Generalizing Geometric Brownian Motion
aa r X i v : . [ q -f i n . M F ] S e p Generalizing Geometric Brownian Motion
Peter Carr, Zhibai Zhang
Department of Finance and Risk EngineeringTandon School of EngineeringNew York University12 Metro Tech CenterBrooklyn NY 11201, USA [email protected]@gmail.com
Abstract
To convert standard Brownian motion Z into a positive process, Geometric Brownian motion (GBM) e βZ t , β > α ≥ β is theinstantaneous volatility as prices become arbitrarily high. Our generalization preserves the positivity, constantproportional drift, and tractability of GBM, while expressing the instantaneous volatility as a randomly weighted L mean of α and β . The running minimum and relative drawup of this process are also analytically tractable.Letting α = β , our positive process reduces to Geometric Brownian motion. By adding a jump to default tothe new process, we introduce a non-negative martingale with the same tractabilities. Assuming a security’sdynamics are driven by these processes in risk neutral measure, we price several derivatives including vanilla,barrier and lookback options. Introduction
Stochastic processes are used in option pricing models for multiple purposes. A very common purpose is smileinterpolation and extrapolation. Given several co-terminal market quotes, the objective here is to produce impliedvolatilities at a continuum of strike prices or delta levels. A second purpose is to value path-dependent contingentclaims such as quantoed forward contracts or barrier options. For both purposes, it is well known that arbitrageis avoided so long as all relative price processes are specified as martingales under the appropriate probabilitymeasure.In general, arbitrages can either be model-based or model-free. An example of a model-free arbitrage is aviolation of put call parity. An example of a model-based arbitrage is when two European-style futures optionshave different implied volatilities in the Black model. A martingale specification produces prices that are free ofboth types of arbitrage. For example, using driftless geometric Brownian motion to describe a futures price underthe futures measure Q leads to both put call parity holding and to equal implied volatilities across strikes andmaturity.Suppose that a market maker uses one martingale specification on an initial date and then uses a differentmartingale specification on a second date. For example suppose that a market maker uses a geometric Brownianmartingale with 10% volatility on the first date and then uses a geometric Brownian martingale with 20% volatilityon the second date. The prices produced on both dates are devoid of model-free arbitrages. For example put callparity will hold on both dates. The prices produced on both dates do produce an arbitrage based on the Blackmodel being correct. For example, if the actual volatility in the Black model is constant at 10%, then the pricesproduced on the second day allow model-based arbitrage. If the actual volatility is instead constant at 20%, thenthe prices produced on the first day allow model-based arbitrage. If the actual volatility is instead constant atsome other value e.g. 15%, then the prices produced on both days allow arbitrage based on the Black modelbeing correct. However, if the Black model is not describing the risk-neutral dynamics of the underlying, then themarket maker’s use of time-inconsistent martingale specifications need not produce any model-based arbitrages.Nonetheless, the use of time-inconsistent martingale specifications does produce a set of values that are devoid ofmodel-free arbitrages.When the only goal is to produce values that are devoid of model-free arbitrages, the only challenge to be metis to be consistent with all of the liquid and transparent quotes. For this purpose, time-inconsistent martingalespecifications offer greater flexibility than a time-consistent specification. A market maker using the Black modelwith the same volatility on both dates is unlikely to be able to match the ATM quote on both dates. In contrast,a market maker using the Black model with the ability to change the volatility on the second date is guaranteedto be able to match the ATM quotes on both dates. In contrast this time-inconsistent Black model does notguarantee the ability to match more than one option price on any given date. When two or more simultaneousquotes differ in maturity, and are devoid of model-free arbitrage, one can match them by moving from the constantvolatility model to the deterministic volatility Black model. However, when two co-terminal quotes differ in strikeand are devoid of model-free arbitrage, one cannot necessarily match them with the deterministic volatility Blackmodel. A different type of martingale specification is required to guarantee a match.In choosing an alternative martingale specification, it is wise to understand the reasons behind the success ofthe Geometric Brownian Martingale as the benchmark process. Once these reasons are understood, it becomesclearer as to which properties of GBM should be kept and which properties should be jettisoned. For example,at first glance, driftless arithmetic Brownian motion (ABM) appears to be an attractive alternative to driftlessGBM due to its simplicity and tractability. However, it is widely agreed that the failure of ABM to preserve thepositivity property of GBM makes it unviable as an alternative. It is widely argued that this positivity propertyof GBM makes it a good first approximation in describing market prices of assets whose owners enjoy limitedliability. However, GBM has state space (0 , ∞ ) while prices of limited liability assets occupy [0 , ∞ ). To capturethe possibility that the price of a limited liability asset can vanish, one can add a jump to default to a GBM, asdone in [5]. 1he GBM remains appropriate as a toy model for a stock index, where it is widely agreed that zero isinaccessible. The inaccessibility of the origin for GBM also makes it a good toy model for an exchange rate,since if X is an exchange rate, X needs to be well defined. For a driftless GBM, its state space and dynamics arepreserved upon inversion of the coordinate and a change of probability measure. In foreign exchange (FX) markets,inverting an FX rate is a natural operation and the change in probability measure corresponds to a change ofnumeraire. It is highly likely that these invariance properties of GBM explain why this stochastic process playssuch a large role in the FX options market. If one wants to address deficiencies of GBM while retaining applicabilityto FX options pricing, it stands to reason that preserving at least some notion of invariance under inversion iscrucial. The purpose of this paper is to propose a process that generalizes GBM while respecting invariance underinversion. Not surprisingly, hyperbolic functions play a large role in our analysis.It is helpful to begin by reviewing some well-known properties of GBM. Consider an arbitrage-free market andlet Q be an equivalent martingale measure. Let Z denote standard Brownian motion on the real line under Q .Consider the process g t = e βZ t , t ≥
0, where β >
0. Clearly, the process g starts at one and stays positive forever.From Itˆo’s formula: dg t g t = β dt + βdZ t , t ≥ . (1)We say the process g has constant proportional drift at rate β and constant proportional variance at rate β .The parameter β is called the volatility. The process g is called Geometric Brownian motion.To obtain a non-negative martingale from g , there are at least three approaches. First, one can change theprobability measure from Q to ˜ Q by setting d ˜ Q d Q = e − β Z T − β T . Second, one can alternatively change the coordinateby setting F t = g t e − β t/ . Both of these approaches to creating a martingale preserve the strict positivity of g .If only non-negativity of the martingale is required, one can alternatively add a jump to default to the g processwith arrival rate β / g t = e βZ t , t ≥ α ≥
0. For our new process, α describes the instantaneous volatility whenever a new low is reached.while β is the instantaneous volatility whenever the process becomes arbitrarily high. Our generalization preservesthe positivity, constant proportional drift, and tractability of GBM, while expressing the instantaneous variancerate at any time as a convex combination of α and β . The model actually allows a third parameter γ which isthe initial instantaneous volatility, and hence is required to lie between α and β .For many options markets, three parameter models are widely used to interpolate and extrapolate impliedvolatilities across strikes. Intuitively, market participants agree that options markets display nonzero skewnessand kurtosis, but there is little discussion about moments higher than the fourth power. Put another way, marketparticipants agree that it is necessary to match some measure of level, slope, and convexity of implied volatilityat the money, but there is little discussion about the third or higher derivative of implied volatility.Unfortunately, our particular three parameter model is not as flexible as some other three parameter modelse.g. SABR with fixed β or ρ . As a result, our three parameter model is only suitable for options markets wherethe implied volatility slice appears to be monotone across strike e.g. SPX or VIX. For non-monotone slices suchas when implied volatilities smile, one must alter the model by adding e.g. stochastic volatility. So long as theimplied volatility slice appears to be monotone across strike price, our three parameters, α ≥ β > γ ≥ α controls the asymptotic implied volatility at low strikes,while the parameter β controls the asymptotic implied volatility at high strikes. The parameter γ is used to meetan at-the-money implied volatility.An overview of this paper is as follows. The next section develops a new special function called the twoparameter exponential function. The following section first uses this special function to construct a positivecontibuous sub-martingale that has a constant drift. Then we introduce a non-negative martingale by addinga jump to default process to the sub-martingale. This martingale has three parameters α ≥ β >
0, and γ between α and β . This is followed by derivations of the transition PDF’s for the new martingale. The penultimate2ection presents closed form valuation formulas for contingent claims written on these martingales. In particular,we examine vanilla options, lookback options and barrier options. The final section provides both a summary ofthe paper and some suggestions for future research. In this section, we construct a new special function which we call a two parameter exponential function. In thenext section, we will use this special function to construct our three parameter martingale. For β >
0, let y = e βx be the standard one parameter exponential function. While the function is defined for β ∈ C and x ∈ C , weconsider it only for β ∈ R + and x ∈ R + . The defining characteristics of e βx are that the ratio of the function’sslope to its height is constant at β > x ≥ x = 0 for all β >
0. Accordingly, our two parameter exponential function will have unit height at x = 0 for all values of its twoparameters α ≥ β >
0. We will show that the ratio of the function’s slope to its height is α ≥ x = 0 andapproaches β > x ↑ ∞ . Since infinitely many functions meet just these criteria, we further require that theratio of the function’s curvature to its height be constant at β > x ≥
0. This property also belongs to theone parameter exponential function and serves to uniquely determine our two parameter exponential function.For x ≥ β >
0, and α ≥
0, we define the two parameter exponential function by: e βxβ − α ≡ β + α β e βx + β − α β e − βx . (2)Thus, the subscripted exponential is a linear combination of the ordinary exponential e βx and its reciprocal. The β − α subscript in e βxβ − α describes the numerator of the fraction multiplying the reciprocal e − βx . The numeratorof the fraction multiplying e βx is always the sum of the asymmetry parameter α and the scaling factor β in theordinary exponential e βx . The common denominator of both fractions is twice this scaling factor β . These rulesuniquely expand the LHS of (2) into the RHS.On our function’s domain x ≥
0, the ordinary exponential e βx in the linear combination is larger than itsreciprocal i.e. e βx ≥ e − βx . If α = 0, the two fractions simplify to one half and the function is increasing andconvex. Increasing α increases the fraction multiplying the larger exponential e βx and decreases the fractionmultiplying the smaller exponential e − βx , while keeping the value of the function at x = 0 fixed at one. As aresult, increasing α causes our special function to slope up faster at every x ≥
0. If α = β , then the two parameterexponential e βx reduces to the one parameter exponential e βx . Thus the subscript β − α on e βxβ − α is also a measureof the deviation of our two parameter exponential function from the one parameter exponential function. Like theone parameter exponential function e βx , the two parameter exponential function e βxβ − α defined by (2) is positive,increasing, and convex in x for all x ≥ β > x of our two parameter exponential function is: ddx e βxβ − α = βe βxα − β , α ≥ , β > , x ≥ , (3)where: e βxα − β ≡ β + α β e βx + α − β β e − βx , α ≥ , β > , x ≥ . (4)At α = 0, e βxα − β is the right arm of the hyperbolic sine and hence positive. Increasing α increases the weight onboth exponentials and hence e βxα − β > α ≥ , β > , x ≥
0. Since β > Our special function f ( x ) solves the ordinary differential equation f ′′ ( x ) = β f ( x ) on x ≥ f (0) = 1 and the Neumann boundary condition f ′ (0) = α . Our function can also be expressed as cosh( βx ) + αβ sinh( βx ) , x ≥ , α ≥ , β > ddx e βxβ − α is positive. Thus, the x -derivative of our two parameter exponential function behaves the sameway as the x -derivative of the ordinary exponential function w.r.t to its scaling factor β . Differentiating our twoparameter exponential function w.r.t. x also switches the sign on the subscript. To convert e βxα − β on the RHS of(3) back into an expression involving its cohort e βxβ − α , one can again differentiate w.r.t. x . In particular: d dx e βxβ − α = β e βxβ − α , α ≥ , β > , x ≥ . (5)Thus, the ratio of the function’s curvature to its height is constant at β > x ≥
0, as previously indicated.There is an alternative way to convert e βxα − β back into an expression involving its cohort e βxβ − α . The appendixshows that: e βxα − β = s(cid:16) e βxβ − α (cid:17) + α − β β . (6)We now use this alternative conversion mechanism to show that our two parameter exponential function sets theratio of its slope to its height at α at x = 0. We will also show in contrast that the ratio of its slope to its heightapproaches β as x ↑ ∞ . These behaviors define the role of each parameter in our two parameter exponentialfunction.Consider the ratio of the slope of our two parameter exponential function to its height: ddx e βxβ − α e βxβ − α = β e βxα − β e βxβ − α , (7)from (3). Using (6) on the RHS of (7), this ratio can also be represented as: ddx e βxβ − α e βxβ − α = β r(cid:16) e βxβ − α (cid:17) + α − β β e βxβ − α = β vuut α − β β (cid:16) e βxβ − α (cid:17) . (8)Bringing β under the square root: ddx e βxβ − α e βxβ − α = vuuuut α (cid:16) e βxβ − α (cid:17) + β − (cid:16) e βxβ − α (cid:17) . (9)Since 1 / (cid:16) e βxβ − α (cid:17) ∈ (0 , α and β . At x = 0, e βxβ − α = 1, so (cid:16) e βxβ − α (cid:17) also= 1 and the ratio ddx e βxβ − α e βxβ − α = α . As x ↑ ∞ , e βxβ − α ↑ ∞ , so (cid:16) e βxβ − α (cid:17) ↓ ddx e βxβ − α e βxβ − α converges to β .Like the one parameter exponential function, our two parameter exponential function has an explicit inverse.To derive it, let: y = e βxβ − α = β + α β e βx + β − α β e − βx , x ≥ , α ≥ , β > . (10)We need to solve for x as a function of y . Multiplying (10) by βe βx leads to a quadratic function of e βx : β + α e βx − βye βx + β − α , x ≥ , α ≥ , β > . (11)4y the quadratic root formula: e βx = βy + p β y − ( β − α ) β + α , x ≥ , α ≥ , β > , (12)where we have chosen + in ± since e βx >
0. Solving for x : x = 1 β ln βy + p α + β ( y − β + α , x ≥ , α ≥ , β > . (13)Hence, for y ≥
1, the function on the RHS of (13) is the explicit inverse of our two parameter exponential function.Notice that from (12): β + α β e βx = y s y − β − α β , (14)where we observe from (10) that β − α β is just the product of the two terms which sum to y . Equation (14) is anexplicit formula that maps y to the first term in the sum (10) defining it. When x = 0 and α = 0, this first termhas the same size of as the second term, but otherwise, the first term is larger. To obtain an explicit formulathat maps y to the smaller term in the sum defining it, notice that multiplying (10) by βe − βx leads to a quadraticfunction of e − βx : β − α e − βx − βye βx + β + α . (15)By the quadratic root formula: e − βx = βy − p β y − ( β − α ) β − α , (16)where now we have chosen − in ± since e − βx <
1. Hence: β − α β e − βx = y − s y − β − α β . (17)This equation is an explicit formula that maps y to the last smaller term in the sum (10) defining it.For the one parameter exponential function y = e βx , x ≥ , β >
0, adding one to the input variable x causes theoutput variable y to grow by the factor e β >
1. We say the exponential function turns addition into multiplication.For our two parameter exponential function defined by (10), adding one to the input variable x causes the outputvariable y to grow as follows. First, split y into its larger term involving e βx given explicitly by (14) and its smallerterm involving e − βx , given explicitly by (17). Next, grow the larger term by a factor e β > e − β ∈ (0 , y . We saythe two parameter exponential function turns addition into a blend of multiplication and division. In this section, we use the two parameter exponential function constructed in the last section to define a new threeparameter non-negative continuous martingale denoted by F t . Recall that to create a driftless GBM F b , onefirst creates an auxiliary positive continuous process g t = e βZ t with constant positive drift of β / F bt F b = g t e − β t/ . We will mimic this construction in the next subsectionby first constructing an auxiliary positive continuous process G with positive constant drift of β /
2. The followingsubsection then corrects for this constant drift by adding a jump to default process.5 .1 Constructing a Positive Continuous Process with Constant Drift
Let 0 be the valuation time and let Z be a standard Brownian motion Z under Q whose value at t = 0 is Z = 0as usual. We allow Z to exist prior to time 0. Let t ≤ Z exists for all t ≥ t . For t ≥ t , let Z t ≡ inf s ∈ [ t ,t ] Z s denote the running minimum of the standard Brownian motion Z under Q . Notice that Z ’s pathmonitoring begins at time t ≤
0, so Z ≤
0. For t ≥ t , let ˇ Z t ≡ Z t − Z t denote Z ’s running drawup process. Let:ˇ G t = e β ˇ Z t β − α , t ≥ t , β > , (18)be a new stochastic process with state space [1 , ∞ ).Recall that setting α to zero reduces the two parameter exponential e βxβ − α , x ≥ , β > e βx , x ≥ , β >
0. The GBM e βZ t and the processes cosh( βZ t ), cosh( β | Z t | ), and cosh( β ˇ Z t ) allgrow in expectation at the rate β /
2. The hyperbolic cosine is a simple average of the increasing exponential e βx , x ≥ , β >
0, and its reciprocal. When the asymmetry parameter α is made positive, this simple averageis replaced with an asymmetric average putting more weight on the increasing exponential. The effect on themean of this skewing is the same as the effect on the mean of the GBM e βZ t if Z behaved asymmetrically justwhen visiting its minimum ¯ Z . In particular, if Z is interpreted as a the limit of a scaled random walk, thenputting greater probability on rising above the minimum raises the mean growth rate of e βZ t above β /
2. Let ˆ Z denote this skewed Brownian motion. The effect on the mean of e β ˆ Zt of this rarely imposed asymmetry can beremoved by multiplying by e α ¯ Z t . We will similarly remove the effect on the mean of ˇ G t of replacing cosh( β ˇ Z t )with e β ˇ Z t β − α , β > , α ≥ G t by e α ¯ Z t .We introduce a new parameter γ which will be used to determine the value of ˇ G t at t = 0. We require that γ be between α and β . For technical reasons, we allow γ = α , but we do not allow γ = β . This allows us to set:ˇ G = s α − β γ − β . (19)The radicand is ≥ G . We next use (13) to set ˇ Z :ˇ Z = 1 β ln β ˇ G + q α + β [ ˇ G − β + α , x ≥ , α ≥ , β > . (20)Since ˇ G ≥
1, ˇ Z ≥
0. At each t ≥
0, ˇ G t ≥ Z t ≥
0. Equation (21)implies that (18) can be explicitly inverted:ˇ Z t = 1 β ln β ˇ G t + q α + β [ ˇ G t − α + β , t ≥ t , α ≥ , β > . (21)We next set Z = − ˇ Z so that Z ≡ Z + ˇ Z = 0. With Z determined at some non-positive value, let: G t = e αZ t t ≥ t , α ≥ , (22)be a super-martingale with state space (0 , G t ∈ (0 ,
1] defined in (22) is increasing in its driver Z t ≤
0, For α >
0, (26) can be explicitly inverted: Z t = 1 α ln G t , t ≥ t . (23)6or α ≥ , β > γ between them, let: G t = G t ˇ G t , t ≥ , (24)be our auxiliary continuous process with state space (0 , ∞ ). We claim that G t = inf s ∈ [ t ,t ] G s . In words, we claimthat the super-martingale G t ∈ (0 ,
1] defined in (22) is just the running minimum of the G process defined in (24).To see why, note that substituting (18) and (22) in (24) implies that for α ≥ , β > γ between them: G t = e αZ t e β ˇ Z t β − α , t ≥ t . (25)Since Z only declines when ˇ Z = 0: inf s ∈ [ t ,t ] G s = e αZ t , t ≥ t , α ≥ , β > , (26)which matches the defining equation (22) for G t . Hence G t is the running minimum of the G process defined in(24). Since G t has state space (0 , G is positive forever. From (24):ˇ G t = G t G t , t ≥ , (27)so ˇ G is the relative drawup process of G .Applying Itˆo’s formula to (18), (3) implies that: d ˇ G t = βe β ˇ Z t α − β d ˇ Z t + β e β ˇ Z t β − α d h ˇ Z i t , t ≥ t . (28)Thus the increments of ˇ G t depend on the increments of ˇ Z t and the squared increments of ˇ Z t . Since Z is a processof bounded variation, it has zero quadratic variation and hence: h ˇ Z i t = h Z − Z i t = h Z i t = t, t ≥ t . (29)Substituting (6) and (29) in (28) implies that the coefficients just depend on e β ˇ Z t β − α : d ˇ G t = β e β ˇ Z t β − α dt + β s(cid:16) e β ˇ Z t β − α (cid:17) + α − β β d ˇ Z t , t ≥ t . (30)Substituting (18) in (30) implies that ˇ G solves the following stochastic differential equation (SDE): d ˇ G t = β G t dt + r α + β h(cid:0) ˇ G t (cid:1) − i d ˇ Z t , t ≥ t . (31)This SDE is univariate since the coefficients for ˇ G t just depend on ˇ G t . Dividing by ˇ G t implies: d ˇ G t ˇ G t = β dt + s α G t + β (cid:20) − G t (cid:21) d ˇ Z t , t ≥ t . (32)Hence, ˇ G solves the above simple SDE when the two drivers are t and ˇ Z . To determine the coefficients of Z t and Z , note that substituting d ˇ Z t = dZ t − dZ t in (32) implies: d ˇ G t ˇ G t = β dt + s α (cid:18) G t (cid:19) + β (cid:20) − G t (cid:21) ( dZ t − dZ t ) , t ≥ t . (33)7ince Z only decreases when ˇ G = 1, the net coefficient of dZ in (33) is zero. As a result, ˇ G also solves the followingSDE: d ˇ G t ˇ G t = − αdZ t + β dt + s α G t + β (cid:20) − G t (cid:21) dZ t , t ≥ t . (34)The coefficient of dZ t in (34) is the instantaneous lognormal volatility of ˇ G , which is a randomly weighted L mean of α and β . This form is clearly just a consequence of (9). Since (cid:16) G t (cid:17) ∈ (0 , G t is just a convex combination of α and β . When Z is at its minimum Z , ˇ Z = 0,and hence ˇ G = 1. At such times, (34) implies that the instantaneous volatility of ˇ G is α . In contrast, as thedifference between Z and its minimum Z approaches infinity, ˇ G also approaches infinity, and (34) implies that theinstantaneous volatility of ˇ G approaches β . These results clearly follow from the behavior of our two parameterexponential function e βxβ − α at x = 0 and at x = ∞ .The dynamics in (34) clearly depend on our first two parameters α and β , which are the respective instantaneousvolatilities of ˇ G at ˇ G ’s extremes of one and infinity. To interpret our third parameter γ , note that squaring bothsides of (19) implies that: ˇ G = α − β γ − β . (35)Cross multiplying and re-arranging: γ ˇ G = α − β + β ( ˇ G ) . (36)Dividing by ˇ G and taking the square root implies: γ = s α G + β (cid:20) − G (cid:21) . (37)Comparing (37) to the volatility in (34) evaluated at t = 0 implies that our third parameter γ is just the initialvolatility of ˇ G .We next determine the dynamics of the G process, which (24) defined as the product: G t = G t ˇ G t t ≥ t , (38)for α ≥ , β > γ between them. Itˆo’s formula implies that: dG t G t = dG t G t + d ˇ G t ˇ G t = αdZ t + d ˇ G t ˇ G t , t ≥ t , (39)since G t = e αZ t . Substituting in (32) implies that G solves the following SDE: dG t G t = β dt + vuut α (cid:18) G t G t (cid:19) + β " − (cid:18) G t G t (cid:19) dZ t , t ≥ t , (40)since G t = G t G t .Like the ˇ G process, the G process has constant proportional drift at rate β . Unlike the SDE (34) for ˇ G ,the SDE (40) for G has coefficients that depend on the auxiliary process G . Since (cid:16) G t G t (cid:17) ∈ (0 , G is also a convex combination of α and β . When G t = G t , the G process behaves locally like aGBM with constant proportional drift rate β and constant proportional variance rate α . As G rises above G t ,the lognormal variance rate moves towards β and asymptotes to this value in the limit as G ↑ ∞ .8ubstituting G = G G in (37) implies that: γ = vuut α (cid:18) G G (cid:19) + β " − (cid:18) G G (cid:19) . (41)Evaluating the coefficient of dZ t in (40) at t = 0, (41) implies that the instantaneous lognormal volatility of G is γ .Hence, the three parameters α, γ , and β can be respectively interpreted as the instantaneous lognormal volatilityof G at each new low, at the initial time, and at infinitely high values of G .The bivariate transition PDF of the pair ( Z, ˇ Z ) is known in closed form and is given in [2]. Since G and ˇ G areeach just univariate, increasing, explicitly invertible transformations of Z and ˇ Z respectively, it follows that thebivariate transition PDF of the pair ( G, ˇ G ) can easily be obtained in closed form.Recall from (25) that: G t = e αZ t e β ˇ Z t β − α , t ≥ t . (42)As β ↓
0, the G process becomes driftless and two parameter exponential function e βxβ − α in (42) converges to thelinear function 1 + αx . As a result, the process G converges to the martingale F in [2] in this limit when F = 1.Setting α = β in (42), the two parameter exponential reduces to the one parameter exponential and hence: G t = e βZ t e β ˇ Z t = e β ( Z t + ˇ Z t ) = e βZ t , t ≥ t . (43)Thus, the G process generalizes the exponential of standard Brownian motion, by adding parameters α and γ .Being a sub-martingale, the G process can be used directly to model spot price (e.g. spot FX rates) and pricederivatives written on G in risk neutral measure. For this purpose, we introduce a new sub-martingale process F t = F G t , t ≥ t , (44)where F > G , F is positive and has a positive drift. Note that thepositivity of the drift of G is not a binding restriction due to the international put-call equivalence [6]. For instance,if a positive process S t has a negative drift, one can use it to model the inverse of a process that has a positive driftvia F t = S t . For derivatives on future price, the underlying security is required to be driven by a martingale in therisk neutral measure for derivative pricing. In the next subsection we introduce a new martingale process from G by adding a jump to default process which has a negative drift. However, one should interpret the sub-martingaleEqn (44) and the new martingale as dynamics of two different securities, instead of spot and future prices of onesecurity. For α ≥ , β >
0, and for γ between them, the G process constructed in the last subsection starts at one and hasconstant positive drift β . In this section, we change the starting point to F > F which starts at F . Let N t be a standard Poisson process with arrival rate β under Q . For F >
0, let: F t = F G t N t =0 , t ≥ t . (45)be a non-negative process started at F >
0. Then F is a Q martingale which drifts up at the constant rate β inorder to compensate for a possible jump to zero. Once F hits zero, it is absorbed there. Let: F t = inf s ∈ [ t ,t ] F s , t ≥ t (46)9e the running minimum of F . Let τ be the exponentially distributed random time at which F jumps to zero.For t ∈ [ t , τ ), (45) implies: F t = F G t . (47)Dividing (47) by (46) implies that for t ∈ [ t , τ ): F t F t = G t G t . (48)As a result, the SDE for F is: dF t = F t − vuut α (cid:18) F t − F t − (cid:19) + β " − (cid:18) F t − F t − (cid:19) dZ t − (cid:18) dN t − β dt (cid:19) , t ≥ t . (49)Substituting (24) in (45) implies that F t can be related to the contemporaneous values of the pair ( Z, ˇ Z ) and N t : F t = F e αZ t e β ˇ Z t β − α N t =0 , t ≥ t . (50)The price relative F t F is a non-negative martingale started at one. From (50), this price relative decomposes intothe product of a positive strict supermartingale started at one, e αZ t N t =0 and a positive strict submartingalestarted at one, namely ˇ G t = e β ˇ Z t β − α .If α = β , then the two parameter exponential function e βxβ − α in (50) reduces to the one parameter exponentialfunction e βx , and hence (50) simplifies to: F t = F e βZ t e β ˇ Z t N t =0 = F e β ( Z t + ˇ Z t ) N t =0 = e βZ t N t =0 , t ≥ t , (51)which is GBM with jump to default. When β →
0, then (50) asymptotes to: F t → F e αZ t (1 + α ˇ Z t ) , t ≥ t , (52)which is a two parameter positive continuous martingale. Setting γ = α further reduces F to the one parameterpositive continuous martingale in [2].From [2], the bivariate transition PDF of the Brownian Minimum and Brownian Drawup: Q t { Z T ∈ dj, ˇ Z T ∈ d ˇ k ; Z t = Z, ˇ Z t = ˇ Z } = b ( j, ˇ k ; w, T − t ) djd ˇ kb ( j, ˇ k ; w, T − t ) ≡ s π ( T − t ) (ˇ k − j + w ) e − (ˇ k − j + w )22( T − t ) , j < w, ˇ k ≥ , (53)where w = Z + ˇ Z and w = Z . Note that in a special case when Z T = Z t , the bivariate transition PDF becomes aunivariate one: ˜ Q t { Z T = Z t , ˇ Z T ∈ d ˇ k ; Z t = Z, ˇ Z t = ˇ Z } = ˜ b (ˇ k ; w, T − t ) d ˇ k ˜ b (ˇ k ; w, T − t ) ≡ s π ( T − t ) (cid:18) e − ˇ k T − t ) − e − (ˇ k + w − w )22( T − t ) (cid:19) , ˇ k ≥ . (54)Next we construct the bivariate transition PDF for the double-exponential process (50). Let F sT be theminimum of F at T conditional on surviving to T . Similarly, let ˇ F sT be the drawup of F at T , conditional onsurviving to T . The bivariate transition PDF of the Brownian Minimum and Brownian Drawup can be used to10erive the bivariate PDF of the pair ( F sT , ˇ F sT ), conditional both on surviving to T and on ( F st , ˇ F t ) = ( F , ˇ F ). For J ∈ (0 , F ], and ˇ K ≥
1, we seek: Q { F sT ∈ dJ, ˇ F sT ∈ d ˇ K | N T = 0 , F st = F , ˇ F t = ˇ F } . In other words, we wish to know the bivariate conditional PDF when we change variables from ( j, ˇ k ) to:( J, ˇ K ) = ( F e αj , e β ˇ kβ − α ) . Let j ( J ) be the inverse of J = F e αj : j ( J ) = 1 α ln (cid:18) JF (cid:19) , J ∈ (0 , F ] . (55)Similarly, let ˇ k ( ˇ K ) be the inverse of ˇ K = e β ˇ kβ − α :ˇ k ( ˇ K ) = 1 β ln β ˇ K + q α + β ( ˇ K − α + β , ˇ K ≥ . (56)The determinant of the Jacobian for this change of variables is: (cid:18) αJ q α + β ( ˇ K − (cid:19) − . (57)Using the standard change of variables formula, it follows that for J ∈ (0 , F ] , ˇ K ≥
1, the conditional bivariatePDF of the pair ( F sT , ˇ F sT ) is given by: Q { F sT ∈ dJ, ˇ F sT ∈ d ˇ K | N T = 0 , F st = F , ˇ F t = ˇ F } = f ( J, ˇ K ; w, T − t ) dJ d ˇ Kf ( J, ˇ K ; w, T − t ) ≡ s π ( T − t ) (cid:0) ˇ k ( ˇ K ) − j ( J ) + w (cid:1) e − ( ˇ k ( ˇ K ) − j ( J )+ w ) T − t ) αJ q α + β ( ˇ K − , (58)and w = j ( F ) + ˇ k ( ˇ F ) . (59)Note that w = Z t , and the reason we use w is that it is written on market observables ˇ F and F while Z t is not.Let F sT = F sT ˇ F sT be the forward price at T conditional on survival to T . The bivariate PDF of the pair ( F sT , ˇ F sT )can be used calculate the conditional transition PDF of F sT : Q { F sT ∈ dF | N T = 0 , F st = F , ˇ F t = ˇ F } = g ( F ; w, T − t ) dF , (60)where g ( F ; w, T − t ) = Z F f (cid:18) J, FJ ; w, T − t (cid:19) dJ (61)= Z F s π ( T − t ) (cid:0) k (cid:0) FJ (cid:1) − j ( J ) + w (cid:1) e − ( k ( FJ ) − j ( J )+ w ) T − t ) αJ r α + β h(cid:0) FJ (cid:1) − i dJ, w is given in (59). When F is only conditioned on surviving to t rather than to T , the transition PDF’s ofboth ( F T , ˇ F T ) and F T are just given by the product of their corresponding transition PDF conditioned on survivalto T and the probability of further surviving to T , which is e − β ( T − t ) : Q { F T ∈ dJ, ˇ F T ∈ d ˇ K | N t = 0 , F st = F , ˇ F t = ˇ F } = f ( J, ˇ K ; w, T − t ) e − β ( T − t ) dJ d ˇ K , Q { F T ∈ dF | N t = 0 , F st = F , ˇ F t = ˇ F } = g ( F ; w, T − t ) e − β ( T − t ) dF . (62)The PDF of F T is an integral over a bounded domain and it cannot be simplified further. We will find that whencommon payoffs are integrated against this PDF, additional quadratures are not introduced. It is for this reasonthat we consider the process F to be tractable.There are two similar constructions of a non-negative martingale which also use jump to default. The cumula-tive hazard process of N is Λ t = e β t which is deterministic. Suppose instead that the cumulative hazard processis ˆΛ t = e − αZ t , which is random. Let ˆ N denote the corresponding counting process and let ˆ F denote the desirednon-negative martingale: ˆ F t = F e − β t e β ˇ Z t β − α N t =0 , t ≥ t (63)is a non-negative martingale started at F >
0. Since Z = 0, this process start off with no chance of jumpingto zero but soon endures the possibility of such a default. More generally, one can start the process Z at somenon-positive number m ≤ Z to say m since Z is still zero. Since ˇ Z t = Z t − m t startsat − m >
0, one must then also adjust its origin: F t = F e − β t e β ( ˇ Z t + m ) β − α ˆ N t =0 , t ≥ t (64)There is yet another construction of a non-negative martingale possibly jumping to zero. Now suppose thatthe cumulative hazard process of N is ˜Λ t = e − αZ t + β t , where we return to ˇ Z t = Z t − Z t with Z = 0. Let ˜ N denote the corresponding counting process and let ˜ F denote the desired non-negative martingale:˜ F t = F e β ˇ Z t β − α ˜ N t =0 , t ≥ t (65)is a non-negative martingale started at F >
0. This process is convenient if an event happens at the first passagetime τ of F to a constant upper barrier H = e βhβ − α where h >
0. In this case, τ is also the first passage time of ˇ Z to h . Since FF is a martingale started at one, the bivariate Laplace transform of Z τ and τ becomes known:= Ee αZ τ − β τ e βhβ − α . (66)One can develop yet other tractable constructions of non-negative martingales by altering the cumulativehazard process yet again and compensating by coordinate change as was done above. In risk neutral measure, non-arbitrage insures that the expected payoff of a security is equal to its current price.In this section we show how our model can be applied in derivative pricing assuming the underlying asset followsthe dynamics of either the sub-martingale Eqn (44) or the martingale Eqn (50) in risk neutral measure. Theformer is used for derivatives written on spot price of a security while the latter is for future price of a security.Since the two processes only differ by the inclusion of a jump to default process, the pricing formulas for them arequite close. For this reason, we only present the derivation of pricing for the martingale dynamics. The results forthe sub-martingale dynamics are labelled by subscripts for clarification. Note since our model tracks the asset’srunning minimum and drawup rate, it is especially useful in pricing barrier type of path-dependent options.12 .1 One-Touch with a lower barrier
We first price a One-Touch with a lower barrier. A One-Touch option pays one dollar if the underlying asset’sprice touches the lower barrier price before maturity, and otherwise expires worthless. Assuming that the presenttime is t and the underlying asset has not defaulted ( N t = 0). The price of a One-Touch with a lower barrier L and maturity T is OT t ( L, T ) = F t ≤ L + F t >L · (cid:0) N T =0 E t (cid:2) F T ≤ L (cid:3) + N T =0 (cid:1) = F t ≤ L + F t >L · (cid:18) e − β T − t )2 E t h Z T ≤ ln L − ln F α i + 1 − e − β T − t )2 (cid:19) , (67)to get the second line, F T = F e αZ T has been used. After substituting the transition PDF on Z T one obtains OT t ( L, T ) = F t ≤ L + F t >L · e − β T − t )2 " N ln L − ln F α − w √ T − t ! − + 1 ! , (68)where w is given in (59) and N is the standard normal distribution function. Taking α = 1 the price of theOne-Touch reduces to that in [2]. This is because essentially the payoff of a One-Touch option is only determinedby the minimum of the underlying, which is driven by the running minimum of a Brownian motion in both cases.A One-Touch written on spot price can be priced similarly with Eqn (44), which is equivalent to dropping theprobability induced by the jump to default process in Eqn (50). The price is then given by OT Spot t ( L, T ) = F t ≤ L + F t >L · E t (cid:2) F T ≤ L (cid:3) = F t ≤ L + F t >L · N ln L − ln F α − w √ T − t ! . (69) A lookback call option matures at T with a floating strike price pays off the difference between the terminal valueof the asset and its minimum, namely the terminal drawup. If default happens ( N T = 0), the option expiresworthless ( F T = F T ). So under the martingale (50) the value of this option at maturity is then LC float,t = N T =0 E t [ F T − F T ] = N T =0 E t (cid:2) F T (cid:0) ˇ F T − (cid:1)(cid:3) = N T =0 E t h F e αZ T (cid:16) e β ˇ Z T β − α − (cid:17)i . (70)The expectation value in Eqn (70) can be evaluated using the bivariate transition PDF of ( Z T , ˇ Z T ) the in Eqn(53) if the security runs into a new minimum after t , or otherwise the univariate transition PDF of ˇ Z T in Eqn(54)if Z T = Z t : E t h F e αZ T (cid:16) e β ˇ Z T β − α − (cid:17)i = F Z Z t −∞ dj Z ∞ d ˇ k s π ( T − t ) (ˇ k − j + w ) e − (ˇ k − j + w )22( T − t ) e αj (cid:16) e β ˇ kβ − α − (cid:17) + F Z ∞ d ˇ k s π ( T − t ) (cid:18) e − ˇ k T − t ) − e − (ˇ k + ˇ w )22( T − t ) (cid:19) e αw (cid:16) e β ˇ kβ − α − (cid:17) , (71)13here ˇ w = w − w . By working out the integral we obtain the price of this option evaluated at tLC float,t = F e αw (cid:20) αβ e β ˇ w N (cid:18) − ˇ w − β ( T − t ) √ T − t (cid:19) − αβ e − β ˇ w N (cid:18) − ˇ w + β ( T − t ) √ T − t (cid:19) + β + αβ N (cid:16) β √ T − t (cid:17) + β − αβ N (cid:16) − β √ T − t (cid:17) + e − β T − t )2 (cid:18) N (cid:18) − ˇ w √ T − t (cid:19) − (cid:19) − e α ˇ w + ( α − β T − t )2 N (cid:18) − ˇ w − α ( T − t ) √ T − t (cid:19) (cid:21) . (72)A lookback call option on spot price can be priced the same way: LC Spot float,t = E t [ F T − F T ] = E t h F e αZ T (cid:16) e β ˇ Z T β − α − (cid:17)i = e β T − t )2 LC float,t . (73)If we instead consider a lookback option with a fixed strike price, then the payoff is determined by the mini-mum/maximum for a put/call lookback option at maturity. Since Eqn (50) tracks minimum and drawup, it canalso be used to evaluate a lookback put option with fixed price. The price is given by LP fixed,t ( K, T ) = N T =0 E t (cid:2) ( K − F T ) + (cid:3) + N T =0 · K ,LP
Spot fixed,t ( K, T ) = E t (cid:2) ( K − F T ) + (cid:3) , (74)where K is the strike price. This can be evaluated by integrating the price of a one-touch barrier with respect tothe barrier, so we will not carry out the derivation for simplicity.We can also engineer another derivative analogous to a lookback call option with a floating strike price, whichpays off the ratio between the terminal price and the minimum price before maturity. Since the underlying assetcan default ( F T = F T = 0), we assume the payoff is zero in that case. The price of this option is given by LC ∗ float,t = N T =0 E t (cid:20) F T − F T F T (cid:21) = N T =0 (cid:18) E t (cid:20) F T F T (cid:21) − (cid:19) = N T =0 (cid:16) E t h e β ˇ Z T β − α i − (cid:17) , (75)and the expectation can be evaluated with the bivariate PDF: E t h e β ˇ Z T β − α i = Z Z t −∞ dj Z ∞ d ˇ k s π ( T − t ) (ˇ k − j + w ) e − (ˇ k − j + w )22( T − t ) e β ˇ kβ − α + Z ∞ d ˇ k s π ( T − t ) (cid:18) e − ˇ k T − t ) − e − (ˇ k + ˇ w )22( T − t ) (cid:19) (cid:16) e β ˇ kβ − α − (cid:17) (76)which can be evaluated similar to Eqn (72), LC ∗ float,t = β + αβ N (cid:16) β √ T − t (cid:17) + β − αβ N (cid:16) − β √ T − t (cid:17) − . (77)Note that the value of LC ∗ float,t is unitless, since the option is written on the drawup ratio. If there is a sizeassociated to the underlying security, it can be multiplied to LC ∗ float,t which gives it a dollar amount. As in Eqn(73), the price for this derivative on spot price is LC ∗ Spot float,t = e β T − t )2 LC ∗ float,t . (78)14 .3 Vanilla and Down-and-In Call Now we price a Down-and-In Call (DIC) option which becomes from worthless to a vanilla call if the lower barrieris hit before maturity. A vanilla call can be viewed as a special case of a Down-and-In barrier call (DIC) with thelower barrier has been hit prior to presence. The value of a DIC option written on F t is given byDIC t ( L, K, T ) = F t ≤ L · N T =0 · C t ( K, T ) + F t >L · N T =0 · E t (cid:2) F T ≤ L ( F T − K ) + (cid:3) , (79)where L is the barrier, K is the strike price, T is maturity and C t is a vanilla call priced at t . Note setting L = F reduces the DIC to a vanilla call. As implied by Eqn (79) if default happens ( N T = 0), the option becomesworthless. To evaluate the expectation value of the second term in (79), we once again apply the bivariate transitionPDF: E t (cid:2) F T ≤ L ( F T − K ) + (cid:3) = E t (cid:20) Z T ≤ α ln LF ( F e αZ T e β ˇ Z T β − α − K ) + (cid:21) = Z α ln LF −∞ dj Z ∞ k ∗ d ˇ k s π ( T − t ) (ˇ k − j + w ) e − (ˇ k − j + w )22( T − t ) (cid:16) F e αj e β ˇ kβ − α − K (cid:17) , (80)where k ∗ ( j ) is determined by k ∗ = max (cid:18) f − (cid:18) SF e αj (cid:19) , (cid:19) , f ( x ) = e βxβ − α . (81)For the dependence of k ∗ on j , the integral above cannot be obtained in closed form, a similar situation as in [2].Nonetheless, the result can be further simplified as E t (cid:2) F T ≤ L ( F T − K ) + (cid:3) = F Z α ln LF −∞ dje αj + β T − t )2 (cid:20) ( β + α ) e β ( j − w ) N (cid:18) j − w − k ∗ + β ( T − t ) √ T − t (cid:19) − ( β − α ) e − β ( j − w ) N (cid:18) j − w − k ∗ − β ( T − t ) √ T − t (cid:19) (cid:21) , (82)which gives rise to the value of the DIC option After replacing Z t with the market observable w , we now have theprice for the DIC option:DIC t ( L, K, T ) = F t ≤ L C t ( K, T ) + F t >L F Z α ln LF −∞ dje αj (cid:20) ( β + α ) e β ( j − w ) N (cid:18) j − w − k ∗ + β ( T − t ) √ T − t (cid:19) − ( β − α ) e − β ( j − w ) N (cid:18) j − w − k ∗ − β ( T − t ) √ T − t (cid:19) (cid:21) . (83)In the special case when L = F , the DIC option reduces to a vanilla call with a price of C t ( K, T ) = F Z −∞ dje αj (cid:20) ( β + α ) e β ( j − w ) N (cid:18) j − w − k ∗ + β ( T − t ) √ T − t (cid:19) − ( β − α ) e − β ( j − w ) N (cid:18) j − w − k ∗ − β ( T − t ) √ T − t (cid:19) (cid:21) , (84)which completes the pricing of a DIC option on Eqn (50). For a DIC option on spot price, Eqn (79) becomesDIC Spot t ( L, K, T ) = F t ≤ L · C FX Spot t ( K, T ) + F t >L · E t (cid:2) F T ≤ L ( F T − K ) + (cid:3) , (85)15hich leads to slight modification on both Eqn (83) and Eqn (84), and the results areDIC Spot t ( L, K, T ) = F t ≤ L C Spot t ( K, T )+ F t >L F e β T − t )2 Z α ln LF −∞ dje αj (cid:20) ( β + α ) e β ( j − w ) N (cid:18) j − w − k ∗ + β ( T − t ) √ T − t (cid:19) − ( β − α ) e − β ( j − w ) N (cid:18) j − w − k ∗ − β ( T − t ) √ T − t (cid:19) (cid:21) ,C Spot t ( K, T ) = F e β T − t )2 Z −∞ dje αj (cid:20) ( β + α ) e β ( j − w ) N (cid:18) j − w − k ∗ + β ( T − t ) √ T − t (cid:19) − ( β − α ) e − β ( j − w ) N (cid:18) j − w − k ∗ − β ( T − t ) √ T − t (cid:19) (cid:21) . (86)Before closing this section, we would like to point out that Eqn (83) is related to several options. For instance,when α = 1 and β = 0 the result reduces to that in [2]. In the special case of a zero strike DIC option ( K = 0),Eqn (83) has closed form expressions: DIC t ( L, , T ) = F (cid:20) (cid:18) LF (cid:19) α + β e − βw N α ln LF − w + β ( T − t ) √ T − t ! − e αw + ( α − β T − t )2 N α ln LF − w − α ( T − t ) √ T − t ! + (cid:18) LF (cid:19) α − β e βw N α ln LF − w − β ( T − t ) √ T − t ! (cid:21) . (87) We proposed a three parameter continuous martingale with state space [0 , ∞ ). This is done by first generating aprocess with a positive drift driven by the running minimum and drawup of a Brownian motion in the Az´ema-Yorsetting, and adding a jump to default process. The process generalizes driftless Geometric Brownian motion byadding two more parameters while preserving its tractability. In particular, its running minimum and drawuprate (the ratio between level and running minimum) are both analytically tractable. The three model parameters α, γ , and β can be respectively interpreted as the instantaneous volatility of the underlying at each new low, atthe initial time, and at infinitely high prices of the underlying. The parameter α controls the implied volatility atlow strikes, while the parameter β controls the implied volatility at high strikes. So long as implied volatility ismonotonic in strike price, the parameter γ can be used to meet an at-the-money implied volatility. It is shown thatin certain limits, this new process can reduce to Geometric Brownian motion and the positive martingale givenin [2]. We also presented the bivariate transition PDF of the process’ running minimum and drawup rate. Byutilizing the transition PDF, we priced several options assuming the dynamics are driven by the three parametermartingale in risk neutral measure. The options include a one-touch option with a lower barrier, lookback optionswith floating and fixed strike prices, vanilla call and a down-and-in call option.Since not all implied volatility slices are monotonic, future research should be directed towards extending themodel by introducing either stochastic volatility or jumps. One can also use the process without jump to defaultto model dynamics that involve a positive drift, for instance, the cumulative return of an investment strategy.Moreover, Girsanov’s theorem can be used to remove the drift of G , at which point a reflection principle becomesavailable. In the interests of brevity, these extensions are best left for future research. Acknowledgement
We are grateful to Matthew Lorig, Vasily Strela, Jane Yu, and especially Travis Fisher, for their comments. Theyare not responsible for any errors. 16 ppendix
1. More about e βxβ − α This technical appendix proves the result (6). For x ≥ α ≥
0, and β >
0, our two parameter exponentialfunction is defined as: e βxα − β ≡ α + β β e βx + α − β β e − βx . (88)Squaring this result implies that: (cid:16) e βxα − β (cid:17) = (cid:18) α + β β (cid:19) e βx + α − β β + (cid:18) α − β β (cid:19) e − βx . (89)Consider the cohort of (88): e βxβ − α ≡ α + β β e βx + β − α β e − βx . (90)Squaring this cohort implies that: (cid:16) e βxβ − α (cid:17) = (cid:18) α + β β (cid:19) e βx − α − β β + (cid:18) α − β β (cid:19) e − βx . (91)Subtracting (91) from (89) implies that: (cid:16) e βxα − β (cid:17) − (cid:16) e βxβ − α (cid:17) = α − β β . (92)Taking the positive square root of each side leads to the desired result: e βxα − β = s(cid:16) e βxβ − α (cid:17) + α − β β . (93)17 eferences [1] Black, F., 1976, “The Pricing of Commodity Contracts”, Journal of Financial Economics , , 167–179.[2] Carr P., 2014, “First Order Calculus and Option Pricing”, Journal of Financial Engineering , 1.[3] Guyon, J., 2014, “Path-Dependent Volatility”, Risk , .[4] Hobson, D. G. and L. C. G. Rogers, 1998, “Complete Models with Stochastic Volatility”, MathematicalFinance , , 27-48.[5] Merton, R.C., 1976, “Option pricing when underlying stock returns are discontinuous”, Journal of FinancialEconomics , , 125-144.[6] Grabbe, J.O., 1983, “The pricing of call and put options on foreign exchange”, Journal of International Moneyand Finance ,2