Overview FX exotics and the relevance of Accumulative Forward Instalment Options computational methods in their Greeks Contact Information pricing and risk management
mathfinance.de Uwe Wystup Title Page Commerzbank Securities - FX Options JJ II and J I
Page 1 of 68 HfB - Business School of Finance and Management
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Quit Abstract Starting with an overview of the current FX derivatives industry we take a look at a few examples where computational methods Overview are crucial to run the daily business. The examples will include in- Accumulative Forward stalment contracts, accumulative forward contracts and the efficient Instalment Options computation of option price sensitivities Greeks Contact Information
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Quit Overview Accumulative Forward Instalment Options Greeks Contact Information 1. Overview
mathfinance.de EUR/USD is one of the most liquid underlying markets Title Page Trading activities in FX are JJ II 1. Spot/Forward (90%) - extremely small margins J I
Page 3 of 68 2. Vanilla Options (9%) - small margins
Go Back 3. Exotic Options (1%) - potentially higher margins
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Quit 1.1. FX Exotics
Overview 1. barrier and touch options Accumulative Forward Instalment Options 2. compound and instalment Greeks Contact Information 3. average rate options 4. forward start and cliquets mathfinance.de
Title Page 5. corridors/fader/accumulative options
JJ II 6. quanto options J I 7. multi-currency options: baskets, bestof, outside Page 4 of 68 barriers Go Back 8. vol- and variance swaps Full Screen 9. structured products Close
Quit 2. Accumulative Forward Overview Accumulative Forward Market of Jan 7 2003, EUR/USD Spot at S0 = 1.0200. Instalment Options Greeks Zero cost contract for T = 1 year. Contact Information Client sells 200k USD at K = 0.9700 every day the
mathfinance.de EUR/USD fixing Fti is between K = 0.9700 and B = 1.0700. Title Page
JJ II Client sells 400k USD at K = 0.9700 every day the
J I EUR/USD fixing Fti is below K = 0.9700.
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Go Back If B = 1.0700 ever trades, then the client stops accu- mulating but keeps 50% of the accumulated amount. Full Screen
Close Total of 255 Fixings.
Quit Overview Accumulative Forward Payoff per 200k USD is Instalment Options Greeks Contact Information X (S − K) II 50%II + 50%II T {Fti J I TV can be computed in closed form (see [7]). Page 6 of 68 What is the market price? Go Back Full Screen Close Quit 2.1. Pricing and Hedging: Method 1 replicate the structure using options we can price over TV Overview A Client buys strip of 0.9700 eur call , RKO 1.07. We price the 3,6,9,12 Accumulative Forward month Instalment Options Greeks month bp Contact Information 3 +50 6 +35 mathfinance.de 9 +30 Title Page 12 +23 JJ II Average of 34 bp over for nominal amount of 255 * 200,000 / 0.9700 J I = 52.58 MIO EUR Page 7 of 68 Go Back Overhedge A = 179 K Full Screen Close Quit B Client sells strip of 0.9700 eur put , KO 1.07. We price the 3,6,9,12 month Overview month bp Accumulative Forward 3 −5 Instalment Options 6 −15 Greeks Contact Information 9 −20 12 −20 mathfinance.de Average of 15 bp under for nominal amount of 255 * 400,000 / 0.9700 Title Page = 105.15 MIO EUR JJ II Overhedge B = 158 K J I Page 8 of 68 Go Back Full Screen Close Quit C Client sells a one-touch 1.0700 (to account for the 50% reduction of his payout if we touch 1.0700) maturity 1 year. Price is 4% under TV. Overview Payoff of the one-touch = 50% * 50 mio * (1.07-0.97) = 2.5 MIO Overhedge C = 2.5 mio * 4% = 100 K Accumulative Forward Instalment Options Total Overhedge = A + B + C = 437 K Greeks Contact Information mathfinance.de Title Page JJ II J I Page 9 of 68 Go Back Full Screen Close Quit Overview 2.2. Pricing and Hedging: Method 2 Accumulative Forward Instalment Options Looking at the cost of vega management Greeks Contact Information A structure has 25K negative volga ... cost 115K (using a butterfly) mathfinance.de B structure has 325K negative vanna between 0.99 and Title Page 1.09 ... cost 285K using the price of a 1 year Risk JJ II Reversal J I C structure has 200K of vega ... cost 20 K of spread Page 10 of 68 (0.1 vol versus mid- market) Go Back Total Overhedge = A + B + C = 420 K EUR Full Screen Close Quit 2.3. Financial Engineering Issues 1. Need fast calculators for TVs, ideally closed-form solutions Overview 2. Automate computation of the hedge and its cost Accumulative Forward 3. Live market data feed: Spot, Termstructure of Interest Rates, Vol- Instalment Options surface Greeks Contact Information 4. For Method 1: shift exotic risk to liquid risk, i.e. using first generation exotics to price 2nd generation exotics mathfinance.de 5. For Method 2: shift exotic risk to liquid risk, i.e. using vanillas to price first generation exotics. Title Page JJ II J I Page 11 of 68 Go Back Full Screen Close Quit 2.4. Pricing a one-touch • pays a fixed amount of a pre-specified currency, if the underlying ever touches a barrier Overview Accumulative Forward • costs between 0% and 100% Instalment Options • the closer the spot at the barrier, the more expensive the one-touch Greeks Contact Information • market price often far away from TV, due to cost of risk management All details in [17]. mathfinance.de Title Page JJ II J I Page 12 of 68 Go Back Full Screen Close Quit Example for market: EUR/USD 17 July 2002 1.0045 EUR 3.33% USD 1.76%, 3 M ATM vol 11.85%, RR 1.25%, BF 0.25% Overview Accumulative Forward Instalment Options Greeks Contact Information mathfinance.de Title Page JJ II Figure 1: Overhedge for one-touch options J I Page 13 of 68 Go Back Full Screen Close Quit The Overhedge calculation • Market price of the option Overview • = TV (theoretical value) Accumulative Forward Instalment Options • +p· vanna of the option · value RR / vanna RR Greeks • +p· volga of the option · value BF / volga BF Contact Information where mathfinance.de • RR: Risk Reversal Title Page • BF: Butterfly JJ II • p: probability that the hedge is needed J I Page 14 of 68 Go Back Full Screen Close Quit Overview Butterfly und Risk Reversal Accumulative Forward Smile for fixed maturity vol Instalment Options Greeks Contact Information RR mathfinance.de BF Title Page JJ II -25% +25% J I put delta ATM call delta Page 15 of 68 Go Back Full Screen Figure 2: Butterfly and Risk Reversal Close Quit Example • 1Y USD/JPY one-touch at 127.00, notional in USD Overview • Market data: 117.00 spot, 8.80% vol, 2.10% USD interest rate, 0.10% Accumulative Forward JPY interest rate, 25delta RR -0.45%, 25delta BF 0.37% Instalment Options • TV: 38.2%, Vanna -9.0, Volga -1.0 Greeks Contact Information Market price is computed as TV = 38.2% • +p· -9.0 · -0.15% / 4.5 mathfinance.de • +p· -1.0 · 0.27% /0.035 Title Page • = 38.2% + p · [0.3% − 7.7%] = 38.2% − p · 7.4% JJ II where J I • p = 100% − 38% Page 16 of 68 • so, overhedge is 62% · −7.4% = −4.7% Go Back • so, market mid price is 38.2% − 4.7% = 33.5% Full Screen • so, bid - ask could be 32%/35% Close • and the hedge: sell 2 RR and 28 BF Quit 3. Instalment Options Joint work with Susanne Griebsch, Goethe University. Overview Accumulative Forward 3.1. What is an Instalment Option? Instalment Options • Like Vanilla Option, but Greeks Contact Information (1) Premium is divided into several payments and is paid periodi- cally on so-called ”instalment dates” (2) Holder has the right to cancel option through the termination mathfinance.de of instalment payments Title Page JJ II J I Page 17 of 68 Go Back Figure 3: Dates for Instalment Payments Full Screen • Other names: continuation option, pay-as-you-go option, a general- Close ization of compound option Quit • n-Instalment Option can be understood as a series of n options de- pending on each other Overview Accumulative Forward Instalment Options Greeks Contact Information mathfinance.de Title Page JJ II Figure 4: Lifetimes of the options Vi J I • Characterized by Page 18 of 68 – n exercise times t1, ..., tn = T (often ti = iT/n for all i), Go Back – n strike prices k1, ..., kn, Full Screen +1 if option i is a call – n put/ call indicators φ1, ..., φn where φi := Close −1 if option i is a put Quit Market data • S0: spot Overview • rd: domestic interest rate Accumulative Forward Instalment Options • rf : foreign interest rate Greeks • σ: volatility Contact Information mathfinance.de Title Page JJ II J I Page 19 of 68 Go Back Full Screen Close Quit 3.2. Advantages of Instalment Options • Traded over-the-counter tailor-made to client needs Overview • Prevention of losses through possibility of termination Accumulative Forward • Helpful in situations where necessity of hedge is uncertain Instalment Options Greeks • Low initial premium is easy to schedule in the firm’s budget Contact Information mathfinance.de Title Page JJ II J I Page 20 of 68 Go Back Figure 5: Comparison of Instalment Option with Vanilla Put: Continuation Full Screen of instalment payments until expiration vs. Continuation of instalment payments until expiration Close Quit 3.3. Example of a Traded Instalment Option • Application area: International Treasury Management Overview • Corporate buys EUR Call/ USD Put 25 Mio EUR notional Accumulative Forward • Strike price: 1.0500 EUR/USD Instalment Options Greeks • Exercise type: European Contact Information • Maturity date: 17 Dec 2003, Delivery settlement on 19 Dec 2003 • Transaction date: 19 Dec 2002 mathfinance.de • EUR USD spot ref: 1.0259 Title Page • Premium and strike prices: 285,500.00 USD JJ II • Decision and Value dates: 31/03/03, 02/04/03, 30/06/03, 02/07/03, J I 30/09/03, 02/10/03 Page 21 of 68 • The corporate has extended the instalment at all dates and finally Go Back sold the EUR call on Nov 19 2003 for a profit of 2.77 MIO EUR (spot Full Screen was at 1.1900). Close Quit 3.4. Pricing of Instalment Options in the Black-Scholes Model • Like Vanilla Options or Compound Options, i.e. discounted expec- tation of payoff function Overview Accumulative Forward • dSt = St[(rd − rf )dt + σdWt] for 0 ≤ t√≤ T 2 Instalment Options St2 = St1 exp((rd − rf − σ /2)∆t + σ ∆tZ), for 0 ≤ t1 ≤ t2 ≤ T ), Greeks ∆t = t2 − t1 Contact Information def + • Payoff at maturity is max(φn(ST − kn), 0) = (φn(ST − kn)) • Date before last instalment date tn−1 buyer pays kn−1 to receive clas- mathfinance.de sical european option, in which the price at tn−1 is described by Title Page def −rd(tn−tn−1) + Vn(s) = VStd(s) = e E[φn[ST − kn] | Stn−1 = s] JJ II J I • Rational buyer only pays instalment rate if VStd ≥ kn−1 shortly before instalment date option is worth max(VStd − kn−1, 0) Page 22 of 68 • Compound option price at time tn−2 is Go Back def −rd(tn−1−tn−2) + Full Screen Vn−1(s) = VCp(s) = e E[φn−1[Vn − kn−1] | Stn−2 = s] Close Quit • Next steps are analogous, compound option Vi with option Vi+1 so that Vi is an option on Vi+1 with strike ki and decision date ti Overview • Exact expression for value function of Instalment Option Accumulative Forward def Instalment Options −rd(ti−ti−1) + Vi(s) = e E[(φi(Vi+1(ti+1)−ki)) | Si−1 = s], for i = 1, ..., n−1. Greeks Contact Information • When carried out for all i ≤ n − 1, result is first instalment which is paid to open the deal at t0 = 0 def mathfinance.de −rd(t1−t0) + p = V1(s) = e E[φ1[V2 − k1] | St0 = s] Title Page • Nested expectations require analysis of multiple integrals JJ II • Numerical computation of multiple integrals is time consuming and J I possibly imprecise Page 23 of 68 Go Back Full Screen Close Quit 3.5. n-variate Cumulative Normal Formula • n-variate cumulative normal function Overview Accumulative Forward Nn(hi; {ρij}1≤j≤n,i JJ II ρij −ρi1ρj1 ρij∗1 = 1 1 (i, j 6= 1 and j 6= i) 2 2 2 2 (1−δi1) (1−δj1) J I • Special case n = 2 was used for compound option formula Page 24 of 68 ! Z h1 Go Back h2 − ρy N2(h1, h2; ρ) = N n(y)dy 2 1 −∞ (1 − ρ ) 2 Full Screen Close Quit " S0 (+) S0 (+) r # ln ¯ + µ t1 ln ¯ + µ t2 t rf t2 S1 S2 1 VCp = e S0N2 √ , √ , σ t1 σ t2 t2 " S0 (−) S0 (−) r # ln ¯ + µ t1 ln ¯ + µ t2 t −rdt2 S1 S2 1 Overview − e k1N2 √ , √ , σ t1 σ t2 t2 Accumulative Forward " S0 (−) # Instalment Options ln ¯ + µ t1 − e−rdt1 k N S1 √ Greeks 2 σ t1 Contact Information n-variate case mathfinance.de ~ • k = (k1, ..., kn) strike prices Title Page ~ JJ II • t = (t1, ..., tn) instalment dates J I ~ • φ = (φ1, ..., φn) put/call indicators Page 25 of 68 Go Back • correlation coefficients of n-variate cumulative normal functions Full Screen q ρij = ti/tj for i, j = 1, ..., n and i < j Close Quit ~ ~ Vn(S0, k,~t, σ, rd, rf , φ) −rf tn = e S0φ1 · ... · φn " S0 (+) S0 (+) S0 (+) # ln + µ t1 ln + µ t2 ln + µ tn S¯1 S¯2 S¯n Overview Nn √ , √ , ..., √ ; {ρij} σ t1 σ t2 σ tn Accumulative Forward Instalment Options −rdtn − e knφ1 · ... · φn Greeks " S0 (−) S0 (−) S0 (−) # ln + µ t1 ln + µ t2 ln + µ tn S¯1 S¯2 S¯n Contact Information Nn √ , √ , ..., √ ; {ρij} σ t1 σ t2 σ tn −rdtn−1 − e kn−1φ2 · ... · φn mathfinance.de " S0 (−) S0 (−) S0 (−) # ln + µ t1 ln + µ t2 ln ¯ + µ tn−1 S¯1 S¯2 Sn−1 Title Page Nn−1 √ , √ , ..., √ ; {ρij} σ t1 σ t2 σ tn−1 JJ II . . " S0 (−) S0 (−) # J I ln ¯ + µ t1 ln ¯ + µ t2 −rdt2 S1 S2 − e k2φn−1φnN2 √ , √ ; ρ12 Page 26 of 68 σ t1 σ t2 Go Back " S0 (−) # ln ¯ + µ t1 −rdt1 S1 − e k1φnN √ Full Screen σ t1 Close Quit 3.6. Binomial Tree Option Pricing Technique • Binomial model was developed by Cox, Ross and Rubinstein Overview • Price movements of log-returns of underlying are modeled as constant p p Accumulative Forward up and down movements (u = exp(σ T/m, d = exp(−σ T/m) in Instalment Options the tree. Greeks Contact Information mathfinance.de Title Page JJ II J I Page 27 of 68 Go Back Full Screen Close Quit 3.7. Algorithm for Pricing Instalment Options by H. Ben- Ameur, M. Breton and P. Fraincois [2] • Approximation of value of Instalment Option at t0 through piecewise Overview linear interpolation, therefore solving dynamic programming equa- Accumulative Forward tion which results in a closed form Instalment Options Greeks • Exercise value is Vn(s) = max(0, φn(ST − kn)) Contact Information h −rd∆t • Holding value at ti is Vi (s) = E[e Vi+1(Sti+1 ) | Sti = s] for i = 0, ..., n − 1 mathfinance.de where Title Page h V0 (s) for i = 0 JJ II h vi(s) = max(0,Vi (s) − ki) for i = 1, ..., n − 1 J I Vn(s) for i = n Page 28 of 68 h • Net holding value Vi (s) − ki Go Back Full Screen Close Quit • a0 = 0 < a1 < ... < ap < ap+1 = +∞ set of points + R0, ..., Rp partition of R in (p + 1) intervals Rj = (aj, aj+1] for j = 0, ..., p Overview • Given approximationsv ˜i of option value vi at aj in step i Accumulative Forward piecewise linear interpolation of this function achieved through Instalment Options p X Greeks i i vˆi(s) = (αj + βjs)Iaj mathfinance.de • Assumingv ˆi+1 is known, calculate expectation in step i Title Page h −rd∆t v˜i (ak) = E[e vˆi+1(Sti+1 )|Sti = ak] JJ II p −r ∆t X i+1 J I d a √ a = e αj E[I j µ∆t+σ ∆tz j+1 ] a Full Screen Close 2 µ = rd − rf − σ /2,v ˜i approximated holding value of Instalment Option Quit • For k = 1, ..., p and j = 0, ..., p first integrals N(xk,1) for j = 0 Overview a √ a Ak,j = E[I j µ∆t+σ ∆tz j+1 ] = N(xk,j+1) − N(xk,j ) for 1 ≤ j ≤ p − 1 a Greeks √ µ∆t+σ ∆tz B = [a e I a √ a ] Contact Information k,j E k j Title Page √ with xk,j = [ln(aj /ak) − µ∆t]/(σ ∆t). JJ II J I Page 30 of 68 Go Back Full Screen Close Quit Procedure 0. Calculate ai Overview 1. Calculatev ˆn(s) for all s Accumulative Forward 2. Calculatev ˜h (a ) for all k in closed form Instalment Options n−1 k Greeks 3. Calculatev ˜n−1(ak) for all k Contact Information 4. Calculatev ˆn−1(s) for all s > 0 mathfinance.de 5. Iterate these steps untilv ˆ1(s0)=Price of Instalment Option at time 0 is calculated Title Page JJ II J I Page 31 of 68 Go Back Full Screen Close Quit 3.8. Comparison of Accuracy and Speed • Results of binomial trees oscillate strongly Overview Accumulative Forward Instalment Options Greeks Contact Information mathfinance.de Title Page Figure 6: Convergence of the value function in the binomial trees imple- JJ II mentation J I • Trivariate formula is the fastest of all considered methods, even for Page 32 of 68 higher numbers of instalments Go Back • Accuracy of trivariate formula now only depends on accuracy of cal- culation of multivariate normal integrals and calculation of roots Full Screen Close • Algorithm of ABF works for equally distant instalment dates Quit Performance Numerical Method Value of VTV Time Overview Binomial Trees n = 4000 1, 69053 1109 sec Accumulative Forward Instalment Options Trivariate Formula 1, 69092 < 1 sec Greeks Algorithm (Article of ABF) p = 4000 1, 69084 168 sec Contact Information Numerical Int. (50000-point Gauss-Legendre) 1, 69087 176 sec Numer. Int. of Cp Formula (Mathematica) 1, 69091 47 sec mathfinance.de Title Page Table 1: S0 = 100, k1 = 100, k2,3 = 3, σ = 20%, rd = 10%, rf = 15%, T = 1, ∆t = 1/3, φ1,2,3 = 1 JJ II J I Page 33 of 68 Go Back Full Screen Close Quit 3.9. Convergence of Identical Premium • Continuous Instalment Option is an american type option, where Overview – Total sum of premiums paid at beginning Accumulative Forward – Difference repaid in case of an option termination Instalment Options • Discounted sum of instalments Greeks n Contact Information X −rdti un = fn e where ti = (i − 1)∆t and n∆t = T i=0 u price of n-Instalment Option with instalment dates t and iden- mathfinance.de n i tical premium fn paid at ti, 0 ≤ i ≤ n − 1 Title Page • With increasing number of instalments n the total premium un in- JJ II creases (increasing optionality) J I • With increasing n, instalment payments decrease Page 34 of 68 • un converges to an upper bound Z T Go Back U = g e−rdsds 0 Full Screen n → ∞ (and ∆t → 0) Close g is the uniform premium for continuous Instalment Option paid between gdt and t + dt g corresponds with limit fn → g Quit ∆t Overview Accumulative Forward Instalment Options Greeks Contact Information mathfinance.de Title Page Figure 7: Convergence of uniform premium in discrete case to continuous JJ II premium J I • How can we describe this upper bound? Page 35 of 68 • Possible approach: Go Back Continuous Instalment Option = Vanilla Call plus American Com- Full Screen pound Put on this call with linearly decreasing strike (w.r.t. time) Close Quit 4. Greeks Joint work with Oliver Reiss, Weierstrass Institute Berlin Overview Accumulative Forward Instalment Options Greeks Contact Information mathfinance.de Title Page JJ II J I Page 36 of 68 Go Back Full Screen Close Quit 4.1. Notation S stock price or stock price process Overview B cash bond, usually with risk free interest rate r Accumulative Forward r risk free interest rate Instalment Options q dividend yield (continuously paid) Greeks Contact Information σ volatility of one stock, or volatility matrix of several stocks ρ correlation in the two-asset market model mathfinance.de t date of evaluation (“today”) Title Page T date of maturity JJ II τ = T − t time to maturity of an option J I x stock price at time t Page 37 of 68 f(·) payoff function Go Back v(x, t, . . . ) value of an option k strike of an option Full Screen l level of an option Close vx partial derivation of v with respect to x (and analogous) Quit The standard normal distribution and density functions are defined by ∆ 1 − 1 t2 n(t) = √ e 2 (2) 2π Overview x ∆ Z Accumulative Forward N (x) = n(t) dt (3) −∞ Instalment Options 2 2 ∆ 1 x − 2ρxy + y Greeks n2(x, y; ρ) = exp − (4) p 2 2(1 − ρ2) Contact Information 2π 1 − ρ x y ∆ Z Z N2(x, y; ρ) = n2(u, v; ρ) du dv (5) −∞ −∞ mathfinance.de Title Page JJ II J I Page 38 of 68 Go Back Full Screen Close Quit 4.2. Common Greeks Delta ∆ vx Overview Gamma Γ vxx Accumulative Forward Theta Θ vt Instalment Options Rho ρ v in the one-stock model Greeks r Contact Information Rhor ρr vr in the two-stock model Rhoq ρq vq mathfinance.de Vega Φ vσ Title Page Kappa κ vρ correlation sensitivity (two-stock model) JJ II J I Page 39 of 68 Go Back Full Screen Close Quit 4.3. Not so common Greeks x Leverage λ v vx sometimes Ω, sometimes called “gearing” 0 Overview Vomma / Volga Φ vσσ Accumulative Forward Speed vxxx Instalment Options Charm v Greeks xt Contact Information Color vxxt Cross / Vanna vxσ mathfinance.de F Forward Delta ∆ vF Title Page Driftless Delta ∆dl ∆eqτ JJ II Dual Theta DualΘ vT k J I Strike Delta ∆ vk k Page 40 of 68 Strike Gamma Γ vkk l Go Back Level Delta ∆ vl l Level Gamma Γ vll Full Screen σ1 Beta β12 ρ two-stock model Close σ2 Quit 4.4. Scale-Invariance of Time Based on the relation Overview v(x1, ..., xn, τ, r, q1, ..., qn, σ11, ..., σnn) = Accumulative Forward τ √ √ v(x1, ..., xn, , ar, aq1, ..., aqn, aσ11, ..., aσnn) (6) Instalment Options a Greeks we obtain Contact Information Theorem 4.1 (scale invariance of time) n n mathfinance.de X 1 X 0 = τΘ + rρ + q ρ + Φ σ , (7) i qi 2 ij ij Title Page i=1 i,j=1 JJ II where Φij denotes the differentiation of v with respect to σij. J I Page 41 of 68 Go Back Full Screen Close Quit 4.5. Scale Invariance of Prices Definition 4.1 (homogeneity classes) We call a value function k-ho- mogeneous of degree n if for all a > 0 Overview n Accumulative Forward v(ax, ak) = a v(x, k). (8) Instalment Options Greeks value function strike-homogeneous of degree 1: strike-defined option value function level-homogeneous of degree 0: level-defined option Contact Information mathfinance.de Title Page JJ II J I Page 42 of 68 Go Back Full Screen Close Quit 4.5.1. Strike-Delta and Strike-Gamma For a strike-defined value function we have for all a, b > 0 Overview abv(x, k) = v(abx, abk). (9) Accumulative Forward We differentiate with respect to a and get for a = 1 Instalment Options Greeks bv(x, k) = bxvx(bx, bk) + bkvk(bx, bk). (10) Contact Information We now differentiate with respect to b get for b = 1 v(x, k) = xvx + xvxxx + xvxkk + kvk + kvkxx + kvkkk (11) mathfinance.de 2 k 2 k = x∆ + x Γ + 2xkvxk + k∆ + k Γ . (12) Title Page If we evaluate equation (10) at b = 1 we get JJ II v = x∆ + k∆k. (13) J I We differentiate this equation with respect to k and obtain Page 43 of 68 k k k ∆ = xvkx + ∆ + kΓ , (14) Go Back 2 k kxvkx = −k Γ . (15) Full Screen Together with equation (12) we conclude Close x2Γ = k2Γk. (16) Quit 4.6. European Options in the Black-Scholes Model Relations among Greeks for European claims in n-dimensions Overview Accumulative Forward dSi(t) = Si(t)[(r − qi) dt + σi dWi(t)], i = 1, . . . , n(17) Instalment Options Cov(Wi(t),Wj(t)) = ρijt, (18) Greeks Contact Information where r is the risk-free rate, qi the dividend rate of asset i or foreign inter- est rate of exchange rate i, σi the volatility of asset i and (W1,...,Wn) a standard Brownian motion (under the risk-neutral measure) with correla- mathfinance.de tion matrix ρ. Let v denote today’s value of the payoff f(S1(T ),...,Sn(T )) at maturity T . Then it is known that v satisfies the Black-Scholes partial Title Page differential equation JJ II n n X 1 X 0 = −v − rv + x (r − q )v + (σ ◦ σT ) x x v . (19) J I τ i i xi 2 ij i j xixj i=1 i,j=1 Page 44 of 68 Go Back Full Screen Close Quit 4.6.1. Relations among Greeks Based on the Log-Normal Distribution The value function v has a representation given by the n-fold integral Z √ Overview −rτ σi τxi+µiτ v = e f ...,Si(0)e ,... g(~x, ρ) d~x, (20) Accumulative Forward Instalment Options 1 2 where µi = r−qi − σi and g(~x, ρ) is the n-variate standard normal density Greeks 2 with correlation matrix ρ. Since we do not want to assume differentiability Contact Information of the payoff f, but we know that the transition density g is differentiable, ∆ √ σi τxi+µiτ we define a change the variables yi = Si(0)e , which leads to mathfinance.de yi ! Z ln − µiτ d~y v = e−rτ f(. . . , y ,... )g Si(0)√ , ρ √ . (21) Title Page i Q σi τ yiσi τ JJ II J I Page 45 of 68 Go Back Full Screen Close Quit 4.6.2. Properties of the Normal Distribution We collect some properties of the multivariate normal density function g. We suppose that the vector X of n random variables with means zero Overview and unit variances has a nonsingular normal multivariate distribution with Accumulative Forward probability density function Instalment Options − 1 n 1 1 T Greeks g(x , . . . , x ; c , . . . , c ) = (2π) 2 |C| 2 exp − x Cx . (22) 1 n 11 nn 2 Contact Information Here C is the inverse of the covariance matrix of X, which is denoted by ρ. mathfinance.de Theorem 4.2 (Plackett’s Identity, 1954) [10] Title Page ∂g ∂2g = . (23) JJ II ∂ρij ∂xi∂xj J I In the two-dimensional case: 2 Page 46 of 68 ∂n (x, y; ρ) ∂ n (x, y; ρ) 2 = 2 , (24) ∂ρ ∂x∂y Go Back extends to the corresponding cumulative distribution function, i.e., Full Screen ∂N (x, y; ρ) ∂2N (x, y; ρ) 2 = 2 = n (x, y; ρ). (25) Close ∂ρ ∂x∂y 2 Quit 4.6.3. Correlation Risk and Cross-Gamma ∆ ∂2g Using the abbreviation gjk = the cross-gamma and correlation risk ∂xj ∂xk are Overview 2 Z Accumulative Forward ∂ v −rτ 1 d~y = e f(. . . , yi,... )gjk Q √(26), Instalment Options ∂Sj(0)∂Sk(0) Sj(0)Sk(0)σjσkτ yiσi τ ∂v Z d~y Greeks = e−rτ f(. . . , y ,... )g √ . (27) i ρjk Q Contact Information ∂ρjk yiσi τ Invoking Plackett’s identity (23) saying that gρjk = gjk leads to mathfinance.de Theorem 4.3 (cross-gamma-correlation-risk relationship) Title Page ∂v ∂2v = Sj(0)Sk(0)σjσkτ . (28) JJ II ∂ρjk ∂Sj(0)∂Sk(0) J I Page 47 of 68 Go Back Full Screen Close Quit 4.6.4. Interest Rate Risk and Delta A similar computation yields Overview Theorem 4.4 (delta-rho relationship) Accumulative Forward ∂v ∂v Instalment Options = −Sj(0)τ , (29) ∂qj ∂Sj(0) Greeks n ! Contact Information ∂v X ∂v = −τ v − S (0) . (30) ∂r j ∂S (0) j=1 j mathfinance.de Title Page JJ II J I Page 48 of 68 Go Back Full Screen Close Quit 4.6.5. Volatility Risk and Gamma Theorem 4.5 (gamma-vega relationship) n Overview ∂v X ∂2v σ = ρ σ σ S (0)S (0)τ . (31) Accumulative Forward j jk j k j k ∂σj ∂Sj(0)∂Sk(0) k=1 Instalment Options Greeks Contact Information mathfinance.de Title Page JJ II J I Page 49 of 68 Go Back Full Screen Close Quit 4.7. Results for European Claims in the Black-Scholes Model (One-Dimensional Case) Overview 1 0 = τΘ + rρ + qρq + σΦ scale invariance of time (32) Accumulative Forward 2 k Instalment Options v = x∆ + k∆ price homogeneity and strikes (33) 2 2 k Greeks x Γ = k Γ price homogeneity and strikes (34) l Contact Information x∆ = −l∆ price homogeneity and levels (35) x2Γ + x∆ = l2Γl + l∆l price homogeneity and levels (36) ρ = −τ(v − x∆) delta-rho relationship (37) mathfinance.de ρ + ρq = −τv rates symmetry (38) Title Page 1 rv = Θ + (r − q)x∆ + σ2x2Γ Black-Scholes PDE (39) 2 JJ II 1 qv = Θ + (q − r)k∆k + σ2k2Γk dual Black-Scholes (strike) J I 2 (40) Page 50 of 68 1 rv = Θ + (q − r + σ2)l∆l + σ2l2Γl dual Black-Scholes (level) Go Back 2 (41) Full Screen ρq = −τx∆ delta-rho relationship (42) k Close ρ = −τk∆ combination of (42) and (33) (43) Φ = στx2Γ gamma-vega relationship (44) Quit 4.8. A European Claim in the Two-Dimensional Black-Scholes Model Relations among the Greeks Overview Accumulative Forward Instalment Options 0 = ρq1 + S1(0)τ∆1, (45) Greeks 0 = ρq2 + S2(0)τ∆2, (46) Contact Information 1 1 0 = q ρ + q ρ + σ Φ + σ Φ + rρ + τΘ, (47) 1 q1 2 q2 2 1 1 2 2 2 r 0 = Θ − rv + (r − q1)S1(0)∆1 + (r − q2)S2(0)∆2 mathfinance.de 1 2 2 1 2 2 + σ1S1(0) Γ11 + ρσ1σ2S1(0)S2(0)Γ12 + σ2S2(0) Γ22, (48) Title Page 2 2 κ = σ1σ2τS1(0)S2(0)Γ12, (49) JJ II 2 2 0 = ρκ − σ1Φ1 + σ1τS1(0) Γ11, (50) 2 2 J I 0 = ρκ − σ2Φ2 + σ2τS2(0) Γ22, (51) 2 2 2 2 Page 51 of 68 0 = σ1Φ1 − σ2Φ2 − σ1τS1(0) Γ11 + σ2τS2(0) Γ22, (52) ρr = −τ (v − S1(0)∆1 − S2(0)∆2) , (53) Go Back 0 = τv + ρq1 + ρq2 + ρr. (54) Full Screen Close Quit 4.9. European Options on the Minimum/Maximum of Two Assets Overview + [φ (η min(ηS1(T ), ηS2(T )) − K)] . (55) Accumulative Forward Instalment Options This is a European put or call on the minimum (η = +1) or maximum Greeks (η = −1) of the two assets S1(T ) and S2(T ) with strike K. As usual, the Contact Information binary variable φ takes the value +1 for a call and −1 for a put. Its value mathfinance.de Title Page JJ II J I Page 52 of 68 Go Back Full Screen Close Quit function has been published in Stulz [1982] [14] and can be written as v(t, S1(t),S2(t), K, T, q1, q2, r, σ1, σ2, ρ, φ, η) (56) −q1τ Overview = φ S1(t)e N2(φd1, ηd3; φηρ1) −q2τ Accumulative Forward +S2(t)e N2(φd2, ηd4; φηρ2) Instalment Options −rτ 1 − φη √ √ − Ke + φN2(η(d1 − σ1 τ), η(d2 − σ2 τ); ρ) , Greeks 2 Contact Information 2 ∆ 2 2 σ = σ1 + σ2 − 2ρσ1σ2, (57) ρσ − σ ρ =∆ 2 1 , (58) 1 σ mathfinance.de ∆ ρσ1 − σ2 ρ2 = , (59) Title Page σ 1 2 ∆ ln(S1(t)/K) + (r − q1 + 2 σ1)τ d1 = √ , (60) JJ II σ1 τ 1 2 J I ∆ ln(S2(t)/K) + (r − q2 + 2 σ2)τ d2 = √ , (61) σ2 τ Page 53 of 68 ln(S (t)/S (t)) + (q − q − 1 σ2)τ d =∆ 2 1 √ 1 2 2 , (62) Go Back 3 σ τ ln(S (t)/S (t)) + (q − q − 1 σ2)τ Full Screen d =∆ 1 2 √ 2 1 2 . (63) 4 σ τ Close Quit 4.9.1. Greeks Delta. Space homogeneity implies that Overview ∂v ∂v ∂v v = S1(t) + S2(t) + K . (64) Accumulative Forward ∂S1(t) ∂S2(t) ∂K Instalment Options read off the deltas: Greeks ∂v Contact Information −q1τ = φe N2(φd1, ηd3; φηρ1), (65) ∂S1(t) ∂v −q2τ = φe N2(φd2, ηd4; φηρ2), (66) mathfinance.de ∂S2(t) ∂v 1 − φη √ √ = −φe−rτ + φN (η(d − σ τ), η(d − σ τ); ρ) . Title Page ∂K 2 2 1 1 2 2 (67) JJ II J I Page 54 of 68 Go Back Full Screen Close Quit Gamma. We use the identities ! ∂ y − ρx N2(x, y; ρ) = n(x)N , (68) ∂x p1 − ρ2 Overview ! Accumulative Forward ∂ x − ρy N2(x, y; ρ) = n(y)N , (69) Instalment Options ∂y p1 − ρ2 Greeks Contact Information and obtain " ! 2 −q1τ ∂ v φe φ d3 − d1ρ1 = √ n(d1)N ησ (70) mathfinance.de 2 p 2 ∂(S1(t)) S1(t) τ σ1 σ2 1 − ρ !# Title Page η d1 − d3ρ1 − n(d3)N φσ , σ σ p1 − ρ2 JJ II 2 " ! 2 −q2τ ∂ v φe φ d4 − d2ρ2 J I = √ n(d2)N ησ (71) 2 p 2 ∂(S2(t)) S2(t) τ σ2 σ1 1 − ρ Page 55 of 68 !# η d2 − d4ρ2 Go Back − n(d4)N φσ , p 2 σ σ1 1 − ρ Full Screen ! 2 −q1τ ∂ v φηe d1 − d3ρ1 = √ n(d3)N φσ . (72) p 2 Close ∂S1(t)∂S2(t) S2(t)σ τ σ2 1 − ρ Quit Kappa. The sensitivity with respect to correlation is directly related to the cross-gamma Overview ∂v ∂2v Accumulative Forward = σ1σ2τS1(t)S2(t) . (73) ∂ρ ∂S1(t)∂S2(t) Instalment Options Greeks Vega. We refer to (50) and (51) to get the following formulas for the vegas, Contact Information ∂v ρv + σ2τ(S (t))2v = ρ 1 1 S1(t)S1(t) (74) ∂σ1 σ1 " ! √ d − d ρ mathfinance.de −q1τ 1 3 1 = S1(t)e τ ρ1φηn(d3)N φσ p 2 σ2 1 − ρ Title Page !# d3 − d1ρ1 + n(d1)N ησ , (75) JJ II p 2 σ2 1 − ρ J I ∂v ρv + σ2τ(S (t))2v = ρ 2 2 S2(t)S2(t) (76) Page 56 of 68 ∂σ2 σ2 " ! √ d − d ρ −q2τ 2 4 2 Go Back = S2(t)e τ ρ2φηn(d4)N φσ p 2 σ1 1 − ρ Full Screen !# d4 − d2ρ2 + n(d2)N ησ . (77) Close p 2 σ1 1 − ρ Quit Rho. Looking at (45), (46) and (53) the rhos are given by ∂v ∂v Overview = −S1(t)τ , (78) ∂q1 ∂S1(t) Accumulative Forward ∂v ∂v Instalment Options = −S (t)τ , (79) ∂q 2 ∂S (t) Greeks 2 2 ∂v ∂v Contact Information = −Kτ . (80) ∂r ∂K mathfinance.de Title Page JJ II J I Page 57 of 68 Go Back Full Screen Close Quit Theta. Among the various ways to compute theta one may use the one based on (47). ∂v 1 h σ1 σ2 i Overview = − q v + q v + rv + v + v . (81) ∂t τ 1 q1 2 q2 r 2 σ1 2 σ2 Accumulative Forward Instalment Options Greeks Contact Information mathfinance.de Title Page JJ II J I Page 58 of 68 Go Back Full Screen Close Quit 4.10. Heston’s Stochastic Volatility Model h p (1)i dSt = St µ dt + v(t)dWt , (82) Overview p (2) Accumulative Forward dvt = κ(θ − vt) dt + σ v(t)dWt , (83) h i Instalment Options (1) (2) Cov dWt , dWt = ρ dt, (84) Greeks Λ(S, v, t) = λv. (85) Contact Information Heston provides a closed-form solution for European vanilla options paying mathfinance.de + [φ (ST − K)] . (86) Title Page As usual, the binary variable φ takes the value +1 for a call and −1 for a JJ II put, K the strike in units of the domestic currency J I Page 59 of 68 Go Back Full Screen Close Quit 4.10.1. Abbreviations a =∆ κθ (87) Overview ∆ 1 u1 = (88) Accumulative Forward 2 Instalment Options ∆ 1 u2 = − (89) Greeks 2 ∆ Contact Information b1 = κ + λ − σρ (90) ∆ b2 = κ + λ (91) q ∆ 2 2 2 mathfinance.de dj = (ρσϕi − bj) − σ (2ujϕi − ϕ ) (92) Title Page ∆ bj − ρσϕi + dj gj = (93) bj − ρσϕi − dj JJ II τ =∆ T − t (94) dj τ J I ∆ bj − ρσϕi + dj 1 − e Dj(τ, ϕ) = 2 d τ (95) Page 60 of 68 σ 1 − gje j ∆ Go Back Cj(τ, ϕ) = (r − q)ϕiτ dj τ a 1 − gje Full Screen + (bj − ρσϕi + d)τ − 2 ln (96) σ2 1 − edj τ Close (97) Quit ∆ Cj (τ,ϕ)+Dj (τ,ϕ)v+iϕx fj(x, v, t, ϕ) = e (98) Z ∞ −iϕy Overview ∆ 1 1 e fj(x, v, τ, ϕ) Pj(x, v, τ, y) = + < dϕ (99) Accumulative Forward 2 π 0 iϕ Z ∞ Instalment Options ∆ 1 −iϕy pj(x, v, τ, y) = < e fj(x, v, τ, ϕ) dϕ (100) Greeks π 0 1 − φ Contact Information P (φ) =∆ + φP (ln S , v , τ, ln K) (101) + 2 1 t t ∆ 1 − φ P−(φ) = + φP2(ln St, vt, τ, ln K) (102) mathfinance.de 2 Title Page This notation is motivated by the fact that the numbers Pj are the cumu- lative distribution functions (in the variable y) of the log-spot price after JJ II time τ starting at x for some drift µ. The numbers pj are the respective densities. J I Page 61 of 68 Go Back Full Screen Close Quit 4.10.2. Value The value function for European vanilla options is given by −qτ −rτ Overview V = φ e StP+(φ) − Ke P−(φ) (103) Accumulative Forward Instalment Options The value function takes the form of the Black-Scholes formula for vanilla Greeks options. The probabilities P±(φ) correspond to N (φd±) in the constant volatility case. Contact Information mathfinance.de Title Page JJ II J I Page 62 of 68 Go Back Full Screen Close Quit 4.10.3. Greeks Spot delta. Overview ∆ ∂V −qτ ∆ = = φe P+(φ) (104) Accumulative Forward ∂St Instalment Options Greeks Dual delta. Contact Information ∂V ∆K =∆ = −φe−rτ P (φ) (105) ∂K − mathfinance.de Gamma. −qτ Title Page ∆ ∂∆ ∂∆ ∂x e Γ = = = p1(ln St, vt, τ, ln K) (106) ∂St ∂x ∂St St JJ II J I Dual Gamma. K K −rτ Page 63 of 68 ∂∆ ∂∆ ∂y e ΓK =∆ = = p (ln S , v , τ, ln K) (107) ∂K ∂y ∂K K 1 t t Go Back Full Screen Close Quit Rho. Rho is connected to delta via equations (43) and (42). ∂V = φKe−rτ τP (φ), (108) ∂r − Overview ∂V = −φS e−qτ τP (φ). (109) Accumulative Forward ∂q t + Instalment Options Greeks Theta. Theta can be computed using the partial differential equation for Contact Information the Heston vanilla option 1 1 2 Vt + (r − q)SVS + σvVvv + vS VSS + ρσvSVvS − qV mathfinance.de 2 2 +[κ(θ − v) − λ]Vv = 0, (110) Title Page where the derivatives with respect to initial variance v must be eval- JJ II uated numerically. J I Page 64 of 68 Go Back Full Screen Close Quit 4.11. Summary • Understand homogeneity-based methods to compute analytical for- mulas of Greeks for analytically known value functions of options in Overview a one-and higher-dimensional market Accumulative Forward Instalment Options • Restricting the view to the Black-Scholes model there are numerous Greeks further relations between various Greeks Contact Information • Saving computation time for the mathematician who has to differ- entiate complicated formulas as well as for the computer, because analytical results for Greeks are usually faster to evaluate than finite mathfinance.de differences involving at least twice the computation of the option’s Title Page value JJ II • Knowing how the Greeks are related among each other can speed up finite-difference-, tree-, or Monte Carlo-based computation of Greeks J I or lead at least to a quality check Page 65 of 68 • Many of the results are valid beyond the Black-Scholes model Go Back • Most remarkably some relations of the Greeks are based on properties Full Screen of the normal distribution refreshing the active interplay between mathematics and financial markets. Close Quit 5. Contact Information Uwe Wystup HfB-Busniness School of Finance and Management Overview Sonnemanstraße 9-11 Accumulative Forward 60314 Frankfurt am Main Instalment Options Germany Greeks and Contact Information Commerzbank Securities Foreign Exchange Options Mainzer Landstrasse 153 mathfinance.de 60327 Frankfurt am Main Title Page Germany JJ II REUTERS Dealing: CBOF (Commerzbank Options Frankfurt) J I This presentation is available at http://www.mathfinance.de/wystup/ Page 66 of 68 papers.html Go Back Full Screen Close Quit References [1] http://www.amex.com/ Overview [2] Ben-Ameur, H. ,Breton, M. and Francois, P. (2002). Pricing Accumulative Forward Instalment Options with an Application to ASX Instalment Warrants. Instalment Options [3] Borodin, A. N. and Salminen, P., Handbook of Brownian Motion – Greeks Facts and Formulae, Birkh¨auser,Basel, 1996. Contact Information [4] Cox, J.C., Ingersoll, J.E. and Ross, S.A. (1985). A Theory of the Term Structure of Interest Rates. Econometrica 53, 385-407. mathfinance.de [5] Curnow, R.N. and Dunett, C.W. (1962). The Numerical Evalua- Title Page tion of Certain Multivariate Normal Integrals. Ann. Math. Statist., 33, 571-579 JJ II [6] Davis M., Schachermayer, W and Tompkins, R. (2001). Pric- J I ing, No-arbitrage Bounds and Robust Hedging of Instalment Options. Page 67 of 68 Quantitative Finance, 1, p. 597-610. Go Back [7] Hakala, J. and Wystup, U. (2002) Foreign Exchange Risk, Risk Publications, London. Full Screen [8] Heston, S. (1993). A Closed-Form Solution for Options with Stochas- Close tic Volatility with Applications to Bond and Currency Options. The Review of Financial Studies, Vol. 6, No. 2. Quit [9] Karsenty, F. and Sikorav, J. (1996). Instalment Plan. Over the Rainbow, Risk Publications, London. [10] Plackett, R. L. (1954). A Reduction Formula for Normal Multivari- Overview ate Integrals. Biometrika. 41, pp. 351-360. Accumulative Forward Instalment Options [11] Press, W. H., Teukolksy, S. A., Vetterling, W. T. and Flan- nery, B. P. (1992). Numerical Recipes in FORTRAN, Second Edition. Greeks Cambridge University Press Contact Information [12] Shaw, W. (1998). Modelling Financial Derivatives with Mathematica. Cambridge University Press. mathfinance.de [13] Shreve, S.E. (1996). Stochastic Calculus and Finance. Lecture notes, Title Page Carnegie Mellon University JJ II [14] Stulz, R. (1982). Options on the Minimum or Maximum of Two Assets. Journal of Financial Economics. 10, pp. 161-185. J I [15] Taleb, N. (1996). Dynamic Hedging. Wiley, New York. Page 68 of 68 [16] Wystup, U (2000). The MathFinance Formula Catalogue. http:// Go Back www.mathfinance.de Full Screen [17] Wystup, U (2003). The Market Price of One-touch Options in For- Close eign Exchange Markets, Derivatives Week Vol. XII, no. 13, London 2003. Quit