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# FIN 4913 FIN 4913

OK State

GPA 3.5

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This 0 page Class Notes was uploaded by Domingo Parker on Sunday November 1, 2015. The Class Notes belongs to FIN 4913 at Oklahoma State University taught by Timothy Krehbiel in Fall. Since its upload, it has received 26 views. For similar materials see /class/232776/fin-4913-oklahoma-state-university in Finance at Oklahoma State University.

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Date Created: 11/01/15

Chapter 15 Nonlinear Risk Market losses are attributable to the combination of exposures to market risks and the occurrence of adverse movements in market risk factors A risk manager can control exposures but has no control over movements of market risk factors The exposure to market risk of positions involving options or positions involving contracts with embedded options is not constant This characteristic is fundamentally different than positions incorporating linear positions in the underlying and linear derivatives forwards swaps futures From portfolio theory A linear combination portfolio of normal distributed random variables risk factors is also distributed normal While the average change in value of the combination is the weighted average of component s average changes the variance of the combination is not The standard deviation of the combination will be less than the weighted average of the component s standard deviations if the correlation between the changes of a pair of components is less than 10 Hedge positions constructed from combination of a position in a linear derivative with a linear position in the underlying could possibly be modeled with a symmetric distribution like the normal However because options and positions with embedded options produce asymmetric payoffs the change in value of positions containing these cannot possibly be modeled by a symmetric distribution like the normal The importance of this cannot be over stated On Assignment 1 the Value at Risk of LS or SL positions constructed from an underlying exposure and a position in a futures contract utilized the result that linear combinations of normal distributed random variables are themselves normally distributed It is not possible to utilize the same sort of methodology to measure the benefit in terms of risk reduction produced from a similar combination of an underlying exposure and a position in an options contract Likewise it is not possible to apply the iid normal methodology to measure the risk inherent in positions involving options or positions with embedded options Swmp e Remms Undenyme spun Swmp e Remms aHhermuney CaH zuuu 15mm mun opuor as afuncuon ofmarketfactors andterms ome opuor contract fr Srv39rv39r vUrvaT Where 5 rree ofunderlymg annual eo unuously eorr oundednsk free rate from mll T n P ual eormrruously eompounded yreld on unde 1yr mnualxzed standard devrauor onoganmmre change m underlymg39s pnee r e pnee r T ewume to errprrauor L m above the rate of change of an opuor pnee m response to change m amarket factor It rs necessary to measure nsk ofaposmonm opaorrs o It is necessary to measure risk of a position with embedded options A Taylor series expansion of the general derivative pricing relationship identi es how the derivative s value changes in response to changes in the market risk factors and time to expiration The second order term for dS is included to explicitly account for the convexity of the option pricing relationship 2 df del xdS2ngr 6 Xdr 6f Xd0aiXdTRn as 2332 er a at 6r 039 50100 0 05 r 70 0 00 25K10010 25 D elta a ls the measure ofthe apaxunzto dS from aposmon m an opuou Delta for outrofrthermoney optlous approaches zero Delta forlnrthermoney optlous approaches lfon 0 and e for r 0 Delta for atrthermoney opuous near 0 pa ons have hlghly unstable olelta Ianhermoney optlous absolute value deltalncreases as tune to expuauou decreases utrofrthe money optlous absolute value olelta decreases as tame to expuauou decreases CaH optlon delta and We to explratlon xs the measure ofthe rate of change m A a As xdenufxed above short dated options have the highestgamma e most unstable delta Gamma Long opuon posmons have posmve g 5mm opuon posmon hav amma s e negative gamma The gammafor sxmxlarly de nedputs and calls 15 equivalent CaH opuon Gamma r md me to expvauon Vega 3 Vega A i 15 ameasure of the responsiveness of me denVauVE39s pnee to a enange m the Volauhty of the underlying pnee or dutant dated opnons 15 large Vega for ear dated opnons 15 small Vega decreases for opnons far from me money 35 opuon Vega and Ume to expwauon a 8 Theta 7 measures the responsrvehess othe dehvatrve39s phee to the passage of tame Theta measures the rate of deeay m the value of dehvatave as tame passes tetheerhorrey ophorrs have the greatest theta Nearterrh opuons h vs the greatesttheta Short ophor posrtmhs eam e a Long ophor posrtmhs pay theta CaH opuon Theta and ume to exptrauon Theta Chapter 14 7 Hedging linear risks A bond example Recall the first order approximation of bond price changes from chapter 1 AP le XPolAy Which states that the change in market value of a fixed income security portfolio in response to a onetime parallel shift in the term structure of interest rates Ay is a function of the positions modified duration D A similar relationship for futures contracts written on fixed income securities is useful in constructing hedge positions Let Go the current contract price AG DG gtlt G0Ay The duration for a fixed income futures contract is the duration for the asset underlying the contract In the case of the Chicago Board of Trade s US Treasury futures contracts this will be the duration of the cheapest to deliver issue for that contract httpwwwcbotcom March 05 10year note G0 11307 11321875 DG 86 years Contract size 100000 Initial margin per contract 850 Notice that changes to both the market value of the underlying position AP and the change in futures contract price AG are a function of the same risk factor Ay The optimal hedge ratio Co var ianceAP AG VarianceAG h in this case can be computed directly from knowledge of the durations of the underlying position and the futures contract CovarianceAPAG DS gtlt P0 DG gtlt GO 02 Ay VarianceAG DG X 6012 02 Ay or Ds XPol h DG X G0 Creating a delta neutral position ie hedge position with zero duration h 2 745 X 116290867 2 418898 86gtlt11321875 Number of contracts 100000000 100000 08898 X 8898 E 890 Initial margin requirements 850 X 890 756500 This methodology can also be used to achieve a target duration interest rate risk exposure for a given position The objective is to combine the existing underlying position with the appropriate futures market position to achieve the desired objective D xP NDG gtltFD1 arg gtltP 15p ij gxP h D G X G Suppose that the portfolio manager would like to decrease interest rate risk exposure but retain some of the potential for upside due to a decrease in the level of interest rates What is the appropriate position in the MAR 05 7 10year note contract to achieve a modi ed duration of 5 yearsgt 745 5 X 1 16290867 02926 86 x11321875 Number of contracts 02926 X W 2926 E 293 100000 Initial margin 293 x 850 249050 Resulting risk exposure VaRc DH gtlt P0 gtlt norm s1n v1 c X 0Ay X M Much of the modeling illustrated in previous courses and carried out in practice is based on a model of risk factor increments distributed identically and independently iid normal For example it is common when considering individual equity returns to assume that the price appreciation component of continuously compounded returns th lnSthSt over horizon h will be identically and independently distributed Nu 62 This assumption implies that for each successive period of length h the X13141 is an independent random draw from the normal distribution The iid assumption has implications for aggregating risk factor increments to longer horizons H gt h For instance if we have a sample of trading day closing equity prices and want to know properties of a 5trading day return X1715 lnSt5St 1nSlSo SzSl S3Sz S4S3 3534 1113130 1113231 lnS3Sz lnS4S3 1113534 The iid property implies that 2 2 039 xl5t 5 X 039 xlll 0r oxl5l oxllt iid distributed variables exhibit the absence of autocorrelation positive autocorrelation p gt 0 Km p X EH1 negative autocorrelation p lt 0 Km p X EH1 In the presence of positive autocorrelation the true standard deviation of Km will be greater than that implied by the iid assumption ie 0xtht gt 0xt1gtlt 5 In the presence of negative autocorrelation the true standard deviation of KM will be less than that implied by the iid assumption ie 0mm lt mm x 2 When computing statistics from sample data to inform risk statements it is important to consider the precision with which the statistic estimates the true population characteristic For instance the Central Limit Theorem applied to the sample mean In indicates that for samples drawn from a normally distributed random variable or as the sample size increases is itself normal and the variance of the distribution of the sample mean decreases with sample size T 639 mNu gt sem J The sampling distribution of the sample variance A2 2 4 2 A A 1 039 N039039 7 gtse039039 7 T1 ZT Notice that for a given sample size T the estimate of standard deviation is more precise than the estimate of the mean Additionally as the sample size is increased the precision of the estimate of standard deviation improves at a faster rate than the precision of the estimate of the mean Common risk factors historical time series euro 7 13998604 wti 7 12868904 st7 128081604 thy 7 14808604 wseum sap sun mm Descriptive statistics for increments of 1510 days euro wti spx increments de ned as logarithmic changes continuously compounded returns t10y increments de ned as simple changes euro count max min mean stdev skewness ku rtosis 1day 1407 0027089 002114 480E05 0005879 984 42749 5day 281 0034379 00419 0000189 0014333 023071 2732 10day 140 0051321 005817 0000342 0020965 017898 28418 5393 O wti count max min mean stdev skewness ku rtosis 1day 4698 014119 04064 0000187 0022219 17203 32062 5day 939 021016 033144 0000586 0053865 07576 72538 10day 469 027682 029578 0001231 0072609 031464 46256 spx count max min mean stdev skewness kurtosis 1day 6213 0087089 007633 0000364 000902 0015626 95031 5day 1242 0098566 018293 0001836 0023567 057046 73057 10day 621 013025 023547 0003671 0033192 081819 90398 t10y count max min mean stdev skewness ku rtosis 1day 6143 00065 00075 105E 05 0000844 021168 95927 5day 1228 00109 00136 516E 05 000197 030407 77887 10day 614 00114 00235 00001 0002899 08591 11519 Precision estimate of mean and standard deviation Autocorrelation of When interpreting daily logarithmic returns remember that simple return 81814 1 eX 7 1 For instance the minimum daily logarithmic return of 74064 for wtilday is equivalent to a simple return of 73340 For all lday return series examined sample kurtosis is greater than 30 Distributions of daily logarithmic returns are highly kurtotic relative to the normal distribution Notice that as the horizon is extended to 5 and 10 days the estimated kurtosis declines The distribution of daily logarithmic returns is fattailed relative to the normal The frequency of extreme price changes is far greater than expected by a normally distributed random variable A simple test of the normality hypothesis The studentized range is a test statistic with known quantiles under the null hypothesis of normality The de nition of the studentized range is simply the sample range divided by the sample standard deviation SR Max 7 MinSD H0 normality H A HOI H0 The studentized range test is a twosided test the null hypothesis is rejected if the test statistic is greater in absolute value than the critical value for the chosen con dence level Fractiles of the distrbution of the Studentized Range from samples of size T Source HA David HO Hartley and ES Pearson quotThe distribution ofthe ratio in a single normal sample of range to standard deviationquot Biometrika 61 1954 pg 491 0005 001 0025 005 01 09 095 0975 099 0995 9 3 449 3 552 3634 3 720 3 772 9 10 2 470 2 510 2 590 2 670 2 770 3570 3 685 3 777 3 875 3 935 10 11 2 530 2 580 2 660 2 740 2 840 3 680 3 800 3 903 4 012 4 079 11 12 2590 2650 2730 2800 2910 3780 3910 4010 4134 4208 12 13 2 650 2 700 2 780 2 860 2 970 3 870 4 000 4110 4 244 4 325 13 14 2 700 2 750 2 830 2 910 3 020 3 950 4 090 4 210 4 340 4 431 14 15 2 750 2 800 2 880 2 960 3 070 4 020 4170 4 290 4 430 4 530 15 16 2 800 2 850 2 930 3 010 3130 4090 4 240 4 370 4 510 4 620 16 17 2 840 2 900 2 980 3 060 3170 4150 4 310 4 440 4 590 4 690 17 18 2880 2940 3020 3100 3210 4210 4380 4510 4660 4770 18 19 2 920 2 980 3 060 3140 3 250 4270 4 430 4 570 4 730 4 840 19 20 2 950 3 010 3100 3180 3 290 4320 4 490 4 630 4 790 4 910 20 30 3 220 3 270 3 370 3 460 3 580 4 700 4 890 5 060 5 250 5 390 30 40 3 410 3 460 3 570 3 660 3 790 4 960 5150 5 340 5 540 5 690 40 50 3 570 3 610 3 720 3 820 3 940 5150 5 350 5 540 5 770 5 910 50 60 3 690 3 740 3 850 3 950 4 070 5 290 5 500 5 700 5 930 6 090 60 80 3 880 3 930 4 050 4150 4 270 5 510 5 730 5 930 6180 6 350 80 100 4 020 4 000 4 200 4 310 4 440 5 680 5 900 6110 6 360 6 540 100 150 4 300 4 360 4 470 4 590 4 720 5 960 6180 6 390 6 640 6 840 150 200 4 500 4 560 4 670 4 780 4 900 6 150 6 380 6 590 6 850 7 030 200 500 5 060 5 130 5 250 5 370 5 490 6 720 6 940 7 150 7 420 7 600 500 1000 5500 5570 5680 5790 5920 7110 7330 7540 7800 7990 1000

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