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E Cient Computation of Option Price Sensitivities Using Homogeneity

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Ecient Computation of Option Price Sensitivities Using Homogeneity and other Tricks Oliver Rei Weierstra-Institute for Applied Analysis and Sto chastics Mohrenstrasse 39 10117 Berlin GERMANY [email protected] Uwe Wystup Commerzbank Treasury and Financial Pro ducts Neue Mainzer Strasse 32{36 60261 Frankfurt am Main GERMANY [email protected] http://www.mathfinance.de Preliminary version, January 25, 1999 Abstract No front-oce software can survive without providing derivatives of options prices with resp ect to underlying market or mo del parameters, the so called Greeks. We present a list of common Greeks and exploit homogeneity prop erties of nancial markets to derive relationships b etween Greeks out of which many are mo del- indep endent. We apply our results to Europ ean style options, rainbow options, path-dep endent options as well as options priced in Heston's sto chastic volatility mo del and show shortcuts to avoid exorbitant and time-consuming computations of derivatives whicheven strong symb olic calculators fail to pro duce. partially aliated to Delft University,by supp ort of NWO Netherlands 1 2 Rei, O. and Wystup, U. 1 Intro duction Based on homogeneity of time and price level of a nancial pro duct we can derive relations for the options sensitivities, the so-called \Greeks". The basic market mo del we use is the Black-Scholes mo del with sto cks paying a continuous dividend yield and a riskless cash b ond. This mo del supp orts the homogeneity prop erties which are valid in general, but its structure is so simple, that we can concentrate on the essential statements of this pap er. We will also discuss how to extend out work to more general market mo dels. We list the commonly used Greeks and their notations. We do not claim this list to b e complete, b ecause one can always de ne more derivatives of the option price function. First of all we analyze the homogeneity of a nancial market, consisting of one sto ck and one cash b ond. The techniques we present are easily expanded to the higher dimensional case as we show later on. In addition we lo ok at the Greeks of Europ ean options in the Black-Scholes mo del. Essentially it turns out, that one only needs to knowtwo Greeks in order to calculate all the other Greeks without di erentiating. Another interesting example is a Europ ean derivative security dep ending on two assets. For such rainbow options the analysis of the risk due to changing correlation of the two assets is very imp ortant. We will showhow this risk is related to simultaneous changes of the two underlying securities. There are several applications of these homogeneity relations. 1. It helps saving time in computing derivatives. 2. It pro duces a robust implementation compared to Greeks via di erence quo- tients. 3. It allows to check the quality and consistency of Greeks pro duced by nite- di erence-, tree- or Monte Carlo metho ds. 4. It admits a computation of Greeks for Monte Carlo based values. 5. It shows relationships b etween Greeks whichwouldn't b e noticed merely by lo oking at di erence quotients. Option Price Sensitivities 3 1.1 Notation S sto ck price or sto ck price pro cess B cash b ond, usually with risk free interest rate r r risk free interest rate q dividend yield continuously paid volatility of one sto ck, or volatility matrix of several sto cks correlation in the two-asset market mo del t date of evaluation \to day" T date of maturity = T t maturity of an option x sto ck price at time t f payo function v x;t;::: value of an option k strike of an option l level of an option v partial derivation of v with resp ect to x and analogous x The standard normal distribution and density functions are de ned by 2 1 1 t 2 p 1 e nt = 2 Z x N x = nt dt 2 1 2 2 1 x 2xy + y p n x; y ; = exp 3 2 2 2 21 2 1 Z Z x y N x; y ; = n u; v ; du dv 4 2 2 1 1 See http://www.MathFinance.de/frontoce.html for a source co de to compute N . 2 4 Rei, O. and Wystup, U. 1.2 The Greeks Delta v x Gamma v xx Theta v t Rho v in the one-sto ck mo del r Rhor v in the two-sto ck mo del r r Rho q v q q Vega v Kappa v correlation sensitivitytwo-sto ck mo del Greeks, not so commonly used: x Leverage v sometimes , sometimes called \gearing" x v 0 Vomma v Sp eed v xxx Charm v x Color v xx F Forward Delta v F dl q Driftless Delta e Dual Theta Dual v T k Strike Delta v k k Strike Gamma v kk l Level Delta v l l Level Gamma v ll 1 two-sto ck mo del Beta 12 2 2 Fundamental Prop erties 2.1 Homogeneity of Time In most cases the price of the option is not a function of b oth the current time t and the maturity time T , but rather only a function of the time to maturity = T t implying the relations =v = v = v = D ual 5 t T This relationship extends naturally to the situation of options dep ending on several intermediate times such as comp ound or Bermuda options. 2.2 Scale-Invariance of Time Wemaywant to measure time in units other than years in which case interest rates and volatilities, which are normally quoted on an annual basis, must b e changed according to the following rules for all a>0. ! a Option Price Sensitivities 5 r ! ar q ! aq p a 6 ! The option's value must b e invariant under this rescaling, i.e., p ;ar;aq; v x; ; r;q;;:::=v x; a;::: 7 a We di erentiate this equation with resp ect to a and obtain for a =1 1 ; 8 0= + r + q + q 2 a general relation b etween the Greeks theta, rho, rhoq and vega. It extends naturally to the case of multiple assets and multiple intermediate dates. 2.3 Scale Invariance of Prices The general idea is that value of securities may b e measured in a di erent unit, just likevalues of Europ ean sto cks are now measured in Euro instead of in-currencies. Option contracts usually dep end on strikes and barrier levels. Rescaling can have di erent e ects on the value of the option. Essentially wemay consider the following typ es of homogeneity classes. Let v x; k b e the value function of an option, where x is the sp ot or a vector of sp ots and k the strike or barrier or a vector of strikes or barriers. Let a b e a p ositive real numb er. De nition 1 homogeneity classes We cal l a value function k -homogeneous of degree n if for al l a n v ax; ak = a v x; k : 9 The value function of a Europ ean call or put option with strike K is then K - homogeneous of degree 1, a p ower call with strike K and cap C is b oth K - homogeneous of degree 2 and C -homogeneous of degree 0, a double barrier call with strike K and barriers B =L; H isK -homogeneous of degree 1 and B - homogeneous of degree 0. We will call an option strike-de ned, if there is just one strike k and the value function is k -homogeneous of degree 1 and level-de ned if there is just one level l and the value function is l -homogeneous of degree 0, e.g. a digital cash call. 2.3.1 Strike-Delta and Strike-Gamma For a strike-de ned value function wehave for all a; b > 0 abv x; k = v abx; abk : 10 6 Rei, O. and Wystup, U. We di erentiate with resp ect to a and get for a =1 bv x; k = bxv bx; bk + bk v bx; bk : 11 x k Wenow di erentiate with resp ect to b get for b =1 v x; k = xv + xv x + xv k + kv + kv x + kv k 12 x xx xk k kx kk 2 k 2 k = x+ x + 2xk v + k + k : 13 xk If weevaluate equation 11 at b =1 we get k v = x+ k : 14 We di erentiate this equation with resp ect to k and obtain k k k = xv + + k ; 15 kx 2 k kxv = k : 16 kx Together with equation 13 we conclude 2 2 k x =k : 17 2.3.2 Level-Delta and Level-Gamma For a level-de ned value function wehave for all a; b > 0 v x; l = v abx; abl : 18 We di erentiate with resp ect to a and get at a =1 0 = v bx; bl bx + v bx; bl bl : 19 x l If we set b =1 we get the relation l x + l =0: 20 Nowwe di erentiate equation 19 with resp ect to b and get at b =1 2 2 0 = v x +2v xl + v l : 21 xx xl ll One the other hand we can di erentiate the relation b etween delta and level-delta with resp ect to l and get l l v x + l + = 0: 22 xl Together with equation 21 we conclude 2 2 l l x +x= l + l : 23 Option Price Sensitivities 7 3 Europ ean Options in the Black-Scholes Mo del In this section we analyze the Europ ean claim in the Black-Scholes mo del dS = S [r q dt + dW ]: 24 t t t This situation has a simple structure which allows us to obtain even more relations for the Greeks.
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