FX OPTION PRICING: RESULTS FROM BLACK SCHOLES, LOCAL VOL, QUASI Q-PHI AND STOCHASTIC Q-PHI MODELS

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1 FX OPTION PRICING: REULT FROM BLACK CHOLE, LOCAL VOL, QUAI Q-PHI AND TOCHATIC Q-PHI MODEL Absrac Krishnamurhy Vaidyanahan 1 The paper suggess a new class of models (Q-Phi) o capure he informaion ha he marke provides hrough he 5-Dela rangles and 5-Dela Risk Reversals. The model is able o capure he sochasic movemens of a full srike srucure of implied volailiies.we argue ha exracing informaion hrough his model and pricing pah-dependen and non-benchmark srike opions is a beer mehodology han using a conan implied volailiy. The model can be used o price exoic opions and hedge hem robusly wih benchmark European opions. 1 Inroducion Pricing derivaive producs is abou compuing he righ probabiliies. For insance, for a foreign exchange vanilla opion, we need o know he probabiliy densiy funcion of he underlying exchange rae a mauriy. This allows calculaing he probabiliy ha spo a mauriy is wihin a given inerval, which for Call opions is greaer han srike and for Pu opions is less han srike. For a barrier opion, we need o compue he join probabiliy ha he exchange rae is wihin a given inerval a mauriy and ha he spo did no ouch he barrier on is pah from opion sar dae o mauriy. Nex we would muliply he appropriae probabiliies wih he payoff of he opion and sum over over all possible oucomes of he exchange rae a mauriy o ge he expeced payoff. The opions value is his payoff discouned o he opion sar dae. We imply he probabiliies from marke prices, which for liquidly raded opions is implied volailiy. The implied volailiy varies for differen opion srikes. The benchmark srikes ha ge raded in he foreign exchange opions are 5-dela (more commonly known as a-he-money-forward or ATMF srike), 5-Dela Call and 5-Dela Pu (also known as 75-dela Call). Figure 1 depics a ypical volailiy smile quoed in he marke. The marke quoes ATMF volailiy (or shor form vol ), 5- Dela rangle and 5-Dela Risk Reversal which are relaed o he 5-Dela Call and Pu volailiies as follows: 5-Dela rangle = (5-Dela Call vol + 5-Dela Pu vol)/ ATMF vol (1) 5-Dela Risk Reversal = 5-Dela Call vol - 5-Dela Pu vol (favouring Calls) () Alernaively, if 5-Dela Pu vol is higher han 5-Dela Call vol, 5-Dela Risk Reversal is said o be favouring Pus and is quoed as (5-Dela Pu vol - 5- Dela Call vol). 1 INE, CRE Building, IIT Bombay, Powai, Mumbai 476. India. vaidyanahan@alumni.iik.ac.in 1

2 Volailiy mile 18.% 17.% 5-Dela Call, 17.% Implied Volailiy 5-Dela-Pu, 15.% 16.% 15.% 14.% 13.% ATMF, 13.5% % 5% 5% 75% rike dela Fig. 1. The figure depics he volailiy smile ha raders quoe for liquid foreign exchange opions. The benchmark srikes correspond o 5-Dela-Call, ATMF i.e. 5- Dela and 5-Dela-Pu. 1.1 Problem of muliple volailiy for same underlying European opions are ofen priced and hedged using he Black-choles (B) model. In B model here is a one-o one relaion beween he price of a European opion and he volailiy parameer σ B. Consequenly, opion prices are ofen quoed by saing he implied volailiy σ B, he unique value of he volailiy which yields he opion s dollar price when used in B. In heory, he volailiy σ B in B model is a consan. In pracice, opions wih differen srikes K require differen volailiies σ B o mach heir marke prices as shown in figure 1. Handling hese marke skews and smiles correcly is criical o foreign exchange desks, since hey usually have large exposures across a wide range of srikes. Ye he inheren conradicion of using differen volailiies for differen opions makes i difficul o successfully manage hese risks using B model. 1. The informaion conen of benchmark opions prices. There are good reasons for incorporaing he prices of benchmark opion prices like 5-Dela rangles and 5-Dela Risk Reversals ino a model for pricing and risk managing foreign exchange opions. ince he adven of he famous Black and choles (1973) opion pricing model and he inroducion of foreign exchange opion conracs, he volume and liquidiy of fx opions has increased exponenially. imulaneously more and more complex, exoic opion specificaions have arisen wih feaures ranging from knock-in and knock-ou barriers, digial opions and range binaries o combinaions of hese and oher feaures wih many differen payoff funcions. While on he one end of he specrum he developmen has gone owards increasingly complex specificaions, here has been a significan increase in liquidiy in he markes of sandard European Call and Pu opions. For almos every exchange rae, here are liquid markes for European opions wih a broad range of mauriies and srike prices in paricular he srikes corresponding o 5-Delas. These benchmark opions make he rading of a new piece of informaion possible - informaion on volailiy.

3 The opion prices conain fundamenal informaion on no jus he volailiy bu also oher informaion like he co-movemen of volailiy wih he underlying. For insance, when he marke pus a higher premium on 5-Dela Pus vis-à-vis 5-Dela Calls, i is implicily saing ha i expecs he volailiy o be higher when spo goes lower han he case when spo goes higher. Therefore a high risk-reversal implies a high correlaion beween po and Volailiy. imilarly, a high srangle implies ha he marke expecs volailiy iself o be volaile. As an aside, if volailiies were no volaile, he opions marke may no exis! Given ha he benchmark opion prices reveal addiional informaion abou he likely dynamics of he underlying, a pricing model should use heir prices as inpu. This should yield an increase in accuracy over he sandard Black-choles model. Then he sandard opions can also be used as addiional hedge insrumens. The model presened here ries o ake hese poins ino accoun. I is designed o incorporae benchmark opion prices and hus he informaion ha hey conain, in order o improve he pricing of more exoic insrumens. 1.3 Relaed Lieraure The deviaion of observed marke prices for opions from heir heoreical counerpars as given by he Black-choles formula has riggered a large lieraure in which boh academics and praciioners alike have ried o improve on he limiaions of he Black choles model. One srand of he lieraure concenraes on he implied ree approach. The aim is o keep as closely as possible o he Black-choles seup while exacly reproducing he opion prices given in he marke. This is achieved by specifying a ime and sae dependen volailiy funcion which does no conain any addiional random componen. Models of his ype are by Rubinsein (1995), Derman and Kani (1994), Derman e.al. (1996) and Dupire (1994). While exacly reproducing he opion prices observed in he marke he implied volailiy models have he drawback ha hey do no allow for idiosyncraic sochasic dynamics in he opion prices. This is in conflic wih empirical observaion and wih he coninuous updaing of he new informaion refleced in he opion prices. The poor resuls in a hedging es performed by Dumas e.al. (1999) are probably also due o his drawback. Dupire (1996) ook a firs sep owards incorporaing sochasic dynamics ino he erm srucure of volailiies, bu again he models realized volailiies and forward conracs on i, and no implied volailiies from opions prices. Derman and Kani (1998) exended heir implied ree approach o allow for sochasic dynamics in he full erm and srike srucure of implied local volailiies. They derive resricions on he drif of he local volailiies ha are necessary for absence of arbirage, and hese resricions involve inegrals over all possible underlying prices and imes before he mauriy of he forward volailiy concerned. The complexiy of hese resricions makes he model hard o handle and we are going o propose a slighly differen approach. Furhermore i is no obvious how in Derman and Kani s model i is ensured ha he implied volailiies saisfy cerain no-arbirage resricions as expiry is approached. The fundamenal problem is, ha Derman and Kani specify wo hings ha may be conradicory: he dynamics of he spo volailiy and he implied volailiies for differen srike prices and mauriies. The developmen of local volailiy models by Dupire (1994), and Derman- Kani (1996) was a major advance in handling smiles and skews. Local volailiy 3

4 models are self-consisen, arbirage-free, and can be calibraed o precisely mach observed marke smiles and skews. Currenly hese models are sill used for managing smile and skew risk. However, he dynamic behavior of smiles and skews prediced by local vol models is exacly opposie of he behavior observed in he markeplace: when he price of he underlying asse decreases, local vol models predic ha he smile shifs o higher prices; when he price increases, hese models predic ha he smile shifs o lower prices. In realiy, exchange raes and marke smiles move in he same direcion. This conradicion beween he model and he markeplace ends o de-sabilize he dela and vega hedges derived from local volailiy models, and ofen hese hedges perform worse han he naive Black-choles hedges. We will follow an approach for modeling fundamenal quaniies like he sochasic process of he volailiy of he underlying as in he radiional sochasic volailiy models of Hull (1987), Heson (1993) or ein and ein (1991). This faciliaes he fiing of he model o observed opion prices and gives he model a larger degree of flexibiliy. Using a model based approach means ha we do no use he marke-based approach applied o he erm srucure of implied volailiies which is similar o he marke models of he erm srucure of ineres raes by Milersen, andmann and ondermann (1995), Brace, Gaarek and Musiela (1997) and Jamshidian (1997). Nor do we model he insananeous condiional forward volailiies as in he effecive volailiy model by Derman and Kani (1998), or forward variances like in Dupire (199) or model he Black-choles implied volailiies. Anoher direcion of research has been on he naure of he underlying price process which was assumed o be a lognormal Brownian moion by Black and choles. Well known papers of his approach are by Hull (1987), Heson (1993), or ein and ein (1991) as discussed earlier. The approach aken here draws upon boh he srands of research hus far. We model he insananeous volailiy of exchange raes raher han implied volailiies as in he marke models. We also incorporae sochasiciy in he underlying volailiy i.e. volailiy of he underlying is assumed o be uncerain as is observed in he marke. ome of he consrains of our model are ha hough hey reproduce he ypical shapes of implied volailiies observed in he markes known as he smile, hey need o be calibraed o he benchmark opions, i.e. hey are compuaionally inensive and here is no closed form soluion for calculaing opion price. Anoher consrain is ha his model has only one addiional facor driving he sochasic volailiy and canno be exended o he muli-facor case. In wha follows we briefly presen he models ha are currenly used in foreign exchange opions: he famous Black-choles model, he Q-Φ model and he ochasic Q-Φ model, and compare he resuls obained from each of hese models. 4

5 The Black-choles Model We firs inroduce he Black-choles Model.1 po Dynamics and he Local Volailiy The Black-choles (1973) approach owards he probabiliy densiy is sraighforward. The basic assumpion is ha he exchange rae follows a lognormal sochasic process: d = µ * d + σ * dw (3) The Lognormal ochasic Process spo d d ime in days spo drif Fig.. The Lognormal ochasic Process. The blue line shows a simulaed pah for he firs 1 days of an exchange rae according o equaion (3). Iniial spo = 1, volailiy σ = 1% and drif µ = 1%. The pink line shows he drif only, i.e. he deerminisic par of equaion (3). The double-arrows indicae he size of he spo move d in he second ime sep. The relaive move of spo d / over he nex ime inerval d is given by a deerminisic par, he so-called drif erm µ*d and a random erm σ*dw. The drif erm is given by he difference of he numeraire and asse ineres raes: µ*d = (r num r ass )*d. The random par is driven by dw = ε * d, where ε is a random number ha is drawn from a sandard normal disribuion N(,1). The model parameer ha governs he dynamics of he Black-choles model is he local or insananeous volailiyσ. We are ineresed in he disribuion of spo a mauriy of he opion. o we perform a Mone Carlo simulaion o creae i. Draw a random numberε from our sandard normal disribuion N(,1), plug i ino equaion (3) and deermine he new spo +1 = + d, which is he saring poin for he nex ime inerval d. If we repea his procedure unil we reach he expiry dae of he opion we generae a single random pah of spo from oday s dae o expiry dae. A knockou barrier is aken ino accoun by sopping any pah ha ouches his level. The informaion of he oucome 5

6 of for many pahs creaed in he same way provides us wih a frequency disribuion of oucomes of spo a expiry. Afer normalisaion, i.e. division by he number of launched pahs, we obain he probabiliy disribuion of spo a expiry.. The Implied or Black-choles Volailiy In a Black-choles world, insead of running a edious Mone Carlo simulaion we can direcly solve he sochasic parial differenial equaion (3) σ B + = * exp µ σ BW (4) Equaion (4) allows jumping a macroscopic ime sep, e.g. = expiry dae, raher han only microscopic seps of he size d. σ B is he implied or Black-choles volailiy. As ε is from N(,1), W σ ε N, σ B, i.e. σ = is disribued ( ) B normally disribued wih sandard deviaion σ B and zero mean. Taking he logarihm of equaion (4) shows why he Black-choles model is also known as he lognormal model. σ B ln µ + σw = (5) Equaion (5) ells us ha ln( ) is normally disribued wih mean µ and sandard deviaion σ B B B σ, i.e. N( σ, σ B ) µ. A simple ransformaion resuls in he PDF of. The Black-choles or implied volailiy σ B is a enor volailiy whereas σ is only valid for he nex infiniesimal ime sep. In he Black-choles world hese wo volailiies numerically agree, i.e. σ = σ B, however, hese volailiies are wo differen parameers as will become more eviden when we consider more sophisicaed models. The derivaion of he PDF of given ha he spo did no ouch he barrier on is pah is a bi more cumbersome bu he poin o be made is ha we arrive a a probabiliy densiy funcion for spo a mauriy using a single model parameer σ..3 Calibraion The calibraion of he Black-choles model is simple. We deermine σ B (=σ) such ha he Black-choles price of an opion is equal o he marke price of his opion. The sandard pracice in he marke is o choose ATMF (a-he-moneyforward) and 5-Dela Call and Pu srike opions for calibraion..4 Pricing Pricing in he Black-choles world is simple oo. For mos opion producs we can provide an analyical formula for he inegral of he produc of probabiliy imes payoff. This makes implemenaion of he algorihms fas and robus..5 No mile For a given enor he volailiy smile is defined as he implied volailiy as a funcion of srike. Thus, per consrucion, he Black-choles model does no exhibi any smile, i.e., he funcion σ B (K) = σ B is consan. B 6

7 3 The Q-Φ Model 3.1 po Dynamics and he Local Volailiy The Q-Φ model allows for ineres raes and hence he drif erm o be ime dependen. More imporanly, however, he local volailiy is assumed o depend on spo and ime. d () * d σ (, ) * dw = µ + (6) ome Wall ree houses incorporae his emporal informaion in order o price pah-dependen opions. However, he dependence of implied volailiy on he srike, for a given mauriy known as he mile effec, is rickier. Many researchers have aemped o enrich he Black-choles model o compue a heoreical smile. Unforunaely hey have o inroduce a non-raded source of risk, jumps in he case of Meron (1976) and sochasic volailiy in he case of Hull and Whie (1987), hus losing he compleeness of he model. Compleeness is of high value since i allows for arbirage pricing and hedging. If we look carefully, he process in he equaion above looks very similar o he lognormal process of he Black-choles model ha we inroduced in equaion (3). Ye as we simply allow he local volailiy o depend on spo and ime we are able o capure oday s volailiy smile. For opion pricing we could again recourse o he Mone Carlo echnique as described before for he Black-choles model. Draw a random numberε from N(,1), plug i ino equaion (6) and deermine he new spo +1 = + d. However, o do so, we require he values of he local volailiy for each aainable ime and spo level, he so-called local volailiy surface. 3. The Local Volailiy urface The Q-Φ model does no provide an analyic funcion for σ(,). Raher we can derive an equaion ha relaes he local volailiy o he derivaives of he opion prices wih respec o srike and ime. C C rass ( T )* C + µ ( T )* K * σ ( K, T ) = T K (7) 1 C * K * K where C denoes he price of a Call opion wih enor T and srike K. If we knew opion prices for all enors and srikes, we could numerically work ou he derivaives in equaion (7) and deermine he local volailiies. In fac, his is he roue ha is aken. Obviously, we need o know how he Q-Φ model compues vanilla opion prices. They are compued as he average of wo Black-choles prices. 1 C = B σ + B σ ) (8) ( 1 ( ) ( ) B( ) denoes he Black-choles opion pricing formula and he wo implied volailiies are given by 7

8 F σ 1 = σ ATMF ( T )* 1m Q ( T ) Φ( T )* ln (9) K Equaion (8) and (9) can be inerpreed in a simple way: The Q-Φ model assumes a wo-poin disribuion for volailiy. For a given srike K he volailiy will be eiher σ 1 or σ wih probabiliy 1/. The meaning of he model parameers Q and Φ is explained laer. 3.3 Calibraion We calibrae he model agains he marke prices for hree opions, ATMF, 5- dela Calls and Pus. We have o fi he hree parameers of he model, σ ATMF, Q, and Φ, such ha he Q-Φ prices i.e., equaion (8) mach he marke prices. Using equaions (8) and (9) yields vanilla opion prices for all enors and srikes and equaion (7) provides us wih he volailiy surface. Local Volailiy urface 3.% 5.%.% 15.% 1.% 5.% sigma ime in days spo % Fig. 3. Q-Φ Local Volailiy urface as a Funcion of Time and po. The graph above is generaed wih he Q-Φ model. I shows a 6M local volailiy surface. We use an iniial spo of = 1, 6M ATMF volailiy σ ATMF = 13.5%, srangle r =.5% and risk reversal RR = % for Pus. The calibraion reurned Q =.36 and Φ = Pricing The volailiy surface is fed ino a Mone Carlo simulaion or a finie difference grid o price exoic opions consisen wih he vanilla marke. Apar from vanilla opions, for which we can recourse o equaion (8) here is no analyical soluion for Q-Φ opion prices. 3.5 The Use of he Wrong Number The implied volailiy is he parameer ha we have o plug ino he Black- choles formula o obain he marke price of an opion. For he Q-Φ model here is no obvious connecion beween he local volailiy and his implied volailiy. The 8

9 local volailiy surface conains he full informaion abou oday s smile. We can use his surface o correcly and consisenly price any vanilla or exoic opion. The implied volailiy, in conras, is simply he wrong number o be pu ino he wrong formula o ge he vanilla prices righ. However, using he knowingly wrong Black- choles formula allows marke paricipans o express he smile in a sandardised way in wo single numbers: he srangle and he risk reversal. 3.6 mile We can easily relae Q and Φ o srangle and risk reversal. As we will show Φ inroduces some asymmery o he model while Q creaes a symmeric smile. A posiive Φ ends o increase boh, σ 1 and σ and for opions wih K>F, i.e. OTM calls, and decreases boh for opions wih K>F. Thus a posiive Φ corresponds o a risk reversal ha favours calls. Obviously, Φ does no have any impac on he price of ATMF opions as ln ( F K )in equaion (9) vanishes in ha case. Equaion (8) and (9) show ha Q does no affec he price of an ATMF opion while i does so for an OTM opion. Look a an opion s Vega o undersand his resul. Vega as a funcion of volailiy is fairly consan for ATMF opions, i.e. ATMF opions have zero Vomma. In oher words, he price of an ATMF opion is an almos linear funcion of volailiy and we have ATMF: = 1 * ( B( σ 1 ) + B( σ )) B( 1 * ( σ 1 + σ )) = B( σ ATMF C ) In conras, for OTM opions Vega increases wih rising volailiy, i.e. hey have posiive Vomma. Alernaively, we say ha he price of an OTM opion is a convex funcion 3 of volailiy. Thus for OTM: C = 1 * ( B( σ ) + B( σ )) B( 1 ( σ + σ )) 1 * Apparenly, Q makes OTM opions more expensive, ha is, Q can be relaed o he srangle. 1 Vomma is he second derivaive of he opions price C wih respec o volailiy describes he curvaure of he opions price as a funcion of volailiy. c x + c y c x + y 3 For a convex funcion c we have ( ( ) ( )) (( ) ) C σ, i 9

10 4 The ochasic Q-Φ model We provide he formulaion and dynamics of he sochasic Q-Φ model. 4.1 po and Volailiy Dynamics rae. The ochasic Q-Φ model assumes he following dynamics for he exchange d d () * d * f (, ) * dw = µ + σ (1a) σ ξ (1b) σ where and dz () * dz f ( ), = κ * = d α () wih + 1* β κ N () (,1) * 1 The random erm for he exchange rae in equaion (1a) is similar o he one σ * f ha we have seen in he Q-Φ model. Here, he analyical funcion ( ), deermines he local volailiy surface. However, in he sochasic model he volailiy iself is a random process wih no drif and a volailiy (of volailiy) ξ. The model has four parameers: σ, α, β, andξ. σ is he iniial value for he volailiy. We will undersand he meaning of he oher parameers in wha follows. We use he Mone Carlo approach o invesigae on he dynamics of equaions (1a) and (1b). We firs draw a random numberε, plug i ino equaion (1a) o deermine he spo afer he firs ime sep +1 = + d. Before coninuing wih he nex ime sep, however, we have o draw a random number κ, plug i ino equaion (1b) o obain he correc parameerσ +1 = σ + dσ o be used for he second ime sep and so on. As far as he local volailiy surface is concerned, he Q-Φ model is fully deerminisic. In conras, in he sochasic Q-Φ model we can only predic he local volailiy for an arbirary poin in he fuure in a probabilisic sense. We could say ha he sochasic volailiy process in equaion (1b) vibraes he deerminisic σ * f in a random manner. Noe ha his isn exac in a mahemaical funcion ( ), sense bu i is a useful way of looking a he mechanics. We approach he full complexiy of he model by firs considering wo special ξ = and he purely cases of he sochasic Q-Φ model, he Quasi Q-Φ model ( ) sochasic model ( β = ). 4. The Quasi Q-Φ Model uppose ha he volailiy of volailiy vanishes ( ξ = ). This means ha volailiy iself is no a random variable. In his case he smile is fully deerminisic. σ * f. The model becomes similar o he The local volailiy surface is given by ( ), 1

11 Q-Φ approach (Compare equaions (6) and (1a)). The srangle is refleced in β which creaes a parabolic, symmeric local volailiy surface around spo. The asymmery of he smile, i.e. he risk reversal, is creaed by α. We have o calibrae σ, α, and β such ha he marke prices of hree opions, ATMF, 5 Calls and Pus, are mached. Local Volailiy urface 35.% 3.% ime in days spo 5.%.% 15.% 1.% 5.%.% sigma 3.%-35.% 5.%-3.%.%-5.% 15.%-.% 1.%-15.% 5.%-1.%.%-5.% Fig. 4. ochasic Q-Φ Local Volailiy urface. The graph above is generaed wih he sochasic Q-Φ model wih vanishing volailiy of volailiy ( ξ = ). For he example we used an iniial spo of = 1, 6M ATMF volailiy σ ATMF = 13.5%, srangle r =.5% and RR = % for Pus. The calibraion reurned σ = 1.67%, β =. 7 and α = Q-Φ versus Quasi Q-Φ model Compare he graphs of he local volailiy in figures 3 and 4. Boh surfaces were fied o he same smile. In spie of he fac ha hey appear o be very differen here is no conradicion here. The local volailiy surface is no direcly observable in he marke. Raher he surface is implied and inerpolaed from hree marke observaions only, he prices of a 6M ATMF opion and a 6M 5 Call and Pu. In fac he wo models agree on he prices of hese hree opions using he respecive local volailiy surfaces. This is a rivial resul as his is simply an inversion of he calibraion process. To undersand in which case he models would give differen resuls, consider he earlier Mone Carlo simulaion exercise. Each pah sars a =, i.e. a he fron of figure 3 or 4 and runs in a random zigzag line o he back of he graph. Along is way we pick up local volailiies and apply hem o calculae he nex random shock for he spo. Consider a one-ouch opion. A pah se ou o run on he surface of figure 3 presumably experiences a differen probabiliy o ouch he rigger han a pah running on he surface of figure 4. To summarise, wo differen models will naurally agree on he prices of he vanilla producs ha are used for he calibraion. Bu hey may give differen answers, albei he difference may be small, 11

12 for producs wih differen payoff funcions, in paricular pah dependen opions, such as knockous and one ouch opions. 4.4 Pure ochasic Model Assume nex ha β = bu ξ >. In his case he risk reversal is sill creaed byα. The srangle, however, is creaed by he sochasic naure of he volailiy. In our simplified picure of he sochasic model he local volailiy vibraes wih an inensiy proporional o he volailiy of volailiy. Thus, producs wih posiive Vomma, such as OTM opions, will become more expensive. Accordingly, he deerminisic volailiy surface has a nonzero slope o accoun for he risk reversal bu does no exhibi any curvaure (see figure 5). We have o calibrae σ, α, andξ o he usual se of marke prices: ATMF, 5 Calls and Pus. Local Volailiy urface 5.%.% 15.% 1.% sigma.%-5.% 15.%-.% 1.%-15.% 5.%-1.%.%-5.% ime in days spo 5.%.% Fig. 5. ochasic Q-Φ Local Volailiy urface. The graph above is generaed wih he sochasic Q-Φ model wih vanishing β wih similar parameers as before. The calibraion reurned σ = 1.68%, and α =., and ξ = 93%. 4.5 Purely ochasic versus Quasi Q-Φ Wha are he differences beween he purely sochasic model and he Quasi Q-Φ model. To work one ou, we pu ourself a he saring poin of a Mone Carlo simulaion, i.e. a = and = 1. uppose ha afer a shor period of ime, say, a couple of days, he random shocks bring spo down o 95. We consider he view on he volailiy surface ahead wih spo a 95. For he surfaces in figure 4 he landscape will be iled, i.e. he Quasi Q-Φ model increases he risk reversal when spo moves down. In conras, he slope of he surface in figure 5 remains unchanged, i.e. he sochasic model predics ha he risk reversal does no change a all. Anoher major difference is due o he naure of he purely sochasic model. As menioned above, due o he sochasic naure of volailiy he model will mark up all posiive Vomma producs. In conras, he Quasi Q-Φ model assumes a 1

13 deerminisic, i.e. saic local volailiy. I canno capure he benefi of posiive Vomma. As an example, he price of a range binary will be higher if he purely sochasic model is used as compared o he Quasi Q-Φ model. 4.6 ochasic Q-Φ or The Mix As described above, we eiher generae a smile by means of a purely deerminisic local volailiy surface or we use a deerminisic risk reversal combined wih a sochasic volailiy process. Boh approaches are consisen wih he marke, i.e. ATMF, 5 Calls and Pus, so we canno deermine which approach is he correc one. Addiional marke informaion is required o find he mos appropriae model. A purely deerminisic volailiy surface enails ha he risk reversal changes quickly when spo moves, whereas in a purely sochasic model he smile shifs sideways, i.e. if he risk reversal does no change a all when spo moves. Thus, if we combine he wo approaches, via he mix raio we can conrol our model s implied change in risk reversal wih a change in spo. In fac we fi he four parameers of he full model, σ, α, β, andξ such ha we mach ATMF, 5 Calls and Pus and he speed of he risk reversal wih a change in spo. Local Volailiy urface 5.%.% ime in days spo 15.% 1.% 5.%.% si g m.%-5.% 15.%-.% 1.%-15.% 5.%-1.%.%-5.% Fig. 6. ochasic Q-Φ Local Volailiy urface. The graph above is generaed wih he sochasic Q-Φ model. For he example we fied he model o an iniial spo of = 1, 6M ATMF volailiyσ ATMF = 13.5%, srangle r =.5%, RR = % for Pus, and change of risk reversal per 1 % move down in spo of.1% The calibraion reurned σ = 1.65%, and α =. 8, β = 5%, and ξ = 8%. β is he only model parameer ha is sensiive o he speed of risk reversal. Thus, in a simplified picure, we would sar he model s calibraion by fiing β such ha i generaes he marke s speed of risk reversal. A he same ime, β accouns for he curvaure of he local volailiy surface in figure 6. However, he curvaure is less pronounced as compared o he one of he surface in figure 4. This means ha he 13

14 srangle due o β is oo small. We have o use he second handle ha he model offers, he volailiy of volailiy ξ, o creae he residual srangle. Noe ha ξ is smaller (ξ = 8%) as compared o he calibraion resul in he purely sochasic model (ξ = 93%). This is due o he fac ha a par of he marke srangle is already creaed by β. The able below summarises he relaionship of he model parameers o he marke observaions. Table 1. Relaionship of model parameers o a-he-money-forward opions, riskreversals, sranges and speed of risk-reversal. peed of Risk ATMF Risk Reversal rangle Reversal σ + Α + Β + + Ξ Pricing To price exoic opions wih he sochasic Q-Φ model we can again use a Mone Carlo simulaion or a finie difference grid. The sochasic naure of he volailiy adds anoher dimension o he problem. In he implemenaion, he algorihm mus no only accoun for he spo moving up or down bu also ha he local volailiy experiences random shocks. Thus, if we are o build a finie difference grid, we have o consruc a cube wih he axes ime, spo and volailiy. Noe ha for he Q-Φ or he Quasi Q-Φ model i is sufficien o provide a wo-dimensional grid wih he axes spo and ime, hus making i easy o implemen. 5 Pricing example The able below summarises he prices of a 11. / 9. 6M range binary. = 1, ATMF volailiyσ ATMF = 13.5%, srangle r =.5%, RR = % for Pus, and change of risk reversal per 1 % move down in spo of.1% using he differen approaches presened above. As he range binary exhibis a posiive Vomma he price based on he purely sochasic model is highes, followed by he ochasic Q-Φ (The Mix), which, even hough smaller, sill has a sochasic volailiy componen and finally Quasi Q-Φ, where volailiy is assumed o be purely deerminisic. We observe ha he speed of risk reversal which is implied by he Quasi Q-Φ model is much higher han he.1% ha is hisorically seen in he marke. Table. Prices of range binaries for he various models considered. B Quasi Q-Φ Purely ochasic ochasic Q- Φ Price 39.7% 4.% 48.6% 47.5% peed of risk reversal --.4%.%.1% 14

15 6 Conclusions In his paper, we propose a new class of models (Q-Φ), ha capures boh sochasic volailiy and skewness. The models we propose are highly racable for pricing and risk managemen. The model parameers are such ha hey can be hedged using he sandard srangles and risk reversals raded in he marke. The model allows for easy implemenaion and he pricing speed is good which is a key aspec for rading and risk managemen. References Alan Brace, Dariusz Gaarek, and Marek Musiela, The marke model of ineres rae dynamics. Mahemaical Finance, 7, Black, F. and choles, M., The Pricing of Opions and Corporae Liabiliies. Journal of Poliical Economy, 81, Breeden, D. and R. Lizenberger, Prices of ae-coningen Claims Implici in Opion Prices. Journal of Business, 51, Derman Emanuel and Kani Iraj, Riding on he smile. Risk, 7, 3 9. Derman Emanuel and Kani Iraj, ochasic implied rees: Arbirage pricing wih sochasic erm and srike srucure of volailiy. Inernaional Journal of Theoreical and Applied Finance, 1, Derman Emanuel, Kani Iraj, and Chriss Neil, Implied rinomial rees of he volailiy smile. Journal of Derivaives, 4, 7. Donald P. Chiras and even Manaser, The informaion conen of opion prices and a es of marke efficiency. Journal of Financial Economics, 6, Duffie, D., ecuriy Markes, ochasic Models. Academic Press, an Diego. Dumas Bernard, Jeff Fleming, and Rober E. Whaley (1999). Implied volailiy funcions: Empirical ess. Journal of Finance, 53, Dupire, B., 199. Arbirage Pricing wih ochasic Volailiy. Proceedings of AFFI Conference in Paris, June 199. Dupire Bruno, Pricing wih a smile. Risk, 7, 18. Eugene F. Fama, 197. Efficien capial markes: A review of heory and empirical work. Journal of Finance, 5, Farshid Jamshidian, LIBOR and swap marke models and measures. Finance and ochasics, pringer, 1, Jeff Fleming, Barbara Osdiek, and Rober E. Whaley, Predicing sock marke volailiy: A new measure. Journal of Fuures Markes, 15,

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