# Simon Acomb NAG Financial Mathematics Day

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1 1 Why People Who Prce Dervatves Are Interested In Correlaton mon Acomb NAG Fnancal Mathematcs Day

2 Correlaton Rsk What Is Correlaton No lnear relatonshp between ponts Co-movement between the ponts Postve correlaton mon Acomb NAG Fnancal Mathematcs Day

3 Correlaton Rsk tochastcs 3 tandard prcng theory s based on some general stochastc descrpton on the dynamcs of the underlyng asset. Wth two, or more assets dt 1 t d Μ t t d 1 n t n t 1 1 = µ dt + σ dz = µ dt + σ dz n n = µ dt + σ dz dz, dz = ρ dt 1 n Need a correlaton matrx s specfy the dynamcs. mon Acomb NAG Fnancal Mathematcs Day

4 Correlaton Rsk Products Wth Correlaton Exposure 4 1. Quanto Opton N UD JPY 0 JPY 1 + Imagne that the share prce s 100JPY and the delta s 1. In ths case fund of the hedge s done n JPY. If FX rates move then the JPY of the contract moves, but the value of the hedge does not. Weak exposure to the correlaton between FX and Equty. Compostve (Cross) Opton N UD X X JPY UD JPY UD JPY 0 JPY 1 + Lke an opton on a ADR. Every tme we trade delta we fund n UD. Volatlty s the volatlty of measured n UD. trong exposure to FX and Equty correlaton. mon Acomb NAG Fnancal Mathematcs Day

5 mon Acomb NAG Fnancal Mathematcs Day Correlaton Rsk Products Wth Correlaton Exposure 1. pread Opton. Best-of, Worst-Of optons 3. Basket Optons 4. Hmalayan Optons 5. Ranbow Optons 5 + K K x Ma 0 + K Mn 0 + K w 0 + K Max w t 0 remanng + + K Mn w Max w 0 0 1

6 Correlaton Rsk Basket Optons 6 Very commonly traded Can be traded wth the OC market together wth short postons n optons on each of the underlyngs a correlaton product Approxmatons for basket varance σ basket, w w ρ σ σ (Note ths s only an approxmaton as assets are lognormally dstrbuted. Increasng the correlaton ncreases the basket volatlty and makes the optons more expensve Often sold on dverse set of underlyngs to make the opton cheaper More underlyngs n the mx makes the opton cheaper. mon Acomb NAG Fnancal Mathematcs Day

7 Correlaton Rsk Best-Of / Worst-Of 7 General dea s that when the correlaton s low (negatve) there s a more dverse range of outcomes, hence there wll always be one asset whch has outperformed and one that has under performed. (Consder the case of perfect negatve correlaton.) Followng exposure of product to a rse n correlaton Call Put Best-Of - + Worst-Of + - Frequently components of structured products Worst of Call (embedded n a guaranteed product) correlaton used to make the product cheaper hort Worst of Put (embedded n reverse convertble) correlaton used to ncrease the coupon mon Acomb NAG Fnancal Mathematcs Day

8 Correlaton Rsk Hmalayan / Ranbow 8 Hmalayan s lke a combnaton of a basket optons and an asan opton - averagng over both tme and asset. Best assets are fxed early and lose ther tme value. Becomes lke a call on the worst postve exposure to correlaton Ranbow s an nterestng product. Lke a basket ncreasng correlaton, ncreases the value of the product ypcally the product s set up wth the hghest weght appled to the best performng asset, and so has some features of a call on the best ncreasng correlaton, decreases the value of the product Overall product has small correlaton exposure, but t can be ether postve, or negatve. mon Acomb NAG Fnancal Mathematcs Day

9 Correlaton Rsk ypcally Investment Bank Exposure 9 ypcal products that nvestment banks sell. Reverse Convertble on the worst Basket optons Hmalayan / Ranbow Calls on worst All leave the seller short correlaton. Dffcult to manufacture a product whch has correlaton exposure n the other drecton. mon Acomb NAG Fnancal Mathematcs Day

10 Correlaton Rsk Measurng Correlaton 10 Frst attempt would be to use the tme seres of two underlyngs and measure the correlaton of the tme seres. Rollng wndow of 6 months daly data Nkke 5 and &P 500 Note enormous range. For ρ=0. and 6 months daly data statstc confdence nterval s [0.0, 0.36]. Are we pckng up samplng error, or uncertanty n correlaton Correlaton s low due to asynchronous effect. mon Acomb NAG Fnancal Mathematcs Day

11 Correlaton Rsk Measurng Correlaton (Weekly Data) 11 Usng a 1 year rollng wndow Impled correlaton taken from optons markets close to Just as volatlty trades at a premum to hstorc, so does correlaton. Need a mechansm that assess ths premum. mon Acomb NAG Fnancal Mathematcs Day

12 Correlaton Rsk Why Do We Need a Postve em Defnte Matrces 1 Negatve Defnte matrx mples that there are portfolos wth negatve varance. (In portfolo theory can we get away wth ths f we nsst that all assets have postve weght?) Monte Carlo Path Generaton Gven uncorrelated random numbers How do we construct random numbers whch have the correlaton matrx W Z Ρ Fnd pseudo-square root so that and Q R = QQ Algorthms for Q requre R to be postve sem-defnte. Z = k q k W k mon Acomb NAG Fnancal Mathematcs Day

13 Correlaton Rsk Why Do We Get Negatve Defnte Matrces 13 One would magne that f we use tme seres over the same perod and estmated from ths the correlaton matrx would be postve defnte by defnton. Data s not synchronous. Dfferent regons have dfferent holdays. What do you do wth the stock that has only ust been ssued. In realty correlaton matrces suffer from ths problem. For prcng purposes would prefer to use correlatons mpled from opton contracts that I can see. For example baskets and dspersons. Most lkely gong to be mxng hstorc and mpled estmates. Many people want to know what happens when correlaton goes up by 10%. mon Acomb NAG Fnancal Mathematcs Day

14 Correlaton Rsk Correlaton Premum 14 Useful transform ρ ρ + α( 1 ρ) If covarance matrx s postve sem-defnte then ths transform wll mantan ths property Gves calbraton method for correlaton. 1.Estmate correlatons by the best hstorcal method you have avalable.from the small number of observatons that you can observe estmate the correlaton premum 3.Apply ths correlaton premum to all hstorc correlatons that you use. mon Acomb NAG Fnancal Mathematcs Day

15 Correlaton Rsk Correlaton mle 15 Consder an Index such as the Eurotoxx made up of 50 underlyng assets hen to reasonable approxmaton σ I =, w w ρ σ σ I = w If volatlty s a functon of (percentage) strke then ρmust also be a functon of ths strke. I = σ ( K) w w ρ ( K) σ ( K) σ ( K), We know that ndex volatlty smles are steeper than sngle stock volatlty smles so not surprsngly we fnd that correlaton s hgher for low strkes than for hgher strkes mon Acomb NAG Fnancal Mathematcs Day

16 Correlaton Rsk Impled Correlaton 16 mon Acomb NAG Fnancal Mathematcs Day

17 Correlaton Rsk Explanaton of Correlaton mle We know that dstrbutons of assets are not lognormal. o make more sense of the nformaton we should be usng the mpled dstrbutons of each stock.. Gven the dstrbutons of each asset does not determne the ont dstrbuton of two assets. here s a extensve theory of ths called copula. 3. If we used local volatlty as a process we would get very dfferent results. Correlaton smle means many dfferent thngs to dfferent people. If tradng correlaton the most obectve way of dong so wll be wth varance swaps. mon Acomb NAG Fnancal Mathematcs Day

18 Correlaton Rsk Products Exposed to Correlaton mle 18 Dsperson of deep out of the money optons Far out of the money Worst of optons Worst of dgtals Way out of the money altplano products Worst of equty default products mon Acomb NAG Fnancal Mathematcs Day

19 Correlaton Rsk Measurng Explct Correlaton Rsk 19 Common to measure correlaton rsk as ρ ρ +10% What about correlatons at 0.95 What happens f the correlaton matrx s no longer postve defnte Use the α rsk methodology ρ ρ + α( 1 ρ) α = 10% ρ = 90% 91% ρ = 0% 8% maller correlatons move the most (n lne of emprcal observatons) Mantans postve defnteness. Make sure that correlaton scenaro s suffcently large to capture any nonlnearty n correlaton. mon Acomb NAG Fnancal Mathematcs Day

20 Correlaton Rsk Hdden Correlaton Rsk 0 Correlaton exposure wthn dervatves exposure s by no-means unque. Portfolo managers have been dealng wth correlaton exposure for a long tme Common to measure market exposure by movng all assets smultaneously. All the same amount caled by a beta to the market If portfolo only had two assets whch where negatvely correlated then ths rsk measure would be n-approprate Portfolo rsk systems such as Barra break down rsk exposure nto other factors. Named factors related to real varables Prncpal components If you have the break down of delta by asset, worthwhle thnkng about whether portfolo manager technques can help better understand the correlaton of a portfolo whether t has dervatves n t, or not. mon Acomb NAG Fnancal Mathematcs Day

21 Correlaton Rsk Hedgng Correlaton Exposure 1 Most retal structured products leave the seller short correlaton. Reverse Clquet on a Basket + B t Max X 1, 0 t Bt 1 Reverse Convertble on a Basket B Mn B 0,1 mall number of products whch leave the seller long correlaton. mon Acomb NAG Fnancal Mathematcs Day

22 Correlaton Rsk Hedgng Correlaton Rsk - Dsperson Broker market traders dspersons n vanlla optons Max w 1 + w K, 0 w1 Max K,0 wmax, K 1 0 Can go long or short and hence long, or short correlaton. On ntaton product has lttle ndvdual vega, but has exposure to correlaton. When share prces move ths wll no longer be the case. Just as the vega of a vanlla opton dsspates when the opton move away from AM, so the correlaton exposure of a basket opton dsspates as you move away from AM. hese types of trades are popular wth ndces. mon Acomb NAG Fnancal Mathematcs Day

23 Correlaton Rsk Hedgng Correlaton Exposure Varance waps 3 Varance waps provde a smple way of tradng varance wthout an explct strke dependency. Can trade a dsperson of varance swaps Long varance swap on the Eurotoxx 50 hort varance swaps on each of the ndvdual consstuents. Can go long or short. Possble on the ndces wth a small number of assets, but can be approxmated on ndces such as the &P wth trackng portfolos As these are strke-less the correlaton exposure does not dsspate as the underlyng assets move. mon Acomb NAG Fnancal Mathematcs Day

24 Correlaton Rsk Correlaton waps 4 A correlaton swap s an OC product whch has a payoff gven by P = N( N 1) < ρ Gves a drect way of tradng correlaton. Dffcult to hedge product Does t really gve the exposure you requre. If you ms-estmate correlaton between two Geometrc Brownan motons P&L gven by terms ncludng E 0 f 1 1σ 1σ ( ˆ ρ ρ ) dt Covarance swaps wll be easer to prce. mon Acomb NAG Fnancal Mathematcs Day

25 Correlaton Rsk Return to Q-Q Maps 5 Is assumpton correlaton constant correct. hould we be usng a copula based theory. mon Acomb NAG Fnancal Mathematcs Day

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