If either of these strategies provided better results, all investors would move to that strategy, increasing the demand for the given product and changing the rates until both strategies gave the same result. The forward rate is therefore related to the two rates from the yield curve as follows:
Equities are also known as shares or stocks. Equities represent ownership in a company for the holder, and a right to the profits once all of the debts have been paid. Equity values reflect all of the factors associated with the business risks, and as a result, are very volatile compared with bonds.
In addition to stocks that give direct ownership in a company, there are also synthetic instruments such as American Depository Receipts (ADRs) that are not equities, but are contracts designed to mirror the cash flow of an underlying equity and therefore have the same risk. It is also possible to invest in an instrument that pays the same return as an equity market index, and also has equity-like risk. These include swaps in which one side pays a fixed rate and receives an amount proportional to the level of the stock market.
Investors use many techniques to value equities, but most of these include a large amount of intuition and “art.” The best way to learn about market risk for equities is to open an Internet brokerage account, invest $1000, and watch it disappear. From a risk-measurement perspective, equities are so complex as to be simple. There are so many factors that affect equity values that they are generally considered simply to be instruments whose future value is random. The future changes in value are typically considered to have a Normal or Log-normal distribution with the same standard deviation as the historical changes in the equity price.
The value changes can be further broken down into systemic and idiosyncratic risks. The systemic risk is the movement in the equity price that occurs because of a general movement in the stock market. The amount by which the stock tends to move with the market is called the beta (β). The idiosyncratic risk (
), describes price movements that are uncorrelated with the market and are due solely to the performance of the individual company. This can be summarized in the following equation for the change in value:
ΔV = V0(βΔM + )
Here, V0 is the current value, ΔM is the change in the market index, and is the uncorrelated idiosyncratic change. For a diversified portfolio of many equities, the idiosyncratic risks tend to cancel each other out, leaving the trader or investor exposed to the sum of the systemic risks of all the stocks:
Here, N is the number of equities, βk is the beta for equity k, and Vo,k is the current value of equity k. This is the approach most commonly used when assessing the overall risk of a portfolio.
Foreign exchange (FX) trading is also known as currency trading. Generally, FX markets are the most liquid of all of the markets; i.e., large volumes are traded, and it is easy to find someone to buy or sell at a price close to the current market value. There are many ways that currencies can be traded, including spot, forwards, foreign securities, and derivatives.
Spot FX trades are simply an exchange of currencies, and the settlement of the exchange typically happens within a day. Forward trades are agreements to exchange specified amounts of each currency at a specified date in the future. Securities with payments denominated in a foreign currency carry both the risk of changes in value of the security (e.g., a drop in the value of a foreign equity), plus the risk of a change in the exchange rates. As an example, consider a U.S. bank holding a bond issued by a Mexican company. The bank could lose money if the company defaults, if the peso interest rates increase, or if the peso devalues compared with the dollar.
The value of an instrument denominated in a foreign currency is simply found by calculating the current value of the instrument in the foreign currency, then exchanging to the local currency at the prevailing rate.
Most trading is conducted in the spot market (also known as the cash market). In the spot market, trades are settled as soon as operationally possible after the trade agreement has been made, typically within a couple of days. In contrast to spot trades, a forward contract is an agreement to buy a security or commodity at a point in the future. At the time of agreeing to the forward contract, the amounts are fixed for the quantity of the security or commodity to be delivered, the delivery price to be paid, and the delivery date. Typically, the delivery date is several months into the future. Here are some examples of forward contracts:
• A bond forward is an obligation to buy or sell a bond at a predetermined price and time.
• An equity forward is an obligation to buy or sell a specific equity at a predetermined price and time.
• An FX forward is an obligation to buy or sell a currency on a future date for a predetermined exchange rate. On that date, there will be simultaneous exchange of the full amount of each currency.
• A gold forward is an obligation to deliver a specified quantity of gold on a fixed date and receive a fixed delivery price.
Forwards are one of the types of derivative contracts because the value of the forward is derived from the value of the current or future spot prices of the underlying security or commodity. After the contract has been agreed, but before the delivery takes place, the value of the contract will change. This change occurs because new contracts are being signed with different delivery prices. As an example, consider entering into a contract to buy 100 ounces of gold in 3 months’ time with a delivery price of $250 per ounce. One month later, assume that a crisis occurs, people want gold, and are willing to pay $300 for an ounce of gold to be delivered in 2 months. In this case, you could guarantee to deliver 100 ounces in 2 months for $300/ounce. In 2 months, you would be certain to make a net profit of $5000. The current value of the contract is therefore worth $5000 discounted at the risk-free rate for 2 months. If the discount rate was 5%, the contract would be worth $4959.
In general, the value of a forward agreement is the number of items to be delivered (N), times the difference between current market delivery price (DC) and the originally agreed delivery price (D0), discounted by the risk-free rate (rf) from the time of delivery (t):
As a slightly more complex example, let us consider a contract to deliver 100 British pounds and buy 150 U.S dollars in 3 months’ time. The value of that contract in U.S. dollars is 150 dollars discounted at the 3-month U.S. interest rate, minus 100 pounds discounted at the 3-month U.K. rate, then exchanged into dollars at the current spot FX rate. This is summarized in the equation below:
The value of this contract will change if either of the interest rates or the FX rate changes.
An alternative way of doing this analysis is to consider entering into a forward exchange from pounds to dollars at the current FX forward rate, then discounting the result according to U.S. interest rates:
Contract Value = (1 + rUS,3mo)3/12 x (150 – 100 x FXUS/UK,3moForward)
Here, 150 is the agreed delivery price, and 100 × FXUS/UK,3moForward is the current market price for 3-month delivery. For there to be no arbitrage opportunities, the forward rate must be related to the current spot rate and interest rates as follows:
Futures are the same as forwards contracts except that they are traded on exchanges. The market risk for futures is essentially the same as forwards, but there are three practical differences:
• Futures are for standardized amounts and standardized delivery dates, and forwards can be for any amount and date.
• The credit risk for futures is reduced because they are exchange traded. If counterparties fail to honor their agreements to make deliveries, the exchange will make the deliveries in their place.
• The credit of risk futures is reduced because the change in the value of the futures contract is offset by compensatory daily payments, so the value of the contract minus the value of the cash you have received is always close to zero.
The value and the market risk for futures are evaluated in the same ways as for forwards.
A swaps contract is an agreement between two counterparties to exchange payments at several specified points in the future. The amount of the payments is determined by a formula in the contract. The formula will typically specify the payments as a function of market factors, such as the short-term interest rates, FX rates, or commodity prices.
Swaps are derivatives because their values are derived from the current and future values of underlying securities. Swaps were first developed in the 1980s, and now there are many types, including the following:
In an interest-rate swap, a regular fixed amount is paid by one counterparty, and a floating amount is paid by the other counterparty. The floating amount is the prevailing short-term interest rates multiplied by a notional loan amount. The notional amount can be thought of as the size of an imaginary loan or bond.
For example, Party A agrees to pay Party B cash flows every 6 months for 5 years. The amount of the cashflow is 4% (annual 8%) times a notional amount of $100 million (i.e., $4 million every 6 months). In return, every 6 months Party B will make a payment equal to $100 million times the LIBOR divided by 2 (to account for the payments being semiannual). LIBOR is the London Interbank Offer Rate; it is the rate at which banks lend to each other. The lending can be in any currency, and London is chosen rather than any other market because of the liquidity of the market. As an example, 6-month U.S. dollar LIBOR is the rate at which banks will lend U.S. dollars to each other for 6 months.
In practice, each time a payment is made, the 6-month LIBOR will be observed and used to fix the payment to be made in 6 months’ time. The counterparties exchange the difference between the cashflows rather than the whole amount. For example, if a payment had just been made, and 6-month LIBOR was 10%, the payment at the end of the next 6-month period from counterparty B to A would be $1 million: (5%–4%) × $100 million.
The value of such a swap can be determined by considering the swap to be the combination of a fixed-rate bond and a floating-rate bond. In the example above, counterparty A has the same cashflows as if it was short a fixed-rate bond and long a floating-rate bond. The value of the swap to counterparty A is therefore the value of a floating-rate bond of $100 million minus an 8% fixed-rate 5-year bond of $100 million.
Typically, the swap payments are modified slightly to reflect any difference in the creditworthiness of the 2 parties. This credit spread is much less than would be used for a full loan of the same notional value because the actual exposure amount is much less than the notional amount. For example, on an interest-rate swap with a $100 million notional amount, the interest payments due will be just a few percent of the $100 million. If the counterparty defaults, the bank will lose the interest payments, not the full $100 million.
A currency swap is a combination of an FX spot and FX forward transaction with principal amounts exchanged at the beginning and end of the transaction. Thus, risk measurement of currency swaps is accomplished by simply disaggregating the swap into a spot and a forward.
A basis swap is the regular payment of one floating amount against a different floating amount. For example, the U.S. Treasury rate against LIBOR.
An equity swap is the regular payment of equity index or an equity value against a floating interest rate, such as LIBOR.
In general, the valuation of a swap can be accomplished by deconstructing the swap into the two underlying instruments on each side of the swap, and then valuing those instruments.
An option is a type of derivative contract. An option gives the holder the right but not the obligation to buy or sell an underlying asset at a future time, at a predetermined price. The predetermined price is called the strike price. The party holding the right to choose is said to be long the option, and the counterparty who sold or wrote the option is said to be short the option.
Options are similar to futures, but with one important difference: the holder can choose whether to exercise the option at the time of contract expiration. Options carry an “aura of mystery” because they can be difficult to value. Adding to this mystery is the fact that there are many option classifications. Therefore, this section will focus on explaining the different types of options and approaches used to price them and to assess the risk. The goal of this section is to give the reader an intuitive feel for the factors that drive pricing and risk.
Options can be classified as one of the following:
• Vanilla options (so called because traders consider them to be as plain as vanilla ice cream). These have very simple contract terms for payments and are relatively easy to value.
• Packages of vanilla options are simply the sum of several vanilla options put together into one deal.
• Exotic options are nonvanilla options that are complex and typically tailored for an individual customer. It is possible to write virtually any form of option contract depending on any observable market factor, such as equity prices, exchange rates, or even the weather.
Options can be categorized according to the rights given to the holder:
• Puts: A put option gives the holder the right to sell the underlying security and receive a predetermined strike price.
• Calls: A call option gives the holder the right to buy the underlying security by paying a predetermined strike price.
The strike price is agreed on at the beginning of the option contract, and is the cash amount to be paid for the underlying security. At the beginning of the contract, the counterparties also agree on the ultimate maturity (or expiration date) and alternatives to exercising the option, including:
• European-style, which can only be exercised at the end date.
• American-style, which can be exercised on any date until maturity.
• Bermudan-style which can only be exercised on certain dates. (The option is so called because Bermuda is partway between Europe and America.)
• Asian-style, which bases payments on an average price over a period. This technique is useful for shallow markets that are highly volatile or exposed to the risk of price manipulation.
Each option style can now be traded in any geography. When the option is exercised, there are two possibilities for settlement: physical delivery and cash settlement. Physical delivery requires that the strike price be paid, and the underlying security should be delivered. Cash settlement requires only that the difference in the value of the security at the time of exercise and the strike price of the option should be delivered.
To give a more intuitive understanding of how options work, let us look at examples of a call and put option.
Consider a European-style call-option contract on IBM stock with maturity on August 21, 2002, and a strike price of $75. If you own this option, then you may buy 100 IBM shares—100 shares being a contract norm—on August 21, 2002, and pay a strike price of $75 each. If, on that day, the stock is worth $82, the option is “in the money,” and you will receive $700 as determined by the following equation:
Call payoff = number of shares x (value – strike)
$700 = 100 x (82 – 75)
If the stock is worth $72 (out of the money), then the option would expire worthless because you would not want to buy the stock for $75.
Consider a European-style put-option contract on IBM stock with maturity on August 21, 2002, and a strike price of $75. If you own this option, then you may sell 100 IBM shares on August 21, 2002 and receive a strike price of $75 each. If, on that day, the stock is worth $72, you would sell the shares for $75 and receive $300. The payoff would be $300:
Put payoff = number of shares x (strike – value)
$300 = 100 x (75 – 72)
If the stock is worth $82, the option would expire worthless because you would not want to sell the stock for $75.
As we have seen in the example above, there can be great value in holding an option, and therefore, a price has to be paid to obtain an option. The first step in determining what price to pay for an option is understanding the potential payoff at the expiration date. The payoff is the value of the option contract at the time of maturity. It depends on the structure of the contract and the value of the underlying security.
Let’s use the symbol V for the value of the payoff, X for the strike price, and S for the value of the underlying stock that is bought or sold.
In general, on the expiration date the value of a call is the maximum of 0 or S – X:
Call: V = max(0, S – X)
If S is more than X, the call is said to be in the money. Otherwise, it is said to be out of the money.
The value of a put at expiration is the maximum of 0 or X – S. If S is less than X, the put is in the money.
Put: V = max(0, X – S)
The logic for the call and put is summarized in Table 4-4.
Figure 4-4 illustrates the payoff as a function of the stock price at the time of the exercise. As the stock price on the x-axis changes, the stock price on the y-axis also changes, but the strike price stays fixed. The payoff is shown in terms of what you have to give to exercise the option and what you get in return. If you exercise the option, the difference between what you get and what you give is the payoff.
In the discussion above, we established the value of the payoff at the time when the option expires, but what is the value of an option at some time before expiration? As an example, consider a call option that has a strike price that is higher than the current stock price. If the call was about to expire, it would have zero value. However if there were several months before expiration, there is a possibility that by the time of expiration, the stock price will move to become greater than the strike price. If that were to occur, the option would be valuable at expiration. If we were given the chance to buy this option several months before expiration, how much should we be prepared to pay for the possibility of it becoming valuable? The answer to this question is provided by option valuation.
TABLE 4-4 Logic and Payoff for Exercising Options
FIGURE 4-4 Option Payoff as a Function of the Price of the Underlying
There are three primary ways to find the value of an option:
• Formulas based on the Black-Scholes equation
• Binomial trees
• Monte Carlo simulations
The Black-Scholes equation was developed by two professors who were later awarded a Nobel Prize for their accomplishment. The equation is relatively easy and fast compared with other pricing approaches because it has several simplifying assumptions. Because of the simplifications, it is only accurate for vanilla puts and calls. However, it is often used as a first approximation to the value for many other types of options.
In its basic form, the Black-Scholes equation calculates the current value of holding an option on a stock. The equation assumes that the option can only be exercised at maturity, and that there are no dividends or transaction costs. It also assumes that the percentage change in the stock price has Normal distribution.
With these restrictions, the value of a call option on a stock is a function of the following five variables:
S = Spot price of the underlying security
T = Time left to maturity of the option
X = Strike price of option
r = Risk-free interest rate to the time of option expiration
σ = Annual volatility for the underlying security.
(Volatility is the standard deviation of the price change as a percentage of the initial price.) Given these definitions, the Black-Scholes equation for the price of a call option (C) is as follows:
where
N(d) is the value of the standard cumulative Normal distribution function for d. In other words, N(d) is the probability of a random number having a value equal to or less than d. The cumulative Normal distribution function is illustrated in Figure 4-5. Notice that as d goes to negative infinity, the probability of the outcome being less than d goes to zero. As d goes to positive infinity, the probability of the outcome being less than d goes to one.
Understanding the cumulative Normal function allows us to understand how changes in the variables affect the value of the call. We stated that the Black-Scholes equation is as follows:
C = SN(d1) – Xe–rT N(d2)
FIGURE 4-5 Cumulative Normal Probability Distribution
The first term, SN(d1), says that value of a call increases with the value of the underlying stock, S. The second term Xe–rT N(d2) says the value of the option is less if the strike price, X, is high, and the value is greater if the time, T, or discount rate, r, is high. The effect of high T or r is to reduce the NPV of the strike price that would need to be paid if the option were exercised.
Let us now consider d1 and d2. Notice that the difference between d1 and d2 is that in d1 the volatility is added, and in d2 it is subtracted. You could roughly think of d1 as being the extent to which the price could move up and d2 the extent to which it could move down. If the volatility, σ, or time, T, increases, d1 becomes larger and d2, becomes smaller; therefore, the difference between N(d1) and N(d2) becomes larger, and the value of the call increases. In general, holding an option becomes more valuable with an increase in uncertainty in the outcome. This increase in value occurs because the payoff is asymmetric. If uncertainty increases, it is more likely that there will be a large change in the stock price, up or down. If the stock price moves up a large amount; we gain a large amount; if the stock price falls, we do not lose as much because the value of the option has a floor of zero.
As the option reaches expiration and T goes to zero, d1 and d2 both become equal to the following:
If S is greater than X, the logarithm will be positive, d1 and d2 will equal positive infinity, and N(d1) and N(d2) will equal one. The value of the call at expiration is then simply:
CT=0 = S – X
If S is less than X, the logarithm will be negative, and N(d1), N(d2), and the call value equal at expiration equals zero. This corresponds with our earlier analysis of the value of an option at expiration.
As a numerical example, consider a call option on a stock that has a current price of $100 and an annual variance of 2.5%. Assume that the risk-free rate is 5%, the call option has a strike price of $100, and the time to expiration is 6 months. The values for these parameters are given in Table 4-5.
To give an appreciation of the effect of parameter changes on option values, Figures 4-6 to 4-10 show how the price of the example option changes when there are changes in the stock price, strike price, risk-free rate, volatility, and time to maturity.
Note in Figure 4-6 that when the stock price becomes low, there is little probability of it ever rising above the strike price, and the option value drops toward zero. However, when the stock price is high, there is little chance of it dropping below the strike price, and the option will almost certainly be exercised. In this case, the value of the option is the stock value minus the net present value of the strike price.
The opposite effect happens if we keep the stock price fixed and move the strike price, as illustrated in Figure 4-7.
TABLE 4-5 Parameter Values for Option Example
FIGURE 4-6 Effect of Stock Price on the Value of a Call Option
Figure 4-8 shows that changes in interest rates have little direct effect on this option. This would not be the case if the underlying for the option was an interest-rate-sensitive instrument, such as a bond.
Figure 4-9 shows that the option value drops as the volatility of the stock price decreases. Ultimately, if the volatility drops to 0, the stock becomes a risk-free investment. In this case, the value of the stock should grow steadily at the risk-free rate to equal $102.53 at the expiration date. With no volatility, the option is therefore certain to be in the money by $2.53. The NPV of $2.53 is $2.47, which is the current value of holding the option with no volatility.
FIGURE 4-7 Effect of Strike Price on the Value of a Call Option
FIGURE 4-8 Effect of Interest Rate on the Value of a Call Option
FIGURE 4-9 Effect of Stock-Price Volatility on the Value of a Call Option
FIGURE 4-10 Effect of Change in Time to Expiration on the Value of a Call Option
Figure 4-10 shows that as the time to maturity increases, the uncertainty increases, and the option becomes more valuable. If there was no time to maturity left, the option would be worthless because for this option we set the strike equal to the stock price.
The Time Value of an Option
The time value of an option is defined as the actual option value minus the intrinsic value. The intrinsic value is the value that would be realized if the option was about to expire. For a call, the intrinsic value is the maximum of zero or (S – X). The time value for a call option is summarized in the following equation:
Call Time Value = C – max[0, (S – X)]
For a put, the intrinsic value is the maximum of zero or (X – S).
Figure 4-11 shows the value of an option for different possible values of the stock price. Three lines are shown, the value of an option with six months to expiration, the value with three months to expiration, and the value if the option had to be exercised immediately. As the time decreases, the time value drops to zero, and the option price falls to the intrinsic price.
The time value is a function of both the time to expiration and the volatility of the stock. The time value increases with uncertainty, and is therefore large if the volatility of the underlying is high and the time to maturity is long. With increased uncertainty, it is more valuable to have a choice between the stock and the strike.
Implied Volatility
In the discussion above, we used values for S, T, r, X, and σ to find the price of the call, C. Alternatively, if we observed options being traded in the market at a given price, we could use the Black-Scholes equation in reverse to find what value of σ would be needed to cause this price. This value is the implied volatility. It is the expected volatility of the stock that is implied by the current trading price of the option. It is interesting to know this implied volatility because it tells us how much volatility the traders are expecting in the stock price. It is therefore a measure of future expectations. Consider pricing an option the day after a crisis has occurred in the market. If we looked at historical data for prices, we would find that the volatility was relatively low; however, the implied volatility would be high. This is because historical volatility describes what has happened, but implied volatility describes what is expected to happen.
FIGURE 4-11 Time Value of an Option
In using the Black-Scholes equation to price an option, we mentioned several simplifying assumptions, such as the assumption that relative changes in stock prices have a Normal distribution. These assumptions mean that the estimated value of the option will not be quite equal to the true value of the option, as shown in Figure 4-12. Similarly, when we use the Black-Scholes equation in reverse to find the implied volatility, the simplifying assumptions cause slight distortions. The two most important distortions are the “volatility smile” and the “volatility skew.”
The volatility smile is the phenomenon that the implied volatility for options that are deep in or deep out of the money is significantly higher than for options at the money. This occurs because true stock-price movements do not have the Normal distribution assumed in the Black-Scholes equation. In reality, extreme movements are more common than would be predicted by the Normal distribution. Therefore, options that are deep out of the money have a higher chance of moving into the money than would be predicted using the Normal distribution, so the true price is higher than that predicted by the Black-Scholes equation. If we were to maintain the assumption that stock prices move Normally, this higher price could only be accounted for by assuming that the volatility was higher. The same mechanism also applies to stocks that are deep in the money.
FIGURE 4-12 Difference between the Actual Option Price and the Theoretical Price
There is a slight difference between options that are in or out of the money, and this produces the volatility skew. The volatility skew is most strongly observed for options on equities. In this case, options that are out of the money appear to have a higher volatility than those in the money. This is because of supply-and-demand effects in the market that tend to increase the value of out of the money options relative to in the money options. Figure 4-13 illustrates the volatility smile and skew. The smile is represented by the fact that a and b are greater than zero; the skew is represented by a being greater than b.
So far, we have discussed the properties of the analytical approaches based on the Black-Scholes equation. As we have seen, these approaches make significant assumptions about the nature of the market and the structure of the option. However, there are two approaches that reduce the number of required assumptions: namely, binomial trees and Monte Carlo evaluation. These approaches are used by traders in cases where the Black-Scholes approaches would be inaccurate, e.g., for exotic options.
Binomial trees use a tree of possible future values for the underlying and the associated option values. The base of the binomial tree is the current market condition. From that base, the tree takes a small step forward in time and has two possible branches for the stock price: it can either go up or down. The required amount of up or down movement in the price and the associated probabilities are tied together using reasoning similar to that used in the Black-Scholes equation. From the end of each branch, another two branches step forward in time. This is repeated until the branches reach the option’s expiration date. At this point, the result is a tree of possible future stock prices.
FIGURE 4-13 Illustration of the Smile and Skew for Implied Volatility
Having established the range of possible stock prices, a typical valuation process is as follows. The payoff for the option is valued for all the possible stock prices at the end time, and the process then moves back through the tree towards the current date. At each time step, the option is valued based on its possible future value along each of the branches and the probability of taking each branch. Figure 4-14 illustrates a simple binomial tree to value the call option in our previous example.
Although Monte Carlo evaluation can be easier to construct than a binomial tree, and more accurate, it does require a large amount of computer time because it typically uses many more possible future values than the binomial tree.
In essence, Monte Carlo evaluation randomly simulates future price paths for the underlying and evaluates the option along each path. The option value is then the NPV of the average option value for each path. Figure 4-15 gives an example of 5 random simulations. Figure 4-16 gives the outcome distribution of 1000 simulations for the stock price on the expiration date, and Figure 4-17 gives the corresponding outcome distribution of the option value. As we would expect, the distribution for the option value is the same as for the stock except that it is shifted by the value of the strike price ($100), and all of the results in which the stock price was less than the strike price are piled onto $0.
With this new understanding of the factors affecting option prices, let us look at how options are used in trading. If a trader has a specific idea about the way the stock price will change, a mixture of the stock plus puts and calls can be used to create a payoff structure that is tailored to profit in the expected market. Many different forms of payoff can be formed by combining options and the underlying instrument. The most common combinations are the addition of an option and the underlying as illustrated in Figure 4-18.
FIGURE 4-14 Illustration of a Two-Step Binomial Tree
The upper tree shows stock-price movements and their associated probabilities starting at the left with a stock price of 100, a 55% chance of moving up to 108, and 45% chance of moving down to 92. The lower tree shows the value of holding the option given the possible future movements of the stock. The value of the option at one point is the probability-weighted value of the option at the next two points, discounted according to the time difference.
FIGURE 4-15 Example of Five Monte Carlo Simulations for Stock Prices
Being long the stock and long a put means the trader benefits if the value of the stock increases. But if the stock value decreases, the trader is protected from any loss because the put can be exercised to give away the stock and receive the strike price. This combination is known as a protection put.
FIGURE 4-16 Distribution of Outcomes for Stock Prices
FIGURE 4-17 Distribution of Outcomes for Option Prices
Being long the stock and short a call means that if the stock rose above the strike price, the trader would expect the holder of the call option to exercise the option and take the stock in return for giving the strike. If the stock remained or fell below the strike, the option would be worthless, and the trader would have gained the initial income for having sold the call. This combination of owning the stock and being short the call option is known as a covered call.
Other common packages of options are described below. These combinations are called spreads because their value depends on the difference, or the spread, in the price between different options. The total price of the package is simply the sum of the prices of the individual options.
FIGURE 4-18 Payoff for Options Combined with the Stock
This spread is created by buying a call with a strike price, X1, below the current price of the underlying, and selling a call with a strike price, X2, above the current level. The cost of the call option with the low strike price will be greater than the profit made by selling the call with a high strike price. Figure 4-19 illustrates the profit from a bull spread as a function of the price of the underlying. Note that we have subtracted the cost of initially buying the option so you can see the overall profit rather than just the final payoff.
For a bull spread, if the stock price is low, the options expire worthless, and the investor loses the initial cost of the options. If the stock is high, both options are in the money, but the option struck at X1 is worth more than the option struck at X2, so the investor makes a profit. It is therefore called a bull spread because it is bought by investors who expect the market to rise.
It may seem odd to have sold the upper call because by selling this call the trader has limited the potential profit. However, by selling the upper call, some income is generated to offset the cost of buying the lower option.
The bear spread is made by going short a call option with a low strike price, X1, and long a call with a high strike price, X2, as in Figure 4-20. It works in the opposite way of the bull spread, and is profitable if the price of the underlying drops.
The investor initially makes a profit because the call with a low strike price is more valuable than the call with a high strike price. If the price of the stock drops below X1, both options expire worthless, and the investor keeps the profit. If the final price of the stock is high, both options are in the money. However, the option struck at X1 is worth more than the option struck at X2, so the investor has to pay out more on the short position than is received on the long position, creating a loss.
The butterfly spread consists of four options: long a call option struck at X1, short two call options struck at X2, and long a call option struck at X3, as illustrated in Figure 4-21. It can be viewed as the sum of the bull and bear spreads. The butterfly spread is profitable if the underlying does not move significantly. However, if there is a significant movement, the investor loses the premium and receives no payoff.
The calendar spread uses two call options with identical strike prices but different maturity dates. The strike prices are chosen to be close to the current price of the stock to maximize the difference in time value. The spread is made by going long a long-term call and short a short-term call. By having identical strike prices, the investor is largely protected from changes in the stock price. The investor will have to pay more for the long-term call than the short-term call because of the difference in the time values. The time value of the short-term call will decay faster than that of the long-term call, increasing the difference in the prices. If the volatility of the stock does not change, the investor will be able to sell the combination after a few weeks for more than the initial purchase price. If the volatility increases, the difference will be even greater, and the investor will make more of a profit. If the volatility decreases, the investor could make a loss. Figure 4-22 illustrates the three cases.
FIGURE 4-19 A Bull Spread
A bull spread made by buying a call option with a low strike price and selling a call with a high strike price. The x-axis shows possible values for the stock. The y-axis shows the corresponding profit from the option.
FIGURE 4-20 A Bear Spread
A bear spread made by selling a call option with a low strike price and buying a call with a high strike price.
FIGURE 4-21 A Butterfly Spread
A butterfly spread made by buying a call option with a low strike price, selling two calls at the money, and buying a fourth call with a high strike price.
FIGURE 4-22 A Calendar Spread
The profitability of the calendar spread is shown as a function of the stock price. If the stock price moves far from the strike price, the time value of both options falls towards zero, and the investor makes a loss. Three cases are shown; if volatility increases, the investor makes an additional profit. If volatility decreases, the investor could make a loss.
The set of exotic options is comprised of any options other than vanilla options. They are tailor-made for specific profit or hedging opportunities, and the payoffs cannot be decomposed into the payoffs from vanilla options; i.e., you cannot recreate the pattern of payoffs from an exotic option by combining a set of vanilla options. Although exotic options are relatively rare, they have an infinite number of possible forms. The more common ones are described below.
Forward starts have a strike price that is to be fixed at a future date. For example, the option may give the right to buy a stock within a year, for whatever the stock price is in six months’ time.
Binary options have discontinuous payouts. For example, the option might be structured to pay $20 if the final stock price is anywhere above $110, but $0 otherwise.
Look-back options pay the maximum (or minimum) price of the underlying over a certain period of time, minus the strike price. The payment is made on the expiration date. At that time, the participants “look back” to see the highest price of the underlying between the initial date of the option contract and the expiration date. These options do not depend on the final price of the underlying (unless the final value also happens to be the peak value). The value of the options is therefore path dependent, meaning that the history of the stock price determines the value of the option, not just the final price.
Barrier options are also path dependent, and have a clause that determines what will happen when the price of the underlying hits a certain barrier. For example, a knock-out barrier option has a clause that says the option pays out and ceases to exist as soon as the stock hits a certain level. A knock-in option says that the option does not exist until the stock hits a certain level. The barrier may be either higher or lower than the current stock price. If the barrier is higher, the options are called “up and out” or “up and in.” If the barriers are lower, the options are called “down and out” or “down and in.”
Compound options are options on options. They give the right to buy an option in the future.
Chooser options allow the holder to specify whether the option should be a call or a put at some point in the future. For example, the contract may say that in 6 months’ time, the owner of the option must choose whether the option should be a put or a call, and that put or call will expire in 18 months’ time. Chooser options are also known as “As You Like It” options.
Before we go on to the mathematics risk for options, it is useful to know some of the common words and terminology used when trading options:
• An outright or naked position means having an unhedged position, such as selling a call option without owning the stock.
• Having a covered call is being short the call and long the stock. This is a useful technique because it limits the risk for the writer of the option. If the stock becomes very valuable, the writer will be able to deliver the stock without first having to find cash to buy the stock.
• A back-to-back position is one that is fully hedged, e.g., being long an option with one counterparty and short the same option with a different counterparty. This eliminates the market risk because any losses on one option are offset by gains on the other. Each option is priced slightly differently to enable the trader to make a profit. Typically, this difference in price is possible because one of the options is being provided to a corporate customer who does not have full access to the market.
Now that we have discussed the different options and their valuation techniques, we can begin to discuss risk measurement for options.
As we discussed above, the value of options is sensitive to changes in the value of the underlying, interest rates, and volatility. This means that changes in market conditions can lead to the risk of losses for banks that hold options. The sensitivity of options to these market changes is described by what is known as the Greeks. They are called Greeks because they are symbolized by the Greek letters delta, gamma, vega, theta, and rho. The Greeks are similar to duration in that they estimate the change in the value of the option if one of the market variables changes. Each of the Greeks is described below.
Delta is the first derivative of the option price (P) with respect to the value of the underlying stock (S):
Delta is the linear approximation of how much the value of the option will change if the value of the underlying changes by $1.00. For example, if delta is 0.5, then a $1.00 move in the stock will cause a 50-cent change in the option value. Often, traders will describe a position as being “delta-hedged.” This means the trader has bought a combination of options and the underlying such that the overall position has a delta equal to 0. This removes most of the exposure to the underlying, but leaves the gamma, vega, rho, and theta risks.
Delta is not an exact description of the change in the value of the option because, as we have seen, the option value is not a simple linear function of the stock price; therefore, the change in the option value is not proportional to the change in the underlying. For example, if the underlying moves up by $1, the option value may increase by 55 cents; but if it falls by $1, the option value may decrease by 45 cents. The difference between the true value change and the change predicted by delta is called gamma risk, and is illustrated in Figure 4-23.
Gamma is the second derivative of value with respect to the price of the underlying, and it describes how much more the price of the option will change beyond the linear approximation of delta:
Gamma helps to describe the nonlinear risk of the option by using the Taylor Series Expansion, as shown below:
Vega describes the option value’s sensitivity to changes in the volatility (σ). It is the first derivative of the option price with respect to implied volatility. It represents how much the option value will change if the volatility of the stock price changes by 100% per year. If vega equals 0.6, the value of the option will increase by 60% if volatility increased by 100%:
FIGURE 4-23 Illustration of Gamma Risk
Rho is the first derivative of the option price in relation to interest rates. It represents how much the option value changes when interest rates change:
Theta is the first derivative of the option in relation to time. It represents how much the option value changes as it moves toward maturity:
The Greeks can be calculated analytically by using calculus to differentiate a pricing equation, such as the Black-Scholes equation. Alternatively, the Greeks can be calculated by pricing the option with one value of the risk factor and then slightly moving the risk factor and pricing again.
For example, delta would be calculated from the following equation, where is a small addition to S. Typically, would be chosen to be approximately equal to the volatility of the risk factor:
Notice that the Greeks give linear approximations to changes due to one risk factor. If several factors change, the total value change can be estimated by summing the Greeks:
While many trading rooms use the Greeks as a primary measure of risk, most trading rooms also use the Value at Risk tool to measure option risk, and this will be discussed in the next chapter.
In this chapter, we detailed the main instruments that banks trade and why they trade them. We also explained how to value each instrument so that its risk can be managed effectively. Next, we will explain how to measure their changes in value and their market risk.