Market Risk for Single Trading Positions


 Kristin Williams
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1 Chapter 6 Market Risk for Single Trading Positions Market risk is the risk that the market value of trading positions will be adversely influenced by changes in prices and/or interest rates. For banks, market risk occurs because traders in the Financial Markets department trade for account and risk of the bank: proprietary trading. Since the credit crisis, however, banks have become more prudent in allowing their traders to take positions. Market risk of trading positions can be measured by sensitivity parameters, by the Value at Risk method, by stress tests, by the extreme value theory and, finally, by the expected shortfall method. In order to manage market risk, banks impose trading limits on their traders. 6.1 Market Risk Sensitivity Indicators The first way to measure market risk is the use of market sensitivity indicators. Market sensitivity indicators indicate the sensitivity of a position in a financial value to a predefined change in the price determining parameter(s) of that financial value. The following table shows an overview of the most commonly used sensitivity indicators. 85
2 alm and risk financial value sensitivity indicator price determining variable Foreign Exchange Value of one point / pip FX rate Interest Rate Derivatives Basis point Value Yield PV01 Delta Options Delta Price of the underlying value Vega Volatility Theta Remaining Term Rho Interest Rate Value of one point / pip The value of one point gives the sensitivity of an FX position for a change in the FX rate with one point or pip. For instance, if an FX trader holds a long position in euro against Sterling for a nominal amount of EUR 10,000,000, the value of one point of this position is 10,000,000 x = GBP 1,000. This means that the trader gains 1,000 pound Sterling for every rise in the EUR/GBP FX rate and loses 1,000 pound if the euro depreciates with one basis point against the pound Sterling. The value of one point is also used as a risk indicator with shortterm interest rate futures (STIR futures). It represents the change of the value of one futures contract, e.g. a short Sterling contract or Eurodollar contract, if the futures price changes with one (basis) point Basis Point Value The basis point value (BPV), also called PV01 or (interest) delta, specifies how much the price of an interest bearing instrument changes if the interest rate changes by 1 basis point (0.01%). The equation for the BPV is: Basis point value = dirty price duration
3 market risk for single trading positions example If the price of a bond is and the bond has a duration of 4.6, the basis point value of this bond is: BPV = 98.7 x 4.6 x = This means that the price of the bond will decrease from to if the interest rate rises by 1 basis point. A disadvantage of the modified duration and the way in which the basis point value is used above, is that it assumes implicitly that all zero coupon rates move in the same direction and magnitude; in other words that the yield curve moves in a parallel way. The effect of a parallel shift is shown in figure 6.1. Figure 6.1 Value of a loan with a face value of EUR 100 million before and after a rise in interest rates by 1 basispoint 87
4 alm and risk Figure 6.1 shows that the value of the above loan as a result of a parallel interest rate rise of 1 basis point has fallen by EUR 47, This is also the basis point value for this bond. In practice however, instead of assuming a parallel interest rate shift, sensitivityanalyses to interest rate movements are made per time interval or bucket. Thus, separate analyses are made of the impact of a change in the one year zero coupon rate, in the two year zero coupon rate, etc. By doing this, it will become clear that the interest rate sensitivity is almost always different in all time buckets: bucket (year) amount present value modified basis point duration value ,000,000 5,788,712 1 / ,000,000 5,579,480 2 / , ,000, ,630 3 / , ,000,000 5, / , ,000,000 87,755,301 5 / ,251 As might be expected, the table shows that the interest rate sensitivity of the bond principally lies in the five year bucket. After all, this is where the largest cash flow appears. The above table is often referred to as a gap report. Financial institutions use these kinds of gap reports in order to determine how their interest rate exposure is spread across the various maturity periods. If a bank has a clear idea about the interest rate movement in a specific part of the yield curve, it can use this detailed information to fine tune its hedge transactions. In addition to the basic point value that presents the change in value of an interest bearing instrument or future cash flow as a result of a change of 1 basis point in the zero coupon rate, there is a comparable indicator. This indicator shows the change in value of an interest bearing instrument or a single cash flow as a result of a change of 1 basis point in the credit spread. This indicator is called the credit BPV or CV01, although some banks still use the term PV01 for this The Greeks We have seen that the level of the option premium is determined by several parameters, which may interfere with each other. The extent to which the option premium changes due to a change in one of these price determining factors is indicated by the Greek letters: delta (and gamma), vega, theta and rho. 88
5 market risk for single trading positions delta Delta shows the relationship between the absolute change in the option price and an absolute change in the price of the underlying value. A delta of 0.6, for instance, means that the option premium increases by 60 euro cents if the price of the underlying value increases by 1 euro. The delta also provides an indication of the chance that the option will be exercised. A low delta means that this chance is small, whilst a high delta means that the chance of exercising is high. For instance, a delta of 0.9 indicates that the probability that an option will be exercised is 90%. The table below shows the development of the delta of a GBP call / USD put option with a strike price of and a remaining term of three months for different GBP/USD FX forward rates (volatility is 15%). gbp/usd intrinsic value time value option premium delta forward rate * * Note that the time value of the inthemoney options is equal to the time value of the equal outofthemoney options With a GBP/USD rate of , the delta can, for instance, be calculated as follows: ( ) / ( ) = Call options have a positive delta (between 0 and 1) and put options have a negative delta (between 0 and 1). The delta for an option that is far otm is close to zero, the delta for atm options is always around 0.50 (+0.50 for calls or for puts) and the delta for an option that is deep itm is almost equal to 1 (or 1). 89
6 alm and risk Figure 6.2 The development of the delta of a call option with various prices for the underlying value (delta as a percentage) Figure 6.2 shows that the development of the delta depends on the remaining term of the option contract. As the remaining term decreases, the development of the delta becomes less gradual. Just before expiry, the delta for atm options changes dramatically as a result of small price movements gamma Figure 6.2 also shows that the delta changes if the price of the underlying value changes. Each time that an option becomes less otm or more itm, the delta increases. The degree to which this happens is represented by gamma. Gamma describes the relationship between the change in the delta and the change in the price of the underlying value. If an option is very far otm, the change in the delta is always small. The same applies for an option that is very far itm. In both cases, the gamma is small. For atm options, however, the gamma is high. This is especially the case if the option is approaching its expiry date. This is shown in figure 6.3. Figure 6.3 The development of the gamma at differing prices for the underlying value 90
7 market risk for single trading positions Figure 6.3 also shows that the gamma increases as the remaining period to maturity of an option contract becomes shorter vega/volatility Vega gives the change in the option price due to a change in volatility of 1% point (for example, from 20% to 21%). Vega decreases if the remaining term of the option becomes shorter. When option traders quote prices for volatility, they take into account the socalled smile effect. This means that they use lower volatilities for atm options than for far itm or otm options. The line reflecting the relationship between exercise price and volatility therefore looks like a smile. This is shown in figure 6.4. Figure 6.4 Volatility Smile theta The option premium decreases as the remaining term for an option becomes shorter. After all, an option that still has only one day left offers much fewer (additional) profit opportunities than an option that still has a year to run. The relationship between the decrease in the option price and a reduction in the remaining term by one day is given by theta. Because the option premium decreases as time passes, the thèta is always a negative number. As time passes, the theta of an option becomes progressively more negative; in other words, the option premium diminishes to a greater extent day by day. For options that have nearly expired, the thèta is the most important parameter with regard to changes in the option premium. 91
8 alm and risk rho Rho gives the sensitivity of the option premium to a change in interest rates of 1 percentage point (for instance, a change from 5% to 6%). The sensitivity of the option premium to changes in the interest rate is related to the deltahedge. 6.2 Value at Risk The value at risk (VaR) method is a way of estimating the size of market risk under normal market conditions. A sensitivity parameter shows how much the value of a position changes as a result of a standard change in the price determining parameter. The value at risk tells you how much the value of that position changes as a result of a specific scenario of the price determining parameter. Therefore, a sensitivity indicator, in fact, merely gives information about the size of a trader s position whilst the value at risk gives an approach for the actual loss that a trader can suffer under the current market conditions. To calculate the VaR, each day market risk managers determine a number of scenarios for the market parameters that determine the value of a position or a portfolio for the next day. Which scenario will ultimately be chosen as VaR scenario depends on the desired confidence interval. This indicates the degree of statistical certainty with which the chosen scenario really can be considered as a worstcase scenario. Market risk managers generally use a confidence interval of 99%. Next, the market risk manager calculates how much the value of a trading position would fall if the VaR scenario would actually come true. The result is referred to as the value at risk of the trading position. The period over which the value at risk is calculated is called the holding period or time to close position. The duration of the holding period depends on the speed with which a position can be closed. Trading positions in liquid markets can be closed quickly. For this reason, the holding period for these positions is set at one day. For single trading positions, the historical VaR method is used. This is a way of determining the VaR scenario where price changes over a specific historical period are used in a straightforward way. For trading positions, banks generally use the last 250 to 400 daily price movements. These historical observations are ranked from the most unfavourable price movement to the most favourable. If a bank wants to use a desired probability percentage of 99%, for instance, it will choose the observation from the list for which only 1% of all observations were even less favourable as the VaR scenario. 92
9 market risk for single trading positions example On 15 June 2009, the market risk system calculated the VaR scenario for the price of Heineken shares using the last 250 daily price changes. The system ranked the 250 most recent daily relative price changes for the Heineken share price. For each observation, a confidence level was calculated. The confidence level of the worst observation is 100%. After all, based on these 250 scenarios, it must be 100% certain that the price on the next trading day will not fall by more than 4%. scenario % Price change /4% /3.5% /3% /2.7% /2.5% % / 1.6% /1.5% + 3% +3.5% Probability 100% 99.6% 99.2% 98.8% 98.4% 98% 97.6% 97.2% 0.8% 0.4% This system, however, was programmed with a probability percentage of 99%. As VaR scenario, therefore, it chooses the scenario with the next higher probability percentage. This is scenario 248, which indicates a price fall of 3%. If the share trader of this bank has a long position of 100,000 Heineken and the current price of the Heineken share is EUR 20, the market risk system calculates the trader s Value at Risk as: 3% x EUR 2,000,000 = EUR 60, Stress tests We have seen that banks use a particular confidence interval to determine the VaR scenario. This means that the largest negative extremes are kept out of the analysis. Furthermore, banks only use the price movements from the last 250 or 350 days. This means that banks that use the VaR method not only ignore the most negative scenarios from the historical period that the observed, but that they also take no account of any disaster scenario that took place earlier in the past. For this reason, banks also use another method in addition to the VaR method to indicate their market risk. This method provides information about the risks under extreme market circumstances or market events. This method is called stress testing. The objective of stress tests is to evaluate if a bank is able to survive exceptional shocks in the financial markets. The loss on a trading position that appears with a stress scenario is called event value at risk. 93
10 alm and risk With a stress test, a bank calculates the effect of one or more possible market events on the value of its trading positions. The scenarios used for a stress test can be drawn up in various ways. The first possibility is to use scenarios that have actually happened, such as nine eleven ( ). However, the disadvantage of this method is that events from the past are highly unlikely to happen again in the same way in the future. Banks have therefore also made up their own hypothetical scenarios for extreme market circumstances. For instance, they assume a change in exchange rates of 10% or an interest rate change of 100 basis points. Many banks use both, historical and fictitious scenarios. Apart from stress tests, banks are required to perform reverse stress tests. The purpose of a reverse stress test is to identify scenarios and circumstances that will cause the banks business model will become unviable. 6.4 Extreme value theory An alternative for stress testing is studying the behaviour of what can happen during unusual market conditions by using a technique that is referred to as extreme value theory. The first step of extreme value theory is to identify the observations during a specific observation period that can be used to characterize the extreme losses. There are two kinds of model for collecting the extreme observations. The first one is the block maxima model. This model divides the observation period in blocks and then takes the maximum loss within each block as a data. For example, if the observation period is one year and the daily results are registered on a daily basis, we can choose the worst outcome for each month as an extreme. This is shown figure 6.5 where the observation period is from March until March the following year. The extreme for each month is indicated by a bold x. 94
11 market risk for single trading positions Figure 6.5 Block Maxima model The second, and more commonly used method, is the peak over treshold (POT) model. In this model all large observations that exceed a certain threshold during the observation period are identified as extremes. For example during the above mentioned observation period every outcome over a daily change in prices or rates of 2% is identified as an extreme. This is shown in figure 6.6. The extremes are again indicated by a bold x. Figure 6.6 Peak over Treshold Model 95
12 alm and risk Once the extremes are identified, a distribution for extreme tail loss is made. This distribution provides information about the market behaviour during extreme situations. The most important problem of the extreme value theory is obviously the fact that there are are little data. This can be solved by decreasing the time period of the blocks in the Block Maxima model or by lowering the threshold in the Peak over Treshold model. However, in that case it is questionable whether the observations in the then larger sample can be considered as extremes. 6.5 Expected shortfall The expected shortfall, also referred to as conditional VaR, (expected) tail loss or average VaR, is defined as the conditional expectation of loss given that the loss is beyond the VaR level. The expected shortfall is the mean of all the potential losses that exceed the VaR. Where VaR asks the question how bad can things get?, expected shortfall asks if things do get bad, what is our expected loss?. example The expected shortfall in the example in paragraph 6.2 can be calculated by taking the mean of losses under the extremes 4% 3.5% and 3%. These losses are respectively , and The expected shortfall is ( ) / 3 = Trading limits A trading limit indicates the maximum open position that a trader is permitted to hold. Trading limits may apply either for an entire department within the dealing room (trading desks) or for individual traders. The trading limit for a trading desk is determined by the committee that is responsible for drawing up the limit control sheet (LCS). The allocation of limits between individual traders at a specific trading desk is the responsibility of the desk s departmental head. Junior traders are generally allowed to hold only small positions. A trader s limit is raised as his experience and as his profitability increases. Banks use two types of trading limits to manage market risk, value at risk (VaR) limits and nominal limits. At any moment in time, a trader must satisfy all his limits. 96
13 market risk for single trading positions Value at risk limit A VaR limit sets a limit to the VaR of a trader, however, it does not set a fixed limit to the nominal position of a trader. example A shares trader has a VaR limit of EUR 500,000. If the VaR scenario for today is a price decrease of 2%, the maximum allowed market value of the shares position, according to this VaR limit, is EUR 25 million. After all, the VaR is then 2% of EUR 25 million = EUR 500,000. For a VaR scenario of 1%, however, the maximum allowed market value of the position would be EUR 50 million. In quiet market conditions, the price changes in the VaR scenarios are relatively small. If a bank would only use a VaR limit, a trader could hold very large trading positions. This is dangerous because, even after a very quiet period, the market can suddenly become extremely volatile and the possible losses could then become very large Nominal limits With the VaR limit, the allowed size of a position is dependent on the current market circumstances. A nominal limit, in contrast, set an absolute maximum on the size of a trading position. Since the credit crisis, banks have become much more careful about using only VaR limits, and they are increasingly using nominal limits in addition to VaR limits. Nominal limits impose a limit to the size of a trading position regardless of market developments. The most simple nominal limit is a positions limit. A positions limit sets an unconditional limit on the market value of a position. An example is an FX trading limit where the EUR/USD FX trader is allowed to hold a position of maximum EUR 5 million long or short. For interest rate positions and options, dedicated limits are used. Finally, sometimes traders are assigned a stress test limit nominal limits for interest positions A gap limit sets a limit to the mismatch position in terms of volume and time. A money market trader is, for instance, only allowed to have a mismatch position in a 97
14 alm and risk single maturity bracket of not more than 100 million. Another exeample is an interest rate derivative trader who is only allowed to take positions not longer than five years. If an FX trader is allowed to trade FX forwards, he is also assigned a gap limit. The same is true for FX swap traders. A basis point value limit sets a limit to the market value of an interest bearing portfolio measured by its basis point value, assuming a parallel move of the yield curve. If the BPV limit for a trader is, for example, EUR 50,000 this means that he is allowed to hold the following positions: market value modified duration bpv 500 mio 1 50, mio , mio 10 50,000 A variant of the basis point value limit is the credit spread sensitivity limit. This is a limit to the market value of a bond portfolio measured by its change in price as a result of a change in the credit spread of the issuer of one basis point. A slope risk limit sets a limit to the market value of an interest bearing portfolio measured by the change in this value as a result of a predefined change in the slope of the yield curve. For instance, a trader may not loose more than EUR 15,000 if the interest rates for the shorter periods, e.g. up to two and a half years, fall with 1 basis point and at the same time the interest rates for the longer periods rise. A trader that holds the position that is shown in the table below, complies with this limit. bucket (year) basis point value result of the predefined scenario EUR 10, EUR 10, EUR 08, EUR 08, EUR 06, EUR 06, EUR 09, EUR 09, EUR 12, EUR 12, Total change in market value EUR 10,
15 market risk for single trading positions greek limits for option positions Greek limits set a limit to the value of an option portfolio measured by its Greek parameters, the delta, gamma, rho and vega. Delta limit and delta hedging The delta limit sets a limit to the sensitivity of an option position to changes in the price of the underlying value. The main business for option traders is trading volatility. However, when an option trader opens a position by buying or selling an option, the value of his position is not only influenced by changes in the volatility but also, amongst other things, by the price movement of the underlying value. In other words: the option trader also has a virtual position in the underlying value. If the delta, for instance is 0.50, this means that an option position behaves in the same manner as a position in the underlying value for half the contract amount. This is called the delta position of the option position. example An option trader has a long position in call options with a contract volume of 100,000 shares. The delta of the options is The current premium of the options is 4. This means that the market value of the options position is 400,000. If the price of the underlying rises with 1 unit, the option premium rises with and the market value of the options position rises with 15,000 to 415,500. The position thus reacts in the same manner to a change in the share price with one unit as a long position of 15,500 in the underlying shares. If the option trader would have sold this call option his position would react, of course, in the opposite way: i.e. as a short position of 15,500 shares. The delta position of this trader is a short position of 15,500 shares. The common opinion amongst the management of Financial Markets Departments, however, is that options traders must leave trading in shares to share traders, in bonds to bond traders, in FX to FX spot traders et cetera. Options traders with banks, therefore, normally are not allowed to be exposed to changes in the price of the underlying value. In other words: their delta limit is set close to zero and they must make sure that the delta of their position is zero. Option traders theoretically can realize a zero delta position by always concluding a call option and a put option with the same delta at the same time. If an option trader, for instance, wants to have a long position in volatility, he can buy either a call 99
16 alm and risk option or a put option. After all, buying an option means buying volatility. However, if the trader would only buy a call option, he would enter into a virtual long position in the underlying value. To offset this delta positions, he could buy a put option with the same (opposite) delta. And if he would only buy a put option, he would enter into a virtual short position in the underlying value. Now he can offset his delta position by buying a call option with the same delta. However, in reality the delta position is neutralized in another way: the socalled delta hedge. To neutralise the effect of price changes of the underlying value, option traders with banks take a position in the underlying value that is exactly the opposite of their delta position. This is called delta hedging. The option trader s position is then said to be delta neutral. The value of the composite position now only changes as a result of changes in volatility, the remaining term of the option and the interest rate. In an ideal world, options traders would also want to make the value of their position independent of changes in the remaining term and in the level of interest rates; however, this is not possible. Fortunately, this is not a great problem because these factors are much less volatile than the price of the underlying value and thus play generally no major disruptive role. Because the delta of an option changes when the price of the underlying value changes, an option trader must constantly adjust his delta position during the term of the option contract in order to keep his position delta neutral. The size of the transactions as a result of the delta hedging depends on the level of the gamma, that represents the changes in delta. For a low gamma, only small transactions are necessary. For a high gamma, however, an option trader must buy or sell more of the underlying value to keep his position delta neutral. example An option trader has sold a GBP call / USD put option to a client with a strike price of The premium for this option is USD per GBP and the size of the option contract is GBP 1,000,000. At the start date of the option contract term, the delta of this option is The current GBP/USD FX forward rate is As an initial delta hedge, the option trader has bought GBP 250,000 against USD. On a later moment, the GBP/USD FX forward rate has risen to As a result, the delta has also increased, for instance to The option trader must now adjust his delta position by buying 0.05 x 1,000,000 = 50,000 British pounds. If, however, on a still later moment, the GBP/USD FX forward rate falls to , and the delta falls to, for instance, 0.18, the option trader must sell 120,000 British pounds. 100
17 market risk for single trading positions The above example shows that if the delta of an option position increases, an option trader must buy the underlying value and if the delta of the position falls he must sell the underlying value. With this, he will constantly suffer small losses. This is because, in contrast to the golden rule, he buys high and sells low. The option premium is partially a compensation for these trading losses. When quoting his option premium, an option trader makes an estimate of the volatility of the underlying value (implied volatility). A high volatility means that the option trader expects that he will have to adjust his delta position frequently and will have to accept great trading losses. Thus, he asks a high option premium. If the option trader estimated the volatility correctly, he earns the margin on the premium that he had calculated. If he underestimated the volatility, he would suffer a loss. In this case, the premium is not sufficient to offset the trading losses resulting from the delta hedge. The delta hedge can also be used to explain the relevance of the interest rate for the option premium. After all, an option trader who has a short position in call options must buy the underlying value in order to perform his delta hedge. This will involve interest costs. Similarly, an option trader who has taken a short position in put options must sell the underlying value. This produces interest income. example An option trader sells a call option on a share with a remaining term of three months. The delta for this option is The three month interest rate is 4%. The current share price is EUR 40. Due to the delta hedge, the trader must buy 0.25 shares for each option contract unit. The interest costs of the delta hedge, therefore, are: 0.25 x EUR 40 x 90/365 x 0.04 = EUR The option trader will include the interest cost of EUR 0.10 in the option premium. Gamma limit and vega limit Even if an options position has a delta of zero, the position can be very risky. This is especially the case if the remaining term is short and if, at the same time, the option is atthemoney. In this case, a small change in the price of the underlying value can lead to a large delta position that, by definition, only can be hedged at a considerable loss. Therefore all traders are assigned a gamma limit. 101
18 alm and risk The vega limit sets a limit to the sensitivity of an option position to changes in the volatility of the underlying value of the options position. Together with the gamma limit the vega limit is the most relevant limit for an option trader. After all, option traders trade in volatility stress test limit and expected shortfall limit A stress test limit or event risk limit sets a limit to the market value of a position as a result of a predefined market disruption. In order to set an event risk limit, the market risk management department must design socalled stress tests. With a stress test, a bank draws up one or more future disaster scenarios in order to be able to assess the risk associated with future extreme mar ket movements. These scenarios could be an actual historical scenario such as nine eleven (when the twin towers came down in New York). The disadvan tage with this method, however, is that events from the past will most probably not occur again in the same way in the future. Market risk management will therefore also usually create its own imaginary disaster scenarios. For example, it will assume a 10% change in the currency exchange rates or an interest rate change of 100 basis points. Market risk management will then calculate the possible losses for the traders as a result from these imaginary disaster scenarios. A stress test limit is a nominal limit because, it is not influenced by the current market conditions. Banks can also set a limit on the expected shortfall. The expected shortfall limit prevents traders from taking very risky positions whilst at the same time they satisfy with their VaR limits. If, for instance, a trader has a oneday 99% VAR is $10 million, there is a danger that the trader will construct a portfolio where there is a 99% chance that the daily loss is less than $10 million and a 1% chance that it is $500 million. The trader is now satisfying the VaR limits but is clearly taking unacceptable risks. By setting a limit to the expected shortfall, banks can limit their risk more effectively than by only using VaR. 102
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