Exponential stock models driven by tempered stable processes
EEXPONENTIAL STOCK MODELS DRIVEN BY TEMPEREDSTABLE PROCESSES
UWE KÜCHLER AND STEFAN TAPPE
Abstract.
We investigate exponential stock models driven by tempered sta-ble processes, which constitute a rich family of purely discontinuous Lévy pro-cesses. With a view of option pricing, we provide a systematic analysis ofthe existence of equivalent martingale measures, under which the model re-mains analytically tractable. This includes the existence of Esscher martingalemeasures and martingale measures having minimal distance to the physicalprobability measure. Moreover, we provide pricing formulae for European calloptions and perform a case study. Introduction
Tempered stable distributions form a class of distributions that have attractedthe interest of researchers from probability theory as well as financial mathemat-ics. They have first been introduced in [18], where the associated Lévy processesare called “truncated Lévy flights”, and have been generalized by several authors.Tempered stable distributions form a six parameter family of infinitely divisibledistributions, which cover several well-known subclasses like Variance Gamma dis-tributions [26, 25], bilateral Gamma distributions [20, 21] and CGMY distribu-tions [6]. Properties of tempered stable distributions have been investigated, e.g.,in [29, 33, 32, 3], and in [23], where some of the results of this paper have beenannounced. For financial modeling they have been applied, e.g., in [4, 7, 27, 16, 2],see also the recent textbook [28].The purpose of this paper is to provide a systematic analysis of the existenceof equivalent martingale measures for exponential stock price models driven bytempered stable processes, under which the computation of option prices remainsanalytically tractable. In particular, we are interested in martingale measures, underwhich the driving process remains a tempered stable process, or at least becomesa Lévy process for which the characteristic function is explicitly known.Equivalent martingale measures of interest, under which the driving process re-mains a tempered stable process, are the Esscher martingale measure and bilateralEsscher martingale measures which minimize the distance to the original probabilitymeasure in a certain sense, for example the minimal entropy martingale measure orthe p -optimal martingale measure. We will examine the existence of these martin-gale measures in detail. Furthermore, we will treat the Föllmer Schweizer minimalmartingale measure. In case of existence, the driving process is the sum of two inde-pendent tempered stable processes under this measure, and thus the model remainsanalytically tractable. For all the just mentioned martingale measures, we will de-rive option pricing formulae. Moreover, we will illustrate our findings by means ofa case study.The remainder of this text is organized as follows: In Section 2 we introduce thestock model. Afterwards, in Section 3 we study Esscher transforms, in Section 4 we Mathematics Subject Classification.
Key words and phrases.
Exponential stock model, tempered stable process, bilateral Esschertransform, option pricing. a r X i v : . [ q -f i n . M F ] J u l UWE KÜCHLER AND STEFAN TAPPE study bilateral Esscher transforms, and in Section 5 we treat the Föllmer Schweizerminimal martingale measure. Section 6 is devoted to option pricing formulae, andin Section 7 we provide the case study.2.
Stock price models driven by tempered stable processes
In this section, we shall introduce the stock price model and review some resultsabout tempered stable processes. The reader is referred to [23] for all results abouttempered stable processes which we recall in this section.Let (Ω , F , ( F t ) t ≥ , P ) be a filtered probability space satisfying the usual condi-tions. We fix parameters α + , λ + , α − , λ − ∈ (0 , ∞ ) and β + , β − ∈ (0 , . An infinitelydivisible distribution η on ( R , B ( R )) is called a tempered stable distribution , denoted η = TS( α + , β + , λ + ; α − , β − , λ − ) , if its characteristic function is given by ϕ ( z ) = exp (cid:18) (cid:90) R (cid:0) e izx − (cid:1) F ( dx ) (cid:19) , z ∈ R where the Lévy measure F is F ( dx ) = (cid:18) α + x β + e − λ + x (0 , ∞ ) ( x ) + α − | x | β − e − λ − | x | ( −∞ , ( x ) (cid:19) dx. (2.1)2.1. Remark. In [22] we have studied exponential stock models driven by bilateralGamma processes, which would occur for β + = β − = 0 . We can express the characteristic function of η as(2.2) ϕ ( z ) = exp (cid:16) α + Γ( − β + ) (cid:2) ( λ + − iz ) β + − ( λ + ) β + (cid:3) + α − Γ( − β − ) (cid:2) ( λ − + iz ) β − − ( λ − ) β − (cid:3)(cid:17) , z ∈ R , where the powers stem from the main branch of the complex logarithm. We callthe Lévy process X associated to η a tempered stable process , and write X ∼ TS( α + , β + , λ + ; α − , β − , λ − ) . (2.3)The cumulant generating function Ψ( z ) = ln E P [ e zX ] exists on [ − λ − , λ + ] and is given by(2.4) Ψ( z ) = α + Γ( − β + ) (cid:2) ( λ + − z ) β + − ( λ + ) β + (cid:3) + α − Γ( − β − ) (cid:2) ( λ − + z ) β − − ( λ − ) β − (cid:3) , z ∈ [ − λ − , λ + ] . All increments of X have a tempered stable distribution, more precisely X t − X s ∼ TS( α + ( t − s ) , β + , λ + ; α − ( t − s ) , β − , λ − ) for ≤ s < t .(2.5)A tempered stable stock model is an exponential Lévy model of the type (cid:26) S t = S e X t B t = e rt (2.6)where X denotes a tempered stable process and S is a dividend paying stock withdeterministic initial value S > and dividend rate q ≥ . Furthermore, B is thebank account with interest rate r ≥ . In what follows, we assume that r ≥ q ≥ .An equivalent probability measure Q ∼ P is a local martingale measure (in short, martingale measure ), if the discounted stock price process ˜ S t := e − ( r − q ) t S t = S e X t − ( r − q ) t , t ≥ (2.7) XPONENTIAL STOCK MODELS DRIVEN BY TEMPERED STABLE PROCESSES 3 is a local Q -martingale. The existence of a martingale measure Q ∼ P ensures thatthe stock market is free of arbitrage, and the price of an European option Φ( S T ) ,where T > is the time of maturity and Φ : R → R the payoff profile, is given by π = e − rT E Q [Φ( S T )] . Lemma.
The following statements are true: (1) If λ + ≥ , then P is a martingale measure if and only if α + Γ( − β + ) (cid:2) ( λ + − β + − ( λ + ) β + (cid:3) + α − Γ( − β − ) (cid:2) ( λ − + 1) β − − ( λ − ) β − (cid:3) = r − q. (2.8)(2) If λ + < , then P is never a martingale measure.Proof. By [22, Lemma 2.6] the measure P is a martingale measure if and only if E P [ e X ] = 1 , and hence, the assertion follows by taking into account (2.4). (cid:3) Existence of Esscher martingale measures
In this section, we study the Esscher transform, which was pioneered in [9].Throughout this section, let X be a tempered stable process of the form (2.3).3.1. Definition.
Let Θ ∈ ( − λ − , λ + ) be arbitrary. The Esscher transform P Θ isdefined as the locally equivalent probability measure with likelihood process Λ t ( P Θ , P ) := d P Θ d P (cid:12)(cid:12)(cid:12)(cid:12) F t = e Θ X t − Ψ(Θ) t , t ≥ (3.1) where Ψ denotes the cumulant generating function given by (2.4). Lemma.
For every Θ ∈ ( − λ − , λ + ) we have X ∼ TS( α + , β + , λ + − Θ; α − , β − , λ − + Θ) under P Θ .Proof. This follows from Proposition 2.1.3 and Example 2.1.4 in [19]. (cid:3)
We define the function f : [ − λ − , λ + − → R as f (Θ) := f + (Θ) + f − (Θ) , where we have set f + (Θ) := α + Γ( − β + ) (cid:2) ( λ + − Θ − β + − ( λ + − Θ) β + (cid:3) ,f − (Θ) := α − Γ( − β − ) (cid:2) ( λ − + Θ + 1) β − − ( λ − + Θ) β − (cid:3) . Theorem.
The following statements are true: (1)
There exists Θ ∈ ( − λ − , λ + ) such that P Θ is a martingale measure if andonly if λ + + λ − > (3.2) and r − q ∈ ( f ( − λ − ) , f ( λ + − . (3.3)(2) Condition (3.3) is equivalent to α + Γ( − β + ) (cid:2) ( λ + + λ − − β + − ( λ + + λ − ) β + (cid:3) + α − Γ( − β − ) < r − q ≤ − α + Γ( − β + ) + α − Γ( − β − ) (cid:2) ( λ + + λ − ) β − − ( λ + + λ − − β − (cid:3) . (3) If conditions (3.2) and (3.3) are satisfied, then Θ is unique, belongs to theinterval ( − λ − , λ + − , and it is the unique solution of the equation f (Θ) = r − q. (3.4) UWE KÜCHLER AND STEFAN TAPPE
Proof.
Let Θ ∈ ( − λ − , λ + ) be arbitrary. In view of Lemmas 3.2 and 2.2, theprobability measure P Θ is a martingale measure if and only if λ + − Θ ≥ , i.e. Θ ∈ ( − λ − , λ + − , and (3.4) is fulfilled. Note that ( − λ − , λ + − (cid:54) = ∅ if and onlyif (3.2) is satisfied. For the functions f + and f − we obtain the derivatives ( f + ) (cid:48) (Θ) = − α + β + Γ( − β + ) (cid:2) ( λ + − Θ − β + − − ( λ + − Θ) β + − (cid:3) , ( f − ) (cid:48) (Θ) = − α − β − Γ( − β − ) (cid:2) ( λ − + Θ) β − − − ( λ − + Θ + 1) β − − (cid:3) for Θ ∈ ( − λ − , λ + − . Noting that β + , β − ∈ (0 , , we see that ( f + ) (cid:48) , ( f − ) (cid:48) > onthe interval ( − λ − , λ + − . Hence, f is strictly increasing on ( − λ − , λ + − , whichcompletes the proof. (cid:3) Remark.
In contrast to the present situation, for bilateral Gamma stock models( β + = β − = 0 ) condition (3.2) alone is already sufficient for the existence of anEsscher martingale measure, cf. [22, Remark 4.4] . Existence of minimal distance measures preserving the class oftempered stable processes
In the literature, one often performs option pricing by finding an equivalentmartingale measure Q ∼ P which minimizes the distance E P [ g (Λ ( Q , P ))] for some strictly convex function g : (0 , ∞ ) → R . Here are popular choices for thefunction g : • For g ( x ) = x ln x we call Q the minimal entropy martingale measure . • For g ( x ) = x p with p > we call Q the p -optimal martingale measure . • For p = 2 we call Q the variance-optimal martingale measure .We refer to [22, Section 5] for further remarks and related literature. While p -optimal equivalent martingale measures do not exist in tempered stable stock mod-els (which follows from [1, Example 2.7]), we have the following result concerningthe existence of minimal entropy martingale measures:4.1. Theorem.
The following statements are true: (1) If λ + < , then a minimal entropy martingale measure exists. (2) If λ + ≥ , then a minimal entropy measure exists if and only if α + Γ( − β + ) (cid:2) ( λ + − β + − ( λ + ) β + (cid:3) + α − Γ( − β − ) (cid:2) ( λ − + 1) β − − ( λ − ) β − (cid:3) ≥ r − q. This result, which has been indicated in [22, Remark 5.4], follows by adjustingthe arguments of the proof of [22, Theorem 5.3] to the present situation, where thestock model is driven by a tempered stable process.In this section, we shall minimize the relative entropy H ( Q | P ) := E P [Λ ( Q , P ) ln Λ ( Q , P )] = E Q [ln Λ ( Q , P )] within the class of tempered stable processes by performing bilateral Esscher trans-forms. Let X be a tempered stable process of the form (2.3). We decompose thetempered stable process X = X + − X − as the difference of two independent sub-ordinators. Their respective cumulant generating functions are given by Ψ + ( z ) = α + Γ( − β + ) (cid:2) ( λ + − z ) β + − ( λ + ) β + (cid:3) , z ∈ ( −∞ , λ + ] , (4.1) Ψ − ( z ) = α − Γ( − β − ) (cid:2) ( λ − − z ) β − − ( λ − ) β − (cid:3) , z ∈ ( −∞ , λ − ] , (4.2)see [23]. Note that Ψ( z ) = Ψ + ( z ) + Ψ − ( − z ) for z ∈ [ − λ − , λ + ] . XPONENTIAL STOCK MODELS DRIVEN BY TEMPERED STABLE PROCESSES 5
Definition.
Let θ + ∈ ( −∞ , λ + ) and θ − ∈ ( −∞ , λ − ) be arbitrary. The bilateralEsscher transform P ( θ + ,θ − ) is defined as the locally equivalent probability measurewith likelihood process Λ t ( P ( θ + ,θ − ) , P ) := d P ( θ + ,θ − ) d P (cid:12)(cid:12)(cid:12)(cid:12) F t = e θ + X + t − Ψ + ( θ + ) t · e θ − X − t − Ψ − ( θ − ) t , t ≥ . Note that the Esscher transforms P Θ from Section 3 are special cases of the justintroduced bilateral Esscher transforms P ( θ + ,θ − ) . Indeed, we have P Θ = P (Θ , − Θ) , Θ ∈ ( − λ − , λ + ) . (4.3)4.3. Lemma.
For all θ + ∈ ( −∞ , λ + ) and θ − ∈ ( −∞ , λ − ) we have X ∼ TS( α + , β + , λ + − θ + ; α − , β − , λ − − θ − ) under P ( θ + ,θ − ) .Proof. This follows from Proposition 2.1.3 and Example 2.1.4 in [19]. (cid:3)
Proposition.
The following statements are true: (1)
If we have − α + Γ( − β + ) ≤ r − q, (4.4) then no pair ( θ + , θ − ) ∈ ( −∞ , λ + ) × ( −∞ , λ − ) with P ( θ + ,θ − ) being a mar-tingale measure exists. (2) If we have − α + Γ( − β + ) > r − q, (4.5) then there exist −∞ ≤ θ +1 < θ +2 ≤ λ + − and a continuous, strictlyincreasing, bijective function Φ : ( θ +1 , θ +2 ) → ( −∞ , λ − ) such that: • For all θ + ∈ ( θ +1 , θ +2 ) there exists a unique θ − ∈ ( −∞ , λ − ) with P ( θ + ,θ − ) being a martingale measure, and it is given by θ − = Φ( θ + ) . • For all θ + ∈ ( −∞ , λ + ) \ ( θ +1 , θ +2 ) no θ − ∈ ( −∞ , λ − ) with P ( θ + ,θ − ) being a martingale measure exists.Proof. We introduce the functions f + : ( −∞ , λ + − → R and f − : ( −∞ , λ − ] → R as f + ( θ + ) := α + Γ( − β + ) (cid:2) ( λ + − θ + − β + − ( λ + − θ + ) β + (cid:3) ,f − ( θ − ) := α − Γ( − β − ) (cid:2) ( λ − − θ − + 1) β − − ( λ − − θ − ) β − (cid:3) . By Lemmas 2.2 and 4.3, the measure P ( θ + ,θ − ) is a martingale measure if and onlyif θ + ∈ ( −∞ , λ + − and f + ( θ + ) + f − ( θ − ) = r − q. (4.6)The function f + is continuous and strictly increasing on ( −∞ , λ + − with lim θ + →−∞ f + ( θ + ) = 0 and f + ( λ + −
1) = − α + Γ( − β + ) > . The function f − is continuous and strictly decreasing on ( −∞ , λ − ] with lim θ − →−∞ f − ( θ − ) = 0 and f − ( λ − ) = α − Γ( − β − ) < . UWE KÜCHLER AND STEFAN TAPPE
Therefore, if we have (4.4), then for no pair ( θ + , θ − ) ∈ ( −∞ , λ + − × ( −∞ , λ − ) equation (4.6) is satisfied. If we have (4.5), then let −∞ ≤ θ +1 < θ +2 ≤ λ + − bethe unique solutions of the equations f + ( θ +1 ) = r − q,f + ( θ +2 ) = r − q − α − Γ( − β − ) , with the conventions θ +1 = −∞ , if r − q = 0 , θ +2 = λ + − , if r − q − α − Γ( − β − ) > − α + Γ( − β + ) ,and define Φ( θ + ) := ( f − ) − ( r − q − f + ( θ + )) , θ + ∈ ( θ +1 , θ +2 ) . (4.7)Then Φ is continuous and strictly increasing with Φ(( θ +1 , θ +2 )) = ( −∞ , λ − ) , whichfinishes the proof. (cid:3) Remark.
The proof of Proposition 4.4 shows that the situation θ +1 = −∞ occurs if and only if r = q and that the situation θ +2 = λ + − occurs if and only if r − q ≤ − α + Γ( − β + ) + α − Γ( − β − ) . All equivalent measure transformations preserving the class of tempered stable pro-cesses are bilateral Esscher transforms; this follows from [23, Proposition 8.1] , seealso [7, Example 9.1] . Hence, we introduce the set of parameters M P := { ( θ + , θ − ) ∈ ( −∞ , λ + ) × ( −∞ , λ − ) | P ( θ + ,θ − ) is a martingale measure } such that the bilateral Esscher transform is a martingale measure. The previousProposition 4.4 tells us that for (4.4) we have M P = ∅ , and that for (4.5) we have M P = { ( θ, Φ( θ )) ∈ R | θ ∈ ( θ +1 , θ +2 ) } . (4.8) Moreover, we remark that condition (4.5) is always fulfilled for r = q . Lemma.
For all ( θ + , θ − ) ∈ ( −∞ , λ + ) × ( −∞ , λ − ) we have H ( P ( θ + ,θ − ) | P )= − α + Γ( − β + ) (cid:16) λ + β + ( λ + − θ + ) β + − + (1 − β + )( λ + − θ + ) β + − ( λ + ) β + (cid:17) − α − Γ( − β − ) (cid:16) λ − β − ( λ − − θ − ) β − − + (1 − β − )( λ − − θ − ) β − − ( λ − ) β − (cid:17) . Proof.
The relative entropy of the bilateral Esscher transform is given by H ( P ( θ + ,θ − ) | P ) = E P ( θ + ,θ − ) [ln Λ ( P ( θ + ,θ − ) , P )]= E P ( θ + ,θ − ) [ θ + X +1 − Ψ + ( θ + )] + E P ( θ + ,θ − ) [ θ − X − − Ψ − ( θ − )] . XPONENTIAL STOCK MODELS DRIVEN BY TEMPERED STABLE PROCESSES 7
Using Lemma 4.3 and [23, Remark 2.7] we obtain E P ( θ + ,θ − ) [ θ + X +1 − Ψ + ( θ + )]= θ + Γ(1 − β + ) α + ( λ + − θ + ) − β + − α + Γ( − β + ) (cid:104) ( λ + − θ + ) β + − ( λ + ) β + (cid:105) = − α + Γ( − β + ) (cid:18) θ + β + ( λ + − θ + ) − β + + ( λ + − θ + ) β + − ( λ + ) β + (cid:19) = − α + Γ( − β + ) (cid:16) ( β + θ + + λ + − θ + )( λ + − θ + ) β + − − ( λ + ) β + (cid:17) = − α + Γ( − β + ) (cid:16)(cid:0) λ + β + + (1 − β + )( λ + − θ + ) (cid:1) ( λ + − θ + ) β + − − ( λ + ) β + (cid:17) = − α + Γ( − β + ) (cid:16) λ + β + ( λ + − θ + ) β + − + (1 − β + )( λ + − θ + ) β + − ( λ + ) β + (cid:17) . An analogous calculation for E P ( θ + ,θ − ) [ θ − X − − Ψ − ( θ − )] finishes the proof. (cid:3) Theorem.
The following statements are true: (1)
If (4.4) is satisfied, then we have M P = ∅ . (2) If (4.5) is satisfied, then there exist θ + ∈ ( −∞ , λ + ) and θ − ∈ ( −∞ , λ − ) such that H ( P ( θ + ,θ − ) | P ) = min ( ϑ + ,ϑ − ) ∈M P H ( P ( ϑ + ,ϑ − ) | P ) . (4.9) Proof.
If (4.4) is satisfied, then by Proposition 4.4 we have M P = ∅ . Now, supposethat (4.5) is satisfied, and let Φ : ( θ +1 , θ +2 ) → ( −∞ , λ − ) be the function fromProposition 4.4. Let f : ( θ +1 , θ +2 ) → R be the function f ( θ ) := − α + Γ( − β + ) (cid:16) λ + β + ( λ + − θ ) β + − + (1 − β + )( λ + − θ ) β + − ( λ + ) β + (cid:17) − α − Γ( − β − ) (cid:16) λ − β − ( λ − − Φ( θ )) β − − + (1 − β − )( λ − − Φ( θ )) β − − ( λ − ) β − (cid:17) . By Proposition 4.4 and Lemma 4.6, for each θ ∈ ( θ +1 , θ +2 ) the measure P ( θ, Φ( θ )) is amartingale measure and we have H ( P ( θ, Φ( θ )) | P ) = f ( θ ) . The function Φ is strictlyincreasing with lim θ ↓ θ +1 Φ( θ ) = −∞ and lim θ ↑ θ +2 Φ( θ ) = λ − , which gives us lim θ ↓ θ +1 f ( θ ) = ∞ and lim θ ↑ θ +2 f ( θ ) = ∞ . Since f is continuous, it attains a minimum and the assertion follows. (cid:3) Remark.
In contrast to bilateral Gamma stock models, it can happen that M P = ∅ , i.e., there is no equivalent martingale measure under which X remains atempered stable process. Moreover, in contrast to bilateral Gamma stock models, thefunction Φ from Proposition 4.4, which is defined in (4.7) by means of the inverseof f − , does not seem to be available in closed form, cf. [22, Remark 6.7] . Next, we consider the p -distances H p ( Q | P ) := E P (cid:34)(cid:18) d Q d P (cid:19) p (cid:35) for p > .As mentioned at the beginning of this section, for tempered stable stock models the p -optimal martingale measure does not exist. However, we can, as provided for theminimal entropy martingale measure, determine the p -optimal martingale measurewithin the class of tempered stable processes. For this purpose, we compute the UWE KÜCHLER AND STEFAN TAPPE p -distance of a bilateral Esscher transform. Since the subordinators X + and X − are independent, for p > and θ + ∈ ( −∞ , λ + p ) , θ − ∈ ( −∞ , λ − p ) the p -distance isgiven by(4.10) H p ( P ( θ + ,θ − ) | P ) = E P (cid:34)(cid:18) d P ( θ + ,θ − ) d P (cid:19) p (cid:35) = e − p (Ψ + ( θ + )+Ψ − ( θ − )) E P (cid:2) e pθ + X +1 (cid:3) E P (cid:2) e pθ − X − (cid:3) = exp (cid:0) − p (Ψ + ( θ + ) + Ψ − ( θ − )) + Ψ + ( pθ + ) + Ψ − ( pθ − ) (cid:1) = exp (cid:16) − α + Γ( − β + ) (cid:104) p (cid:2) ( λ + − θ + ) β + − ( λ + ) β + (cid:3) − (cid:2) ( λ + − pθ + ) β + − ( λ + ) β + (cid:3)(cid:105) − α − Γ( − β − ) (cid:104) p (cid:2) ( λ − − θ − ) β − − ( λ − ) β − (cid:3) − (cid:2) ( λ − − pθ − ) β − − ( λ − ) β − (cid:3)(cid:105)(cid:17) . A similar argumentation as in Theorem 4.7 shows that, provided condition (4.5)holds true, there exists a pair ( θ + , θ − ) minimizing the p -distance (4.10), and in thiscase we also have θ − = Φ( θ + ) , where θ + minimizes the function θ (cid:55)→ H p ( P ( θ, Φ( θ )) | P ) . (4.11)Numerical computations for concrete examples suggest that θ p → θ for p ↓ ,where for each p > the parameter θ p minimizes (4.11), and θ minimizes θ (cid:55)→ H ( P ( θ, Φ( θ )) | P ) . (4.12)This is not surprising, since it is known that, under suitable technical conditions,the p -optimal martingale measure converges to the minimal entropy martingalemeasure for p ↓ , see, e.g. [10, 11, 30, 14, 1, 17].5. Existence of Föllmer Schweizer minimal martingale measures
In this section, we deal with the existence of the Föllmer Schweizer minimal mar-tingale measure in tempered stable stock models. This measure has been introducedin [8] with the motivation of constructing optimal hedging strategies. Throughoutthis section, we fix a finite time horizon
T > and assume that λ + ≥ . Then theconstant c = c ( α + , α − , β + , β − , λ + , λ − , r, q ) = Ψ(1) − ( r − q )Ψ(2) − , (5.1)is well-defined. For technical reasons, we shall also assume that the filtration ( F t ) t ≥ is generated by the tempered stable process of the form (2.3). As in [22, Lemma 7.1],we show that the discounted stock price process ˜ S is a special semimartingale.Let ˜ S = S + M + A be its canonical decomposition and let ˆ Z be the stochasticexponential ˆ Z t = E (cid:18) − (cid:90) • c ˜ S s − dM s (cid:19) t , t ∈ [0 , T ] , (5.2)where we recall that for a semimartingale X the stochastic exponential Y = E ( X ) defined as E ( X ) t := exp (cid:18) X t − X − (cid:104) X c , X c (cid:105) (cid:19) (cid:89) s ≤ t (1 + ∆ X s ) e − ∆ X s , t ≥ is the unique solution of the stochastic differential equation dY t = Y t − dX t , Y = 1 , XPONENTIAL STOCK MODELS DRIVEN BY TEMPERED STABLE PROCESSES 9 see, e.g. [13, Theorem I.4.61]. The (possibly signed) measure ˆ P with density d ˆ P d P := ˆ Z T (5.3)is the so-called Föllmer Schweizer minimal martingale measure (in short,
FS min-imal martingale measure ).5.1.
Theorem.
The following statements are equivalent: (1) ˆ Z is a strict martingale density for ˜ S . (2) ˆ Z is a strictly positive P -martingale. (3) We have − ≤ c ≤ . (5.4)(4) We have α + Γ( − β + )[( λ + − β + − ( λ + ) β + ] (5.5) + α − Γ( − β )[( λ − + 1) β − − ( λ − ) β − ] ≤ r − q and α + Γ( − β + )[( λ + − β + − ( λ + − β + ] (5.6) + α − Γ( − β )[( λ − + 1) β − − ( λ − + 2) β − ] ≤ − ( r − q ) . If the previous conditions are satisfied, then under the FS minimal martingale mea-sure ˆ P we have (5.7) X ∼ TS(( c + 1) α + , β + , λ + ; ( c + 1) α − , β − , λ − ) ∗ TS( − cα + , β + , λ + − − cα − , β − , λ − + 1) . Proof.
We only have to show the equivalence (3) ⇔ (4), as the rest follows byarguing as in the proof of [22, Theorem 7.3]. We observe that (5.4) is equivalent tothe two conditions Ψ(1) ≤ r − q and Ψ(1) − Ψ(2) ≤ − ( r − q ) , and, in view of the cumulant generating function given by (2.4), these two conditionsare fulfilled if and only if we have (5.5) and (5.6). (cid:3) Remark.
Relation (5.7) means that under ˆ P the driving process X is the sumof two independent tempered stable processes. There are the following two boundaryvalues: • In the case c = 0 we have X ∼ TS( α + , β + , λ + , α − , β − , λ − ) under ˆ P ,i.e., the FS minimal martingale measure ˆ P coincides with the physical mea-sure P . Indeed, the definition (5.1) of c and Lemma 2.2 show that P alreadyis a martingale measure for ˜ S . • In the case c = − we have X ∼ TS( α + , β + , λ + − , α − , β − , λ − + 1) under ˆ P ,i.e., the FS minimal martingale measure ˆ P coincides with the Esscher trans-form P , see Theorem 3.3. Indeed, the definition (5.1) of c shows that equa-tion (3.4) is satisfied with Θ = 1 . As outlined at the end of [22, Section 7], under the FS minimal martingale mea-sure ˆ P we can construct a trading strategy ξ which minimizes the quadratic hedgingerror. The arguments transfer to our present situation with a driving tempered sta-ble process. Option pricing in tempered stable stock models
In this section, we present pricing formulae for European call options. Afterperforming a measure change Q ∼ P as in Section 3 or 4, that is, Q = P Θ or Q = P ( θ, Φ( θ )) for appropriate parameters, we may assume that the driving process X is a tempered stable process of the form (2.3) under the martingale measure Q . We fix a strike price K > and a maturity date T > . Then the price of aEuropean call option with these parameters is given by π = e − rT E Q [( S T − K ) + ] . First, we shall derive an option pricing formula in closed form by following an ideafrom [15, Section 8.1]. In the sequel, F α + ,β + ,λ + ; α − ,β − ,λ − denotes the TS( α + , β + , λ + ; α − , β − , λ − ) -distribution function and ¯ F α + ,β + ,λ + ; α − ,β − ,λ − := 1 − F α + ,β + ,λ + ; α − ,β − ,λ − . Proposition.
Suppose that λ + > . Then, the price of the call option is givenby (6.1) π = S e (Ψ(1) − r ) T ¯ F α + T,β + ,λ + − α − T,β − ,λ − +1 (ln( K/S )) − e − rT K ¯ F α + T,β + ,λ + ; α − ,β − T,λ − (ln( K/S )) . Proof.
By the definition of the likelihood process (3.1) we obtain π = e − rT E Q [( S T − K ) + ] = e − rT E Q [( S e X T − K ) + ]= S e − rT E Q [ e X T { X T ≥ ln( K/S ) } ] − e − rT K Q ( X T ≥ ln( K/S ))= S e − rT E Q (cid:34) e X T { X T ≥ ln( K/S ) } d Q d Q (cid:12)(cid:12)(cid:12)(cid:12) F T (cid:35) − e − rT K Q ( X T ≥ ln( K/S ))= S e − rT e Ψ(1) T Q ( X T ≥ ln( K/S )) − e − rT K Q ( X T ≥ ln( K/S )) , which, in view of Lemma 3.2, provides the formula (6.1). (cid:3) Remark.
Note that applying the option pricing formula (6.1) requires know-ledge about the densities of tempered stable distributions, which are generally notavailable in closed form. Therefore, we will turn to the option pricing formula (6.2)below, which is based on Fourier transform techniques. However, we remark thatformula (6.1) also holds true in the bilateral Gamma case β + = β − = 0 , for whichthe densities are given in terms of the Whittaker function, see [20, Section 4] . In the sequel, we will use the following option pricing formula (6.2), which isbased on Fourier transform techniques. Let X is a tempered stable process of theform (2.3) under P , let Q ∼ P be a martingale measure as in Section 3, 4 or 5, thatis, Q = P Θ , Q = P ( θ, Φ( θ )) or Q = ˆ P , and denote by ϕ X T the characteristic functionof X T under the martingale measure Q .6.3. Proposition.
We suppose that • λ + > , if we have (2.3) under Q . In this case, let ν ∈ (1 , λ + ) be arbitrary. • λ + > , if we have (5.7) under Q . In this case, let ν ∈ (1 , λ + − bearbitrary.Then the price of the call option is given by π = − e − rT K π (cid:90) iν + ∞ iν −∞ (cid:18) KS (cid:19) iz ϕ X T ( − z ) z ( z − i ) dz. (6.2) XPONENTIAL STOCK MODELS DRIVEN BY TEMPERED STABLE PROCESSES 11
Proof.
The stock prices are given by S t = S exp (cid:0) ( r − q ) t + ˜ X t (cid:1) , t ≥ , where ˜ X denotes the Lévy process given by ˜ X t = X t − ( r − q ) t for t ≥ . Moreover,the Fourier transform of the payoff function w ( x ) = ( e x − K ) + is given by ˆ w ( z ) = − K iz +1 z ( z − i ) , z ∈ C with Im z > , see Table 3.1 in [24]. Furthermore, the characteristic function (2.2) is analytic on thestrip { z ∈ C : Im z ∈ ( − λ + , λ − ) } by the analyticity of the power function z (cid:55)→ z β on the main branch of the complex logarithm for β ∈ (0 , . By our parameterrestriction on λ + , and possibly taking into account (5.7), we deduce that ϕ X T isanalytic on a strip of the form { z ∈ C : Im z ∈ ( a, b ) } with a < − and b > .Consequently, [24, Theorem 3.2] applies and provides us with the call option price π = e − rT π (cid:90) iν + ∞ iν −∞ e − iz (ln S +( r − q ) T ) ϕ ˜ X T ( − z ) ˆ w ( z ) dz = − e − rT π (cid:90) iν + ∞ iν −∞ S − iz e − iz ( r − q ) T ϕ X T ( − z ) e iz ( r − q ) T K iz +1 z ( z − i ) dz = − e − rT K π (cid:90) iν + ∞ iν −∞ (cid:18) KS (cid:19) iz ϕ X T ( − z ) z ( z − i ) dz, which proves (6.2). (cid:3) Remark.
Proposition 6.3 does not apply for the boundary case λ + = 1 (or λ + = 2 , respectively), although Q might be a martingale measure in this situation,cf. Lemma 2.2. The point is that in this case the characteristic function ϕ X T is notanalytic on a strip of the form { z ∈ C : Im z ∈ ( a, b ) } with a < − and b > , andhence, the option pricing formula from [24] does not apply. Remark.
The option pricing formula (6.2) can also be derived from [5, Sec-tion 3.1] . Taking into account the characteristic function (2.2) and relation (2.5), applyingProposition 6.3 yields the following pricing formulae: • Performing the Esscher transform from Section 3, for an arbitrary ν ∈ (1 , λ + − Θ) we obtain(6.3) π = − e − rT K π (cid:90) iν + ∞ iν −∞ (cid:18) KS (cid:19) iz exp (cid:16) α + T Γ( − β + ) (cid:2) ( λ + − Θ + iz ) β + − ( λ + − Θ) β + (cid:3) + α − T Γ( − β − ) (cid:2) ( λ − + Θ − iz ) β − − ( λ − + Θ) β − (cid:3)(cid:17) z ( z − i ) dz. • Performing the bilateral Esscher transform from Section 4, for an arbitrary ν ∈ (1 , λ + − θ ) we obtain(6.4) π = − e − rT K π (cid:90) iν + ∞ iν −∞ (cid:18) KS (cid:19) iz exp (cid:16) α + T Γ( − β + ) (cid:2) ( λ + − θ + iz ) β + − ( λ + − θ ) β + (cid:3) + α − T Γ( − β − ) (cid:2) ( λ − − Φ( θ ) − iz ) β − − ( λ − − Φ( θ )) β − (cid:3)(cid:17) z ( z − i ) dz. • Performing option pricing under the FS minimal martingale measure fromSection 5, for an arbitrary ν ∈ (1 , λ + − we obtain(6.5) π = − e − rT K π (cid:90) iν + ∞ iν −∞ (cid:18) KS (cid:19) iz exp (cid:16) ( c + 1) α + T Γ( − β + ) (cid:2) ( λ + + iz ) β + − ( λ + ) β + (cid:3) + ( c + 1) α − T Γ( − β − ) (cid:2) ( λ − − iz ) β − − ( λ − ) β − (cid:3) − cα + T Γ( − β + ) (cid:2) ( λ + − iz ) β + − ( λ + − β + (cid:3) − cα − T Γ( − β − ) (cid:2) ( λ − + 1 − iz ) β − − ( λ − + 1) β − (cid:3)(cid:17) z ( z − i ) dz, where the constant c is given by (5.1).7. A case study
In order to illustrate our previous results, we shall perform a case study in thissection. Figure 1 shows historical values of the German stock index DAX fromJanuary 3, 2011 until December 28, 2012, and the corresponding log returns. Thesedata are available at . − . − . − . . . . Figure 1.
The left plot shows the values of the German stockindex DAX from January 3, 2011 until December 28, 2012. Theright plot shows the corresponding log returns.In the sequel, the time t is measured in trading days. The data set consists of observations, which corresponds to a period of two years. Figure 2 below showsa histogram for the log returns. In order to estimate the parameters from thesehistorical data by the method of moments, we determine the empirical moments upto order , which are given by m = 1 . · − , (7.1) m = 2 . · − , (7.2) m = − . · − , (7.3) m = 2 . · − . (7.4)Then the parameters with a driving Wiener process X ∼ N( µ, σ ) are estimated as µ = 1 . · − and σ = 1 . · − . (7.5)The fitted density is shown in the left plot of Figure 2.It has already been documented in several case studies that the Black Scholesmodel does not provide a good fit to observed log returns of financial data, and thisalso shows up here. Therefore, we consider a tempered stable process X of the form XPONENTIAL STOCK MODELS DRIVEN BY TEMPERED STABLE PROCESSES 13 −0.06 −0.04 −0.02 0.00 0.02 0.04 0.06 −0.06 −0.04 −0.02 0.00 0.02 0.04 0.06
Figure 2.
Histogram for the log returns together with the fittednormal distribution in the left plot and the fitted tempered stabledistribution in the right plot.(2.3). Recall that for β + = β − = 0 we would have a bilateral Gamma process. Weslightly deviate from this situation by choosing β := β + = β − = 0 . . (7.6)In order to estimate the remaining parameters by the method of moments, we haveto solve the system of equations α + ( λ − ) − β − α − ( λ + ) − β − c ( λ + ) − β ( λ − ) − β = 0 α + ( λ − ) − β + α − ( λ + ) − β − c ( λ + ) − β ( λ − ) − β = 0 α + ( λ − ) − β − α − ( λ + ) − β − c ( λ + ) − β ( λ − ) − β = 0 α + ( λ − ) − β + α − ( λ + ) − β − c ( λ + ) − β ( λ − ) − β = 0 , (7.7)where c , c , c , c are given by c = m / Γ(1 − β ) ,c = ( m − m ) / Γ(2 − β ) ,c = ( m − m m + 2 m ) / Γ(3 − β ) ,c = ( m − m m − m + 12 m m − m ) / Γ(4 − β ) , see [23, Section 6] for further details. The solution of (7.7) is given by α + = 1 . , α − = 0 . , λ + = 122 . , λ − = 100 . . (7.8)The right plot in Figure 2 shows the fitted tempered stable density. Recall that thedensities of tempered stable distributions are generally not available in closed form.For the right plot in Figure 2 we have used the inversion formula f ( x ) = 12 π (cid:90) R exp (cid:16) − ixz + α + Γ( − β + ) (cid:2) ( λ + − iz ) β + − ( λ + ) β + (cid:3) + α − Γ( − β − ) (cid:2) ( λ + + iz ) β − − ( λ − ) β − (cid:3)(cid:17) dz, which follows from (2.2) and [31, Lemma 28.5, Proposition 2.5.xii].In the sequel, we suppose that under the real-world probability measure P theprocess X is a tempered stable process of the form (2.3) with (7.6) and estimatedparameters (7.8). As the time t is measured in trading days, the interest rate r denotes the daily interest rate. We suppose that it is given by r = 0 . / , whichcorresponds to an annualized interest rate r a of . Moreover, we suppose that q = 0 , i.e., the stock does not pay dividends.Based on these data, we will illustrate our results from Sections 3–5 concerningthe existence of equivalent martingale measures. −2.0 −1.5 −1.0 −0.5 0.0 − − + − − − −2.0 −1.5 −1.0 −0.5 0.0 . . . . . Figure 3.
The left plot shows the function f from Section 3 to-gether with the interest rate r as dashed line. The right plot showsthe graph of Φ together with the graph of Θ (cid:55)→ − Θ as dashed line.First, we consider the Esscher martingale measure from Section 3. The left plotin Figure 3 shows the function f : [ − λ − , λ + − → R on the interval [ − , ,together with the interest rate r as dashed line. As this plot indicates, condition(3.3) is fulfilled and the solution of equation (3.4) is given by Θ = − . . (7.9)Therefore, the Esscher martingale measure P Θ exists. Alternatively, this can beseen by inspecting the right plot in Figure 3, which shows the function Φ fromProposition 4.4 on the interval [ − , , together with the graph of Θ (cid:55)→ − Θ asdashed line. The graph of Φ represents all martingale measures P ( θ, Φ( θ )) whichpreserve the class of tempered stable processes, and the dashed line represents allEsscher transforms P Θ = P (Θ , − Θ) . Therefore, the intersection point corresponds tothe just determined Esscher martingale measure P Θ . −2.0 −1.5 −1.0 −0.5 . . . . −2.0 −1.5 −1.0 −0.5 . . . Figure 4.
The left plot shows the relative entropies on the interval [ − , , where the x -axis is θ and the y -axis is H ( P ( θ, Φ( θ )) | P ) . Theright plot shows the -distances on the interval [ − , , where the x -axis is θ and the y -axis is H ( P ( θ, Φ( θ )) | P ) .Next, we treat the existence of the minimal bilateral Esscher martingale mea-sures from Section 4. The left plot in Figure 4 shows the relative entropies θ (cid:55)→ H ( P ( θ, Φ( θ )) | P ) on the interval [ − , ; it indicates that the minimal entropy mar-tingale measure within the class of bilateral Esscher transforms is attained for θ = − . . (7.10) XPONENTIAL STOCK MODELS DRIVEN BY TEMPERED STABLE PROCESSES 15
The right plot in Figure 4 shows the -distances θ (cid:55)→ H ( P ( θ, Φ( θ )) | P ) on the interval [ − , ; it indicates that the variance-optimal martingale measure within the classof bilateral Esscher transforms is attained for θ = − . . (7.11) − . − . − . − . . . . Figure 5.
The function r a (cid:55)→ c ( r a ) defined according to (7.12)on the interval [0 , . , where the x -axis is the annualized interestrate r a . The shaded area indicates the values of r a for which theFS minimal martingale measure exists.Finally, we treat the existence of the FS minimal martingale measure from Sec-tion 5. For this purpose, it will be useful to consider the annualized interest rate r a . Figure 5 shows the function r a (cid:55)→ c ( α + , α − , β + , β − , λ + , λ − , r a / , q ) (7.12)defined according (5.1) with varying annualized interest rate r a on the interval [0 , . . According to Theorem 5.1, the FS minimal martingale measure exists ifand only if − ≤ c ≤ , that is, the values of c belong to the shaded area inFigure 5. We see that the FS minimal martingale measure exists if and only if . ≤ r a ≤ . , that is, the annual interest rate is between . and . . In particular, in our model with an annual interest rate of the FSminimal martingale measure does not exist.In the sequel, we shall illustrate our results from Section 6 concerning optionpricing formulae.The left plot in Figure 6 shows the prices of European call options with currentstock price S = 7500 , date of maturity T = 2 and strike prices K varying from to . We have computed these prices with the minimal entropy martingalemeasure, i.e. with formula (6.4), where θ = θ is given by (7.10), and where themodel parameters are given by (7.6) and (7.8). The right plot in Figure 6 showsthe difference between these prices and the corresponding Black Scholes prices.Figure 7 shows the implied volatility surface with current stock price S = 7500 ,maturity dates T varying from to , and strike prices K varying from to . For this procedure, we have computed option prices with the minimalentropy martingale measure, i.e. with formula (6.4), where θ = θ is given by(7.10), and where the model parameters are given by (7.6) and (7.8), and invertedthe Black Scholes formula for the standard deviation σ . We observe a volatilitysmile for T = 2 , which flattens out for longer times of maturity and converges to thestandard deviation σ of the Black Scholes model, which we have estimated in (7.5). − − Figure 6.
The left plot shows the prices of European call optionswith S = 7500 , T = 2 and K ∈ [7000 , , where the x -axis isthe strike price K , computed with the minimal entropy martingalemeasure. The right plot shows these prices minus the correspondingBlack Scholes prices.For T = 10 the implied volatility curve shown in Figure 7 behaves almost like theconstant function which is equal to σ estimated in (7.5); this flat behaviour does notchange for larger times of maturity T . Our empirical observation is not surprising,as we have shown in [23, Theorem 4.10] that, for a tempered stable process X anda Brownian motion W with the same mean and variance, the distributions of X t and W t are close to each other for large time points t ; see also [29, Theorem 3.1.ii]for an investigation of the long time behaviour of tempered stable processes.It is well known that, when estimating the model parameters, reasonable con-fidence intervals for the mean µ can only be achieved for a very large number ofobservations. Therefore, it is important that the model behaves stable with respectto calibration errors. In order to demonstrate the stability of our pricing rules,we have computed option prices for various values of the mean µ = m and thestandard deviation σ = (cid:112) m − m . For this procedure, we have calculated therespective parameters α + , α − , λ + , λ − > by solving the system of equations (7.7)with m , m given by (7.3), (7.4) and β ∈ (0 , given by (7.6), and computed the M a t u r i t y T S t r i k e P r i c e K I m p li ed V o l a t ili t y Figure 7.
The implied volatility surface computed with the min-imal entropy martingale measure. The parameters for the call op-tion are S = 7500 , T ∈ [2 , and K ∈ [7000 , . XPONENTIAL STOCK MODELS DRIVEN BY TEMPERED STABLE PROCESSES 17 M ean −5e−04 0e+00 5e−04 1e−03 S t anda r d D e v i a t i on O p t i on P r i c e Figure 8.
Option prices for the mean µ ∈ [ − . , . andthe standard deviation σ ∈ [0 . , . . The parameters forthe call option are S = 7500 , T = 10 and K = 7700 .option prices with the minimal entropy martingale measure, i.e. with formula (6.4)and θ = θ . Figure 8 shows the computed option prices for µ varying from − . to . , and σ varying from . to . , with current stock price S = 7500 ,date of maturity T = 10 and strike price K = 7700 . The surface behaves locallyflat and shows that the model is stable with respect to minor calibration errors.8. Conclusion
In this paper, we have provided a systematic analysis of the existence of equiv-alent martingale measures for exponential stock price models driven by temperedstable processes, under which the computation of option prices remains analyticallytractable.In this section, we shall review our results and provide a comparison with theresults derived in [28]. The textbook [28] deals with financial models driven byseveral types of tempered stable processes. Its studies encompass the CTS, GTS,KRTS, MTS, NTS, and RDTS processes. We refer to [28] for further details, butpoint out that the tempered stable distributions considered in this paper correspondto the generalized classical tempered stable (GTS) distributions with mean m = Γ(1 − β + ) α + ( λ + ) − β + − Γ(1 − β − ) α − ( λ − ) − β − , (8.1)see formula (3.4) on page 68 in [28], which seems to have a small typo. Note thatthe calculation of the mean in (8.1) is also consistent with formula (2.12) in [23].As pointed out in Remark 4.5, all measure transformations preserving the classof tempered stable processes are bilateral Esscher transforms. This is due to theresult that for X ∼ TS( α +1 , β +1 , λ +1 ; α − , β − , λ − ) under a probability measure P and X ∼ TS( α +2 , β +2 , λ +2 ; α − , β − , λ − ) under another probability measure Q , the measures P and Q are equivalent if andonly if α +1 = α +2 , α − = α − , β +1 = β +2 and β − = β − . In Section 5.3.3 in [28], such aresult has also been shown for GTS-processes, and the characteristic triplet of thelogarithm of the Radon-Nikodym derivative has been determined. Using bilateral Esscher transforms, we have investigated several martingale mea-sures under which the driving process remains a tempered stable process. Thesemartingale measures have been the Esscher martingale measure in Section 3 (whichlater turned out to be a special case of bilateral Esscher martingale measures), andthe minimal entropy martingale measure as well as the p -optimal martingale mea-sure in Section 4. Furthermore, we have provided a criterion for the existence of theFöllmer Schweizer minimal martingale measure in Section 5. In case of existence,the driving process turned out to be the sum of two independent tempered stableprocesses under the new measure, thus providing an analytically tractable model.In Section 6, we have provided the option pricing formulae (6.3)–(6.5), whichapply to the martingale measures that we have studied in the aforementioned sec-tions. These formulae are based on Fourier transform techniques and follow from aresult in [24], which has also been provided in [5]. An option pricing formula of thiskind can also be found in Section 7.5 in [28]; see formula (7.10) on page 152.In our case study in Section 7, we have estimated the parameters of the temperedstable process from historical data of the German stock index DAX. Based onour previous results, we have determined appropriate martingale measures andhave used these in order to compute option prices and implied volatility surfaces.In Section 7.5.2 in [28], the authors have proceeded differently. Namely, they donot consider the real-world probability measure, they rather calibrate the risk-neutral parameters of the tempered stable process from available option price data.Thus, they assume that the driving process is also a tempered stable process underthe martingale measure, which means that the martingale measure is a bilateralEsscher martingale measure. However, comparing our Figure 7 with Figure 7.1 onpage 154 in [28], we observe similar results concerning the implied volatility surfaces:For short maturity dates we have a volatility smile, which flattens out for longermaturities.The class of bilateral Gamma distributions, which occurs for β + = β − = 0 ,is a limiting case within the class of tempered stable distributions. Stock pricemodels driven by bilateral Gamma processes have been examined in [22]. Comparingour results from this paper with those from [22], we see that for tempered stableprocesses with β + , β − ∈ (0 , we obtain more restrictive conditions concerningthe existence of appropriate martingale measures than for driving bilateral Gammaprocesses. This is not surprising, as our investigations in [23] have shown that, inmany respects, the properties of bilateral Gamma distributions differ from those ofall other tempered stable distributions. Acknowledgement
The authors are grateful to two anonymous referees for valuable comments andsuggestions.
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E-mail address : [email protected] Leibniz Universität Hannover, Institut für Mathematische Stochastik, Welfen-garten 1, D-30167 Hannover, Germany
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