MEANVARIANCE HEDGING WITH RANDOM VOLATILITY JUMPS


 Rafe Andrews
 2 years ago
 Views:
Transcription
1 MEANVARIANCE HEDGING WIH RANDOM VOLAILIY JUMPS PAOLO GUASONI AND FRANCESCA BIAGINI Abstract. We introduce a general framework for stochastic volatility models, with the risky asset dynamics given by: dx t ω, η = µ t ηx t ω, ηdt + σ t ηx t ω, ηdw t ω where ω, η Ω H, F Ω F H, P Ω P H. In particular, we allow for random discontinuities in the volatility σ and the drift µ. First we characterize the set of equivalent martingale measures, then compute the meanvariance optimal measure P, using some results of Schweizer on the existence of an adjustment process β. We show examples where the risk premium λ = µ r σ follows a discontinuous process, and make explicit calculations for P. Introduction he introduction of the meanvariance approach for pricing options under incomplete information is due to Föllmer and Sondermann [7], who first proposed the minimization of quadratic risk. heir work, as well as that of Bouleau and Lamberton, focused on the case when the price of the underlying asset is a martingale. he more general semimartingale case was considered by: Duffie and Richardson [6], Schweizer [18, 19, 2, 21], Hipp [1], Monat and Stricker [13], Schäl [17], and a definitive solution was provided by Schweizer and Rheinländer [16], and Gourieroux, Laurent, and Pham [14], with different methods. In the meantime, the random behavior of volatility turned out to be a major issue in applied option pricing, and Hull and White [11], Stein and Stein [22] and Heston [9] proposed different models with stochastic volatility. In fact, such models are special cases of incomplete information, and can be effectively embedded in the theoretical framework developed by mathematicians. A particularly appealing feature of meanvariance hedging is that European options prices are calculated as the expectations of their respective payoff under a possibly signed martingale measure P, introduced by Schweizer [21]. he optimal strategy can also be found in terms of this measure, therefore it is not surprising that considerable effort has been devoted to its explicit calculation. 1
2 In this paper, we address the problem of calculating P in presence of volatility jumps or, more generally, when the socalled market price of risk λ = µ r follows a possibly discontinuous process. σ We consider the following market model, where each state of nature ω, η belongs to the product space Ω H, endowed with the product measure P Ω P H : 1 { dst ω, η = µ t ηs t ω, ηdt + σ t ηs t ω, ηdw t ω t B t = exp r sds and r is a constant. We assume the existence on H of a set A of martingales with the representation property: this somewhat technical condition is in fact satisfied in most models present in the literature. First we characterize martingale measures in terms of A, then, exploiting some results of Schweizer, we isolate the density of the meanvariance optimal martingale measure. When volatility follows a diffusion process such as in the Heston or Hull and White models, only to mention two of them, this result was already obtained by Laurent and Pham with stochastic control arguments. It turns out that the jump size distribution is a critical issue: in fact, finite distributions are easily handled by n martingales, where n is the cardinality of the jump size support. On the contrary, an infinite distribution for the jump size requires a more general approach involving random measures. Moreover we show calculations for sample models where volatility jumps are random both in size and in time of occurrence. For all of them, we calculate the density of the meanvariance optimal measure, and the law of the jumps under P. In the last section, we calculate the meanvariance hedging strategy for a call option, exploiting the change of numeraire technique of El Karoui, Geman and Rochet [8], as well as the general formula in feedback form of Rheinländer and Schweizer [16]. 1. Preliminaries For all standard definitions on stochastic processes, we refer to Dellacherie and Meyer [4] Stopping imes. We recall here some definitions and properties of stopping times, which will come useful in the sequel. In this subsection, we assume that all random variables are defined on some probability space Ω, F, P. 2
3 Proposition 1.1. Let τ be a realvalued, Borel random variable, and F τ the smallest filtration under which τ is a stopping time. We have that: 2 F τ t = τ 1 B[, t] τ 1 t, where BA is the family of Borel subsets of A. Proof. By definition, F τ t = σ{τ s}, s t. Since F τ t is a σfield and contains all sublevels of τ within [, t], it necessarily contains the inverse images of all Borel sets of [, t]. he set τ 1 t, is the complement of τ 1 [, t], and belongs to the σfield. he reverse inclusion is trivial. Finally, the righthand side in 2 is easily seen to be a σfield. Remark 1.2. An immediate consequence of Proposition 1.1 is the rightcontinuity of Ft τ. Moreover, Ft τ = τ 1 B[, t τ 1 [t,. his means that the augmentation of Ft τ is continuous if and only if the law of τ is diffuse. Corollary 1.3. he filtration generated by the random variable τ t coincides with F τ if and only if τ is an optional time, that is, if {τ < t} F t for all t. Definition 1.4. A stopping time τ is totally inaccessible if it is strictly positive and for every increasing sequence of stopping times τ 1,..., τ n, such that τ n < τ for all n, P lim n τ n = τ, τ < =. otally inaccessible stopping times can be characterized as follows: Proposition 1.5. Let τ a strictly positive stopping time. he following properties are equivalent: i τ is totally inaccessible; ii here exists a uniformly integrable martingale M t, continuous outside the graph of τ such that M = and M τ = 1 on {τ < }. Proof. See Dellacherie and Meyer [5], VI.78. Proposition 1.6. If the law of a stopping time is diffuse, then it is totally inaccessible with respect to F τ. Proof. See Dellacherie and Meyer [4], IV.17. Definition 1.7. Let A be an adapted process, with A = and locally integrable variation. he compensator of A is defined as the unique predictable process Ã such that A Ã is a local martingale. Remark 1.8. In particular, the compensator of an increasing process is itself an increasing predictable process. Proposition 1.9. Let τ be a stopping time with a diffuse law. hen the compensator of the process 1 {τ t} with respect to F τ is given by where F x = P τ x. Ã t = log1 F t τ 3
4 Proof. By assumption, M t = 1 {τ t} Ãt is a local martingale, and by Remark 1.8, Ã t is an increasing process. herefore we have: sup M s 1 s t and M t is in fact a martingale see for instance Protter, page 35, heorem 47. We look for a compensator of the form Ãt = a t τ, with a : R + R +. Hence: It follows that: E [ 1 {τ s} 1 {τ t} Ft ] = E [as τ a t τ F t ] P t < τ s P t < τ s t = E [ 1 {t<τ s} a τ a t + 1 {t<τ} a s a t ] P t < τ his equality can be rewritten as an integral equation: F s F t = s t a x a t df x + s a s a t df x It is easy to check by substitution that the unique solution to this integral equation is given by a x = log1 F x. By the uniqueness, this is the only compensator of 1 {τ t} Martingale Representation. We denote by M 2 the set of squareintegrable martingales on a probability space Ω, F, P endowed with a filtration F t satisfying the usual hypotheses. Definition 1.1. A set A = {X 1,... X n } M 2 is strongly orthogonal if X i X j M 2 for i, j {1... n}. Definition A set A of strongly orthogonal martingales X 1,..., X n has the representation property if I = M 2, where { n } t I = X : X t = HsdX i s i for all H predictable such that [ t ] E Hs i 2 d[x i ] s < 1 i n t i=1 In this paper we will need the following result: Proposition Let A = X 1,..., X n a set of strongly orthogonal martingales. If there is only a martingale measure for A, then A has the representation property. Proof. See Dellacherie and Meyer [5], VIII.58 or Protter [15], IV.37. 4
5 1.3. Random measures. For completeness, we provide some basic definitions on random measures: we refer to [12] for a complete treatment of the subject. Let H, F, P be a probability space. Definition A random measure on R + R is a family ν = νη; dt, dx : η H of nonnegative measures on R + R, BR + BR, satisfying νη; {} R = identically. We set H = H R + R, with σfields Õ = O BR and P = P BR, where O and P are respectively the optional and predictable σfield on H R +. Definition For any Õmeasurable function hη; t, x we define the integral process h ν as hη, s, xνη; ds, dx if hη, s, x νη; ds, dx < h ν t η = [,t] H [,t] H + otherwise Analogously, a random measure ν is optional resp. predictable if the process h ν is optional resp. predictable for every optional resp. predictable function h. Definition An integervalued random measure is a random measure that satisfies: i νη; {t} R 1 identically; ii for each A BR + BR, ν, A take its values in N; iii ν is optional; iv Pσfinite, i.e. there exists a Pmeasurable partition A n of H such that 1 An ν is integrable. By proposition 1.14 of [12] we obtain the following characterization of integervalued random measures. Proposition If ν is an integervalued random measure, there exists a random set D = [ n ], where [ n ] n N is the sequence of graphs of stopping times n, and a real optional process β such that νη; dt, du = Σ s 1 D η, sɛ s,βs ηdt, du If n n N satisfies [ n ] [ m ] = for n m, it is called an exhausting sequence for D. Proposition Let X be an adapted càdlàg R d valued process. hen: ν X η; dt, dx = s 1 { Xs }ɛ {s, Xs η}dt, dx defines an integervalued random measure on R + R d. 5
6 Proof. See [12], Chapter II, Proposition Definition A multivariate point process is an integervalued random measure ν on R + R such that νη; [, t] R < for all η, t R +. Proposition Let ν be an optional Pσfinite random measure. here exists a random measure ν p, called compensator of ν, and unique up to a P null set, such that: i ν p is a predictable random measure; ii E [h ν p ] = E [h ν ] for every nonnegative P measurable function h on Ω. Also, if h ν is increasing and locally integrable, then h ν ν p is a local martingale. Proof. See heorem [12], Chapter II, heorem 1.8. Let ν = Σ s 1 D η, sɛ s,βsηdt, du be an integer random measure and ν p its compensator. Definition We denote by G loc ν the set of all Pmeasurable realvalued function h such that the process h t η = hη, t, βη1 D η, t hη, s, xνη; s dx has the property that [Σ s. h s 2 ] 1 2 is locally integrable and increasing. 2. If h G loc ν we call stochastic integral with respect to ν ν p and denote by h ν ν p any purely discontinuous local martingale X such that h and X are indistinguishable. Definition A compensated random measure ν ν p has the representation property on H iff every square integrable martingale M t on H, F H, P H can be written as for some Pmeasurable process k. H M t = M + k ν ν p t We recall that from proposition 1.28, first chapter of [12] follows that h ν ν p t = h ν t h ν p t for every Pmeasurable process h such that h ν t is a locally integrable increasing process. Consequently, we obtain in theorem 1.21 that the integral h ν ν p, which is apriori stochastic, coincides with the Stieltjes integral h ν h ν p. It follows that every local martingale on H has finite variation. heorem Assume that ν is a multivariate point process. hen, ν ν p has the representation property with respect to the smallest filtration under which ν is optional. Proof. See theorem 4.37 of [12]. 6
7 2. he Market Model We introduce here a simple model for a market with incomplete information. We have two complete filtered probability spaces: Ω, F Ω, Ft Ω, P Ω and H, F H, Ft H, P H. We denote by W t a standard Brownian Motion on Ω, and assume that Ft Ω is the P Ω H augmentation of the filtration generated by W. Our set of states of nature is given by the product space Ω H, F Ω F H, P Ω P H. We have a riskfree asset B t and a discounted risky asset X t = S t B t, with the following dynamics: { 3 dx t ω, η = µ t η r t X t ω, ηdt + σ t ηx t ω, ηdw t ω B t = exp t r sds where r is a deterministic function of time. We assume that the equation for X admits P H a.e. a unique strong solution with respect to the filtration F W or, equivalently, that there exists a unique strong solution with respect to the filtration F t = Ft W F H. his is satisfied under fairly weak assumptions: for example, it is sufficient that µ and σ are P H a.e. bounded. Denoting by F X the filtration generated by X, we also assume that Ft X = Ft W Ft H. In particular, at time all information is revealed through the observation of the process X. he following proposition helps checking whether this condition is satisfied: Proposition 2.1. Let X t be defined as in 3. hen, if i µ t is F X t measurable, ii F H t F X t. then F X t = F W t F H t. Proof. By definition of X t, we immediately have that Ft X Ft Ω Ft H. o see that the reverse inclusion holds, observe first that by 3: 4 W t = t µ s σ s ds + t dx s σ s X s By i, the first term above is Ft X measurable. For the second, note that t n σs, η 2 Xs 2 ds = X ti+1 X ti 2 lim sup i t i+1 t i where the limit holds in probability, uniformly in t. his proves that F Ω t F X t, and by ii the proof is complete. A simpler version of this model was introduced by Delbaen and Schachermayer [3], while it can be found in the above form in Pham, Rheinländer, and Schweizer [14]. 7 i=1
8 In this framework, we study the problem of an agent wishing to hedge a certain European option HX expiring at a fixed time. Hedging performance is defined as the L 2 norm of the difference, at expiration, between the liability and the hedging portfolio. More precisely, we look for a solution to the minimization problem: 5 min E [ HX c R c G θ 2] θ Θ where G t θ = t θ s dx s and Θ = { θ LX, G t θ S 2 P } Here LX denotes the space of Xintegrable predictable processes, and S 2 the space of semimartingales Y decomposable as Y = Y + M + A, where M is a squareintegrable martingale, and A is a process of squareintegrable variation. his problem is generally nontrivial, since the agent has not access to the filtration F, but only to F X. Indeed, Reihnländer and Schweizer [16] and, independently, Gourieroux, Laurent and Pham [14], proved that problem 5 admits a unique solution for all H L 2 P, under the standing hypothesis: CL G Θ is closed Moreover, in [21] it is shown that the option price, i.e. the optimal value for c, and the optimal hedging strategy θ can be computed in terms of P, the varianceoptimal martingale measure. In fact, the option price is given by c = Ẽ [H]. Definition 2.2. We define the sets of signed martingale measures M 2 s, and of equivalent martingale measures M 2 e: { M 2 s = Q P : dq } dp L2 P, X t is a Qlocal martingale M 2 e = { Q M 2 s : Q P : } Definition 2.3. he variance optimal martingale measure is the unique solution P if it exists to the minimum problem: [ dq ] 2 6 min E Q M 2 s dp If M 2 s is nonempty, then P always exists, as it is the minimizer of the norm in a convex set: the problem is that it may not be positive definite, thereby leading to possibly negative option prices. However, if X t has continuous paths, and under the standard assumption NA M 2 e 8
9 Delbaen and Schachermayer [3] have shown that P M 2 e. Since we are dealing with continuous processes, and we will always assume NA, we need not worry about this issue. In this paper, using a representation formula from [1], we compute explicitly P for some sample models. 3. Representation of Martingale Measures Here we summarize some of the main results of [1] in order to show that the set M 2 e of equivalent martingale measures admits a convenient representation. We denote by λ t = µ t r t of risk. σ t the socalled market price heorem 3.1. If there exists a ndimensional martingale M on H such that: i [M i, M j ] for all i j; ii M has the representation property for Ft H ; then we have for every Q M 2 e: dq dp = E λ t ηdw t E k t ω, ηdm t where k t is such that E λ tηdw t E k t tω, ηdm t is a square t integrable martingale and k t M t > 1. By heorem 3.1, a martingale measure is uniquely determined by the process k which appears in its representation. In particular, k = corresponds to the minimal martingale measure ˆP introduced by Föllmer and Schweizer [7]. he next proposition identifies P. Proposition 3.2. In the same assumptions as heorem 3.1, we have: d P dp = E λ t dw t E k t ηdm t where k t is a solution of the following equation exp E k λ2 t ηdt t ηdm t = [ E exp ] λ2 t ηdt such that E λ tdw t E k t t ηdm t is a square integrable martingale. t Remark 3.3. Proposition 3.2 shows how the change of measure works on Ω and H. In fact, provided that ˆP exists, we can write: d P dp = d P d ˆP d ˆP dp 9
10 where d ˆP dp = E λ t dw t and d P d ˆP = E exp [ exp λ2 t ηdt ] λ2 t ηdt Since in our model d P does not depend on ω, we have: d ˆP d P H dp H = E [ d P dp ] [ F d H = E P ] d ˆP d ˆP dp F H = d P [ d ˆP E d ˆP dp ] F H = d P d ˆP his provides a rule of thumb for changing measure from P to P via ˆP. First change P to ˆP by a direct use of Girsanov theorem: this amounts to replacing µ with r in 3, and is the key of riskneutral valuation. P H is not affected by this step. In principle, one could repeat the same argument from ˆP to P, but this involves calculating the k t η. As we show with an example in the last section, this task may prove hard even in simple cases. A more viable alternative is calculating P H with the above formula. his avoids dealing with k directly, although its existence is still needed through Proposition 3.2. A sufficient condition for the existence of ˆP is the Novikov condition, namely: [ 1 ] E exp λ 2 t dt < 2 and is satisfied by all the examples in the last section. 4. Volatility jumps and Random Measures Although most markets models considered in the literature can be embedded in a framework consistent with heorem 3.1, there are some remarkable exceptions. For example, continuously distributed jumps in volatility can generate filtrations where no finite set of martingales has the representation property see the examples in the next section. In these cases, we can still represent martingales in terms of integrals with respect to a compensated random measure ν ν p, thereby obtaining an analogous of heorem 3.1 and Proposition 3.2. Note that the following results are complementary to those in the previous section, but do not directly generalize them: in fact any model with volatility following a diffusion process is covered in the previous section, and not in the present one. 1
11 heorem 4.1. If there exists a compensated, integervalued, random measure ν ν p on E R + R such that: i F H coincides with the smallest filtration under which ν is optional; ii ν ν p has the representation property on H, F H, P H. hen we have for every Q M 2 e: dq dp = E λ t dw t E k ν ν p where k t is such that k ν t > 1 and E λ tdw t t E k ν νp t is a square integrable martingale. As in the previous section, we need these lemmas: Lemma 4.2. In the same assumptions as heorem 4.1, every square integrable martingale M t on the space Ω E, F Ω F H, P Ω P H with respect to F t can be written as M t = M + t h s dw s + k ν ν p t Proof. Let us denote the set of martingales for which the thesis hold by M. We want to show that M = M 2 Ω H. By representation property, every square integrable martingale M t ω on Ω H depending only on ω belongs to M, since it can be written as M t = M + t h sdw s. Analogously, every N t η belongs to M, since it is of the form N t = N + k ν ν p t, where k is a Pmeasurable process and k ν t is locally integrable. Denoting P t ω, η = N t ηm t ω, we have that P t is a square integrable martingale on Ω H. Setting P t = Pt d + Pt c, where Pt c and Pt d are the continuous and purely discontinuous parts of P, we have that P t = Pt d = M t N t. By 1.2, it follows that Pt d = P d + Mk ν ν p t. Also, by Itô s formula, Pt c = P c + t N sh s dw s. his shows that any linear combination i M iωn i η belongs to M and, by a monotone class argument, it is easy to see that M is dense in M 2. Hence, for every Z t M 2 there exist a sequence Xt n of squareintegrable martingales such that Xt n = X + t hn s dw s + k n ν ν p t. By the identity: [ ] E [X n ] = E h n s 2 ds + E [ ] ks n 2 νs p it follows that h n and k n are Cauchy sequences respectively in [ ] {h t η predictable: E h 2 sds < } and {k t η, x predictable: E [ k 2 s ν p s ] < } Since these spaces are complete, the proof is finished. 11
12 Lemma 4.3. Let Z be a strictly positive, squareintegrable random variable and denote Z t = E [Z F t ]. hen, if Z t = Z + H ν ν p t, we have: H Z t = Z E ν ν p t. Z Proof. If there exists a martingale M t = M + K ν ν P t such that Z t = Z E M t, then it is unique. In fact, if N t = N + H ν ν P t and Z t = Z E N t, we immediately have M t = N t. Since M t and N t are purely discontinuous martingales by definition 1.2, they must coincide up to evanescent sets. In particular, we have that M t = Z t Z t. For all t >, Z t Z t coincides with the jumps of the purely discontinuous martingale H Z ν νp t, therefore M t exists and is given by M t = logz + H Z ν νp t. Proposition 4.4. In the same assumptions as heorem 4.1, we have: d P dp = E λ t dw t E k ν ν p where k t is a solution of the following equation exp E k ν ν p λ2 t ηdt = [ E exp ] λ2 t ηdt such that E λ tdw t k t E ν ν p is a square integrable martingale. t Proof. he proof is formally analogous to that of heorem 1.16 of [1], by the previous results and the representation property of the compensated random measure ν ν p. 5. Examples We now show how the results in the previous sections provide convenient tools for calculating P and thus pricing options in models where volatility jumps. We start with a simple model where jumps occur at fixed times, and can take only two values. We then discuss the more general cases of jumps occurring at stopping times, and with arbitrary distributions Deterministic volatility jumps. In discretetime fashion, the following model was introduced in RiskMetrics Monitor [23] as an improvement of the standard lognormal model for calculating Value at Risk. We set H = {, 1} n and denote η = {a 1,..., a n }. a 1...., a n are Bernoulli IID random variable, so that H is endowed with the product 12
13 measure from {, 1}. Ft H contains all information on jumps up to time t, therefore it is equal to the parts of {a i } ti t. Setting t i = i, the n+1 dynamics of µ and σ is given by: { µ t = 1 { t<t1 }µ + n i=1 1 {t i t<t i+1 }µ ai σ t = 1 { t<t1 }σ + n i=1 1 {t i t<t i+1 }σ ai In fact, all we need for meanvariance hedging is the dynamics for λ: n λ 2 t = λ {ti t<t i+1 }λ 2 a i i=1 It is easy to check that a martingale with the representation property on H is given by: M t = t i t 1 {a i =} p, where p = P a 1 =. We are now ready to see how the change of measure works: in fact, by heorem 3.1, we have that: d P d ˆP = d P H dp H = exp λ2 1 a1 + +a n n exp λ2 n p exp + 1 p exp λ2 n Since the density above can be written as: d P H n λ exp a 2 1 i = 1 a n i λ2 dp H p exp + 1 p exp i=1 λ2 n n a1 + +a n n λ2 1 n λ2 1 n n n it follows that under P the variables a 1,..., a n are still independent, and p is replaced by: p = p p exp λ2 n exp λ2 1 n + 1 p exp λ2 1 n 5.2. Random volatility jumps. Consider the following model, where µ and σ are constant, until some unexpected event occurs. In other words: { µ t = µ 1 1 {t<τ} + µ 2 1 {t τ} σ t = σ 1 1 {t<τ} + σ 2 1 {t τ} In fact, all we need is the dynamics for λ: λ 2 t = λ α1 {t τ} where α = λ 2 2 λ 2 1. he event τ which triggers the jump is a totally inaccessible stopping time. hat is to say, any attempt to predict it by means of previous information is deemed to failure. α represents the jump size, and it may be deterministic or random. We now solve the problem in three cases: α deterministic, α Bernoulli, and α continuously distributed. 13
14 α deterministic. Since our goal is to find the varianceoptimal martingale measure, we start exhibiting a martingale with the representation property for F H, which in this case is the filtration generated by τ. Proposition 5.1. Let τ be a stopping time with a diffuse law, and Ã the compensator of 1 {τ t}. hen the martingale M t = 1 {τ t} Ãt has the representation property. Proof. Let Q be a martingale measure for M, and ÃQ the compensator of 1 {τ t} in Q. Both M t and 1 {τ t} ÃQ are Qmartingales, therefore their difference ÃQ Ã is also a martingale. However, it is also a finite variation process, therefore it must be identically zero. Since ÃQ = Ã, by Proposition 1.9, the c.d.f. s of τ under P and Q are equal. his implies that Q = P, F τ a.e. heorem 1.12 concludes the proof. We now compute P H, that is the law of τ under P. For simplicity, assume that τ has a density, and denote it by f t. We have: f t = d P d ˆP f t = exp { λ2 sηds 1 = f t = exp λ c 1 + αt f t if t < 1 c exp λ c 1 f if t [ where c = E exp ] λ2 sηds. In this simple example we also calculate k t explicitly, although the computational effort required suggests that in more complex situations it may not be a good idea to do so. First, we see how stochastic integrals with respect to M look like. Recall that, by proposition 1.9 Ãt = at τ, where a : R + R +. Lemma 5.2. Let k t be a F τ measurable process. hen we have: k t dm t = k τ τ1 {τ } τ k t tda t Proof. By Corollary 1.3, we have that any F τ measurable process can be written as k t t τ. Hence: k t t τdm t =k τ τ1 {τ } =k τ τ1 {τ } =k τ τ1 {τ } τ τ k t t τda t τ = k t t τda t = k t tda t 14
15 he above lemma shows that k t s needs only be defined for s = t, so from now on we shall unambiguously write k t instead of k t t. Now we can compute k t : Proposition 5.3. k t is the unique solution of the following ODE: { k t = α + a t + αk t + a tkt 2 7 k = exp λ c where c = exp λ 2 2 exp αt df t + exp λ F and F t = P τ t. Proof. By lemma 5.2, and the generalization of Itô s formula for processes with jumps, we have: τ E k t ηdm t = E k τ 1 {τ } k t da t = Hence, by 3.2, we have: = 1 + k τ 1 {τ } exp τ k t da t τ 1 + kτ 1 {τ } exp k t da t = exp λ2 1 τ α [ E exp ] λ2 t dt [ aking logarithms of both sides, and setting c = E exp ] λ2 t dt, we get: ln τ 1 + k τ 1 {τ } k t da t = λ 2 1 τ α ln c Differentiating with respect to τ, for τ we obtain equation 7. Remark 5.4. Equation 7 is a Riccati ODE, and can be solved in terms of the function a t. Depending on the form of a t, explicit solutions may or may not be available α Bernoulli. In this case, α is a Bernoulli random variable, independent of τ, with values {α, α 1 }. We also set A = {α = α }, B = {α = α 1 }, and p = P B. Since the support of α is no longer a single point, a martingale will not be sufficient for representation purposes. In fact, two martingales do the job, as we prove in the following: Proposition 5.5. Let N t = 1 {τ t} 1 B p. hen the set of two martingales {M, N} has the representation property. 15
16 Proof. First we check that M and N are orthogonal. his is easily seen, since MN = Na τ. We now prove that the martingale measure is unique. Let Q be a martingale measure for {M, N}. As shown in the proof of Proposition 5.1, the distribution of τ under Q must be the same as under P. However, we also need that QB = P B, otherwise N would not be a martingale. he change from P to P is a change in the joint law of τ, α. Under P this is a product measure, since τ and α are independent. However, we cannot expect that the same holds under P. For t we have: d P H = 1 A exp λ 2 1 tα + 1 B exp λ 2 1 tα 1 dp H c herefore the law of τ under P is given by: f t = 1 c p exp λ 2 1 tα p exp λ 2 1 tα ft And the conditional law of α with respect to τ is given by: P B τ dt = = p exp λ 2 1 tα 1 p exp λ 2 1 tα p exp λ 2 1 tα In particular, it is immediately seen that α is independent of τ if and only if it degenerates in the previous case. When α is Bernoullian, calculating k involves solving a system of two Riccati ODEs, which is somewhat cumbersome. More generally, if the support of α is made of n points, it is reasonable that n martingales are required for representation purposes. As a result, the values of k would be the solutions of a system of n ODEs α continuously distributed. In this case the support of α is an infinite set, therefore heorem 3.1 is no longer applicable. In fact we need its random measure analogous, given by heorem 4.1. If the filtration F λ generated by λ t coincides with the one generated by µ t and σ t, we can assume F H = F λ. By Proposition 1.17, there exists a random measure ν associated to λ, and given by: νη; dt, dx = ɛ {τ,αη} dt, dx Since this is a multivariate point process, and F H coincides with the smallest filtration under which ν is optional, by heorem 1.22 the compensated measure ν ν p has the representation property on H. Also, k ν ν p is a purely discontinuous martingale for all Pmeasurable processes k, therefore [W, k ν ν p ] =. his means that the assumptions of heorem 4.1 are satisfied, and P is given by Proposition
17 Suppose that α has a density, say gx. We have: d P H = ht, x = exp λ2 1 x t dp H c Denoting by jt, x and jt, x the joint densities of τ, α under P and P respectively, we have: jt, x = ht, xjt, x = ht, xftgx ft = ft ht, xgxdx gx t = jt, x ft = ht, xgx ht, xgxdx If α N δ, v, the density of τ under P is given by: + 1 f t = f t c exp λ 2 1 α t α δ n dα = v = 1 λ c exp 2 1 δ t t2 v where nx is the standard normal density function. It is easy to check that the conditional density of α given τ is of the form: gx t = 1 exp 1 x α + tv 2 2πv 2 v herefore α is conditionally normal under P, with distribution N δ + tv, v. Remark 5.6. In the specific case of λ t being normally distributed, it can be shown that G Θ is not closed, therefore Proposition 3.2 does not hold. However, in [1] the above formula is proved relaxing this assumption. It is also shown that in this example G Θ is closed if and only if the support of α is bounded from above Multiple random jumps. Leaving α deterministic for simplicity, we now study the following model: n λ 2 t = λ {t τi }α i i=1 We assume that τ i+1 τ i are IID random variables, with common density fx. Denote by H the space [, ] n, endowed with the image measure of the mapping τ 1,..., τ n τ 1,..., τ n, and with the natural filtration F t generated by {τ 1 t,..., τ n t}. A martingale with the predictable representation property is given by M t = n i=1 1 {t τ i }. In this case, the density of P is given by: d P H = exp λ2 dp H 17 n i=1 α i τ i c f t
Riskminimization for life insurance liabilities
Riskminimization for life insurance liabilities Francesca Biagini Mathematisches Institut Ludwig Maximilians Universität München February 24, 2014 Francesca Biagini USC 1/25 Introduction A large number
More informationARBITRAGEFREE OPTION PRICING MODELS. Denis Bell. University of North Florida
ARBITRAGEFREE OPTION PRICING MODELS Denis Bell University of North Florida Modelling Stock Prices Example American Express In mathematical finance, it is customary to model a stock price by an (Ito) stochatic
More informationSensitivity analysis of utility based prices and risktolerance wealth processes
Sensitivity analysis of utility based prices and risktolerance wealth processes Dmitry Kramkov, Carnegie Mellon University Based on a paper with Mihai Sirbu from Columbia University Math Finance Seminar,
More informationMathematical Finance
Mathematical Finance Option Pricing under the RiskNeutral Measure Cory Barnes Department of Mathematics University of Washington June 11, 2013 Outline 1 Probability Background 2 Black Scholes for European
More informationOptimal Investment with Derivative Securities
Noname manuscript No. (will be inserted by the editor) Optimal Investment with Derivative Securities Aytaç İlhan 1, Mattias Jonsson 2, Ronnie Sircar 3 1 Mathematical Institute, University of Oxford, Oxford,
More informationSensitivity analysis of European options in jumpdiffusion models via the Malliavin calculus on the Wiener space
Sensitivity analysis of European options in jumpdiffusion models via the Malliavin calculus on the Wiener space Virginie Debelley and Nicolas Privault Département de Mathématiques Université de La Rochelle
More information4: SINGLEPERIOD MARKET MODELS
4: SINGLEPERIOD MARKET MODELS Ben Goldys and Marek Rutkowski School of Mathematics and Statistics University of Sydney Semester 2, 2015 B. Goldys and M. Rutkowski (USydney) Slides 4: SinglePeriod Market
More informationHedging of Life Insurance Liabilities
Hedging of Life Insurance Liabilities Thorsten Rheinländer, with Francesca Biagini and Irene Schreiber Vienna University of Technology and LMU Munich September 6, 2015 horsten Rheinländer, with Francesca
More informationIntroduction to ArbitrageFree Pricing: Fundamental Theorems
Introduction to ArbitrageFree Pricing: Fundamental Theorems Dmitry Kramkov Carnegie Mellon University Workshop on Interdisciplinary Mathematics, Penn State, May 810, 2015 1 / 24 Outline Financial market
More informationCOMPLETE MARKETS DO NOT ALLOW FREE CASH FLOW STREAMS
COMPLETE MARKETS DO NOT ALLOW FREE CASH FLOW STREAMS NICOLE BÄUERLE AND STEFANIE GRETHER Abstract. In this short note we prove a conjecture posed in Cui et al. 2012): Dynamic meanvariance problems in
More informationNonArbitrage and the Fundamental Theorem of Asset Pricing: Summary of Main Results
Proceedings of Symposia in Applied Mathematics Volume 00, 1997 NonArbitrage and the Fundamental Theorem of Asset Pricing: Summary of Main Results Freddy Delbaen and Walter Schachermayer Abstract. The
More informationFair Valuation and Hedging of Participating LifeInsurance Policies under Management Discretion
Fair Valuation and Hedging of Participating LifeInsurance Policies under Management Discretion Torsten Kleinow Department of Actuarial Mathematics and Statistics and the Maxwell Institute for Mathematical
More informationProperties of BMO functions whose reciprocals are also BMO
Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a nonnegative BMOfunction w, whose reciprocal is also in BMO, belongs to p> A p,and
More informationMoreover, under the risk neutral measure, it must be the case that (5) r t = µ t.
LECTURE 7: BLACK SCHOLES THEORY 1. Introduction: The Black Scholes Model In 1973 Fisher Black and Myron Scholes ushered in the modern era of derivative securities with a seminal paper 1 on the pricing
More information7: The CRR Market Model
Ben Goldys and Marek Rutkowski School of Mathematics and Statistics University of Sydney MATH3075/3975 Financial Mathematics Semester 2, 2015 Outline We will examine the following issues: 1 The CoxRossRubinstein
More informationAsian Option Pricing Formula for Uncertain Financial Market
Sun and Chen Journal of Uncertainty Analysis and Applications (215) 3:11 DOI 1.1186/s446715357 RESEARCH Open Access Asian Option Pricing Formula for Uncertain Financial Market Jiajun Sun 1 and Xiaowei
More informationA spot price model feasible for electricity forward pricing Part II
A spot price model feasible for electricity forward pricing Part II Fred Espen Benth Centre of Mathematics for Applications (CMA) University of Oslo, Norway Wolfgang Pauli Institute, Wien January 1718
More informationLecture 8: Random Walk vs. Brownian Motion, Binomial Model vs. LogNormal Distribution
Lecture 8: Random Walk vs. Brownian Motion, Binomial Model vs. Logormal Distribution October 4, 200 Limiting Distribution of the Scaled Random Walk Recall that we defined a scaled simple random walk last
More informationLOGNORMAL MODEL FOR STOCK PRICES
LOGNORMAL MODEL FOR STOCK PRICES MICHAEL J. SHARPE MATHEMATICS DEPARTMENT, UCSD 1. INTRODUCTION What follows is a simple but important model that will be the basis for a later study of stock prices as
More informationThe BlackScholes Formula
FIN40008 FINANCIAL INSTRUMENTS SPRING 2008 The BlackScholes Formula These notes examine the BlackScholes formula for European options. The BlackScholes formula are complex as they are based on the
More informationInvariant Option Pricing & Minimax Duality of American and Bermudan Options
Invariant Option Pricing & Minimax Duality of American and Bermudan Options Farshid Jamshidian NIB Capital Bank N.V. FELAB, Applied Math Dept., Univ. of Twente April 2005, version 1.0 Invariant Option
More informationJungSoon Hyun and YoungHee Kim
J. Korean Math. Soc. 43 (2006), No. 4, pp. 845 858 TWO APPROACHES FOR STOCHASTIC INTEREST RATE OPTION MODEL JungSoon Hyun and YoungHee Kim Abstract. We present two approaches of the stochastic interest
More informationThe Effect of Management Discretion on Hedging and Fair Valuation of Participating Policies with Maturity Guarantees
The Effect of Management Discretion on Hedging and Fair Valuation of Participating Policies with Maturity Guarantees Torsten Kleinow HeriotWatt University, Edinburgh (joint work with Mark Willder) Marketconsistent
More information第 9 讲 : 股 票 期 权 定 价 : BS 模 型 Valuing Stock Options: The BlackScholes Model
1 第 9 讲 : 股 票 期 权 定 价 : BS 模 型 Valuing Stock Options: The BlackScholes Model Outline 有 关 股 价 的 假 设 The BS Model 隐 性 波 动 性 Implied Volatility 红 利 与 期 权 定 价 Dividends and Option Pricing 美 式 期 权 定 价 American
More informationOn BlackScholes Equation, Black Scholes Formula and Binary Option Price
On BlackScholes Equation, Black Scholes Formula and Binary Option Price Abstract: Chi Gao 12/15/2013 I. BlackScholes Equation is derived using two methods: (1) riskneutral measure; (2)  hedge. II.
More informationFINANCIAL MARKETS WITH SHORT SALES PROHIBITION
FINANCIAL MARKETS WITH SHORT SALES PROHIBITION by Sergio Andres Pulido Nino This thesis/dissertation document has been electronically approved by the following individuals: Protter,Philip E. (Chairperson)
More informationBubbles and futures contracts in markets with shortselling constraints
Bubbles and futures contracts in markets with shortselling constraints Sergio Pulido, Cornell University PhD committee: Philip Protter, Robert Jarrow 3 rd WCMF, Santa Barbara, California November 13 th,
More informationMonte Carlo Methods in Finance
Author: Yiyang Yang Advisor: Pr. Xiaolin Li, Pr. Zari Rachev Department of Applied Mathematics and Statistics State University of New York at Stony Brook October 2, 2012 Outline Introduction 1 Introduction
More informationThe BlackScholes Model
The BlackScholes Model Liuren Wu Zicklin School of Business, Baruch College Options Markets (Hull chapter: 12, 13, 14) Liuren Wu The BlackScholes Model Options Markets 1 / 19 The BlackScholesMerton
More informationIn which Financial Markets do Mutual Fund Theorems hold true?
In which Financial Markets do Mutual Fund Theorems hold true? W. Schachermayer M. Sîrbu E. Taflin Abstract The Mutual Fund Theorem (MFT) is considered in a general semimartingale financial market S with
More informationThe BlackScholesMerton Approach to Pricing Options
he BlackScholesMerton Approach to Pricing Options Paul J Atzberger Comments should be sent to: atzberg@mathucsbedu Introduction In this article we shall discuss the BlackScholesMerton approach to determining
More informationLOGTYPE MODELS, HOMOGENEITY OF OPTION PRICES AND CONVEXITY. 1. Introduction
LOGTYPE MODELS, HOMOGENEITY OF OPTION PRICES AND CONVEXITY M. S. JOSHI Abstract. It is shown that the properties of convexity of call prices with respect to spot price and homogeneity of call prices as
More informationON LIMIT LAWS FOR CENTRAL ORDER STATISTICS UNDER POWER NORMALIZATION. E. I. Pancheva, A. GacovskaBarandovska
Pliska Stud. Math. Bulgar. 22 (2015), STUDIA MATHEMATICA BULGARICA ON LIMIT LAWS FOR CENTRAL ORDER STATISTICS UNDER POWER NORMALIZATION E. I. Pancheva, A. GacovskaBarandovska Smirnov (1949) derived four
More informationOn the decomposition of risk in life insurance
On the decomposition of risk in life insurance Tom Fischer HeriotWatt University, Edinburgh April 7, 2005 fischer@ma.hw.ac.uk This work was partly sponsored by the German Federal Ministry of Education
More informationLiquidity costs and market impact for derivatives
Liquidity costs and market impact for derivatives F. Abergel, G. Loeper Statistical modeling, financial data analysis and applications, Istituto Veneto di Scienze Lettere ed Arti. Abergel, G. Loeper Statistical
More informationLecture 13: Martingales
Lecture 13: Martingales 1. Definition of a Martingale 1.1 Filtrations 1.2 Definition of a martingale and its basic properties 1.3 Sums of independent random variables and related models 1.4 Products of
More informationSimple Arbitrage. Motivated by and partly based on a joint work with T. Sottinen and E. Valkeila. Christian Bender. Saarland University
Simple Arbitrage Motivated by and partly based on a joint work with T. Sottinen and E. Valkeila Saarland University December, 8, 2011 Problem Setting Financial market with two assets (for simplicity) on
More informationHedging Options In The Incomplete Market With Stochastic Volatility. Rituparna Sen Sunday, Nov 15
Hedging Options In The Incomplete Market With Stochastic Volatility Rituparna Sen Sunday, Nov 15 1. Motivation This is a pure jump model and hence avoids the theoretical drawbacks of continuous path models.
More informationEuropean Options Pricing Using Monte Carlo Simulation
European Options Pricing Using Monte Carlo Simulation Alexandros Kyrtsos Division of Materials Science and Engineering, Boston University akyrtsos@bu.edu European options can be priced using the analytical
More informationMartingale Pricing Applied to Options, Forwards and Futures
IEOR E4706: Financial Engineering: DiscreteTime Asset Pricing Fall 2005 c 2005 by Martin Haugh Martingale Pricing Applied to Options, Forwards and Futures We now apply martingale pricing theory to the
More informationAdaptive Online Gradient Descent
Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650
More informationINDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS
INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS STEVEN P. LALLEY AND ANDREW NOBEL Abstract. It is shown that there are no consistent decision rules for the hypothesis testing problem
More informationThe BlackScholes pricing formulas
The BlackScholes pricing formulas Moty Katzman September 19, 2014 The BlackScholes differential equation Aim: Find a formula for the price of European options on stock. Lemma 6.1: Assume that a stock
More informationINTERNATIONAL COMPARISON OF INTEREST RATE GUARANTEES IN LIFE INSURANCE
INTERNATIONAL COMPARISON OF INTEREST RATE GUARANTEES IN LIFE INSURANCE J. DAVID CUMMINS, KRISTIAN R. MILTERSEN, AND SVEINARNE PERSSON Abstract. Interest rate guarantees seem to be included in life insurance
More informationLECTURE 9: A MODEL FOR FOREIGN EXCHANGE
LECTURE 9: A MODEL FOR FOREIGN EXCHANGE 1. Foreign Exchange Contracts There was a time, not so long ago, when a U. S. dollar would buy you precisely.4 British pounds sterling 1, and a British pound sterling
More informationPricing of an Exotic Forward Contract
Pricing of an Exotic Forward Contract Jirô Akahori, Yuji Hishida and Maho Nishida Dept. of Mathematical Sciences, Ritsumeikan University 111 Nojihigashi, Kusatsu, Shiga 5258577, Japan Email: {akahori,
More informationτ θ What is the proper price at time t =0of this option?
Now by Itô s formula But Mu f and u g in Ū. Hence τ θ u(x) =E( Mu(X) ds + u(x(τ θ))) 0 τ θ u(x) E( f(x) ds + g(x(τ θ))) = J x (θ). 0 But since u(x) =J x (θ ), we consequently have u(x) =J x (θ ) = min
More informationDecomposition of life insurance liabilities into risk factors theory and application
Decomposition of life insurance liabilities into risk factors theory and application Katja Schilling University of Ulm March 7, 2014 Joint work with Daniel Bauer, Marcus C. Christiansen, Alexander Kling
More informationCHAPTER 5. Product Measures
54 CHAPTER 5 Product Measures Given two measure spaces, we may construct a natural measure on their Cartesian product; the prototype is the construction of Lebesgue measure on R 2 as the product of Lebesgue
More informationLecture Note of Bus 41202, Spring 2012: Stochastic Diffusion & Option Pricing
Lecture Note of Bus 41202, Spring 2012: Stochastic Diffusion & Option Pricing Key concept: Ito s lemma Stock Options: A contract giving its holder the right, but not obligation, to trade shares of a common
More informationNumeraireinvariant option pricing
Numeraireinvariant option pricing Farshid Jamshidian NIB Capital Bank N.V. FELAB, University of Twente Nov04 Numeraireinvariant option pricing p.1/20 A conceptual definition of an option An Option can
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationLECTURE 15: AMERICAN OPTIONS
LECTURE 15: AMERICAN OPTIONS 1. Introduction All of the options that we have considered thus far have been of the European variety: exercise is permitted only at the termination of the contract. These
More informationFrom Binomial Trees to the BlackScholes Option Pricing Formulas
Lecture 4 From Binomial Trees to the BlackScholes Option Pricing Formulas In this lecture, we will extend the example in Lecture 2 to a general setting of binomial trees, as an important model for a single
More informationEssays in Financial Mathematics
Essays in Financial Mathematics Essays in Financial Mathematics Kristoffer Lindensjö Dissertation for the Degree of Doctor of Philosophy, Ph.D. Stockholm School of Economics, 2013. Dissertation title:
More informationMath 526: Brownian Motion Notes
Math 526: Brownian Motion Notes Definition. Mike Ludkovski, 27, all rights reserved. A stochastic process (X t ) is called Brownian motion if:. The map t X t (ω) is continuous for every ω. 2. (X t X t
More informationStephane Crepey. Financial Modeling. A Backward Stochastic Differential Equations Perspective. 4y Springer
Stephane Crepey Financial Modeling A Backward Stochastic Differential Equations Perspective 4y Springer Part I An Introductory Course in Stochastic Processes 1 Some Classes of DiscreteTime Stochastic
More informationFACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z
FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization
More informationValuation and Hedging of Participating LifeInsurance Policies under Management Discretion
Kleinow: Participating LifeInsurance Policies 1 Valuation and Hedging of Participating LifeInsurance Policies under Management Discretion Torsten Kleinow Department of Actuarial Mathematics and Statistics
More informationLecture. S t = S t δ[s t ].
Lecture In real life the vast majority of all traded options are written on stocks having at least one dividend left before the date of expiration of the option. Thus the study of dividends is important
More informationCS 522 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options
CS 5 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options 1. Definitions Equity. The common stock of a corporation. Traded on organized exchanges (NYSE, AMEX, NASDAQ). A common
More information2 The Term Structure of Interest Rates in a Hidden Markov Setting
2 The Term Structure of Interest Rates in a Hidden Markov Setting Robert J. Elliott 1 and Craig A. Wilson 2 1 Haskayne School of Business University of Calgary Calgary, Alberta, Canada relliott@ucalgary.ca
More informationA Genetic Algorithm to Price an European Put Option Using the Geometric Mean Reverting Model
Applied Mathematical Sciences, vol 8, 14, no 143, 7157135 HIKARI Ltd, wwwmhikaricom http://dxdoiorg/11988/ams144644 A Genetic Algorithm to Price an European Put Option Using the Geometric Mean Reverting
More informationThe HenstockKurzweilStieltjes type integral for real functions on a fractal subset of the real line
The HenstockKurzweilStieltjes type integral for real functions on a fractal subset of the real line D. Bongiorno, G. Corrao Dipartimento di Ingegneria lettrica, lettronica e delle Telecomunicazioni,
More informationHedging bounded claims with bounded outcomes
Hedging bounded claims with bounded outcomes Freddy Delbaen ETH Zürich, Department of Mathematics, CH892 Zurich, Switzerland Abstract. We consider a financial market with two or more separate components
More informationLecture 6: Option Pricing Using a Onestep Binomial Tree. Friday, September 14, 12
Lecture 6: Option Pricing Using a Onestep Binomial Tree An oversimplified model with surprisingly general extensions a single time step from 0 to T two types of traded securities: stock S and a bond
More informationThe Behavior of Bonds and Interest Rates. An Impossible Bond Pricing Model. 780 w Interest Rate Models
780 w Interest Rate Models The Behavior of Bonds and Interest Rates Before discussing how a bond marketmaker would deltahedge, we first need to specify how bonds behave. Suppose we try to model a zerocoupon
More informationOn exponentially ane martingales. Johannes MuhleKarbe
On exponentially ane martingales AMAMEF 2007, Bedlewo Johannes MuhleKarbe Joint work with Jan Kallsen HVBInstitut für Finanzmathematik, Technische Universität München 1 Outline 1. Semimartingale characteristics
More informationMEAN VARIANCE HEDGING AND OPTIMAL INVESTMENT IN HESTON S MODEL WITH CORRELATION
MEAN VARIANCE HEDGING AND OPTIMAL INVESTMENT IN HESTON S MODEL WITH CORRELATION ALEŠ ČERNÝ AND JAN KALLSEN Abstract. This paper solves the mean variance hedging problem in Heston s model with a stochastic
More informationTHE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING
THE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING 1. Introduction The BlackScholes theory, which is the main subject of this course and its sequel, is based on the Efficient Market Hypothesis, that arbitrages
More informationON DETERMINANTS AND SENSITIVITIES OF OPTION PRICES IN DELAYED BLACKSCHOLES MODEL
ON DETERMINANTS AND SENSITIVITIES OF OPTION PRICES IN DELAYED BLACKSCHOLES MODEL A. B. M. Shahadat Hossain, Sharif Mozumder ABSTRACT This paper investigates determinantwise effect of option prices when
More informationOption Pricing. Chapter 9  Barrier Options  Stefan Ankirchner. University of Bonn. last update: 9th December 2013
Option Pricing Chapter 9  Barrier Options  Stefan Ankirchner University of Bonn last update: 9th December 2013 Stefan Ankirchner Option Pricing 1 Standard barrier option Agenda What is a barrier option?
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationOn a comparison result for Markov processes
On a comparison result for Markov processes Ludger Rüschendorf University of Freiburg Abstract A comparison theorem is stated for Markov processes in polish state spaces. We consider a general class of
More informationGuaranteed Annuity Options
Guaranteed Annuity Options Hansjörg Furrer Marketconsistent Actuarial Valuation ETH Zürich, Frühjahrssemester 2008 Guaranteed Annuity Options Contents A. Guaranteed Annuity Options B. Valuation and Risk
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete
More informationSome Research Problems in Uncertainty Theory
Journal of Uncertain Systems Vol.3, No.1, pp.310, 2009 Online at: www.jus.org.uk Some Research Problems in Uncertainty Theory aoding Liu Uncertainty Theory Laboratory, Department of Mathematical Sciences
More informationLecture 12: The BlackScholes Model Steven Skiena. http://www.cs.sunysb.edu/ skiena
Lecture 12: The BlackScholes Model Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena The BlackScholesMerton Model
More informationQUANTIZED INTEREST RATE AT THE MONEY FOR AMERICAN OPTIONS
QUANTIZED INTEREST RATE AT THE MONEY FOR AMERICAN OPTIONS L. M. Dieng ( Department of Physics, CUNY/BCC, New York, New York) Abstract: In this work, we expand the idea of Samuelson[3] and Shepp[,5,6] for
More informationDiusion processes. Olivier Scaillet. University of Geneva and Swiss Finance Institute
Diusion processes Olivier Scaillet University of Geneva and Swiss Finance Institute Outline 1 Brownian motion 2 Itô integral 3 Diusion processes 4 BlackScholes 5 Equity linked life insurance 6 Merton
More informationOption Pricing. 1 Introduction. Mrinal K. Ghosh
Option Pricing Mrinal K. Ghosh 1 Introduction We first introduce the basic terminology in option pricing. Option: An option is the right, but not the obligation to buy (or sell) an asset under specified
More informationChapter 2: Binomial Methods and the BlackScholes Formula
Chapter 2: Binomial Methods and the BlackScholes Formula 2.1 Binomial Trees We consider a financial market consisting of a bond B t = B(t), a stock S t = S(t), and a calloption C t = C(t), where the
More informationShape Optimization Problems over Classes of Convex Domains
Shape Optimization Problems over Classes of Convex Domains Giuseppe BUTTAZZO Dipartimento di Matematica Via Buonarroti, 2 56127 PISA ITALY email: buttazzo@sab.sns.it Paolo GUASONI Scuola Normale Superiore
More information1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
More informationPricing Barrier Options under Local Volatility
Abstract Pricing Barrier Options under Local Volatility Artur Sepp Mail: artursepp@hotmail.com, Web: www.hot.ee/seppar 16 November 2002 We study pricing under the local volatility. Our research is mainly
More informationPractical and theoretical aspects of marketconsistent valuation and hedging of insurance liabilities
Practical and theoretical aspects of marketconsistent valuation and hedging of insurance liabilities Łukasz Delong Institute of Econometrics, Division of Probabilistic Methods Warsaw School of Economics
More informationBarrier Options. Peter Carr
Barrier Options Peter Carr Head of Quantitative Financial Research, Bloomberg LP, New York Director of the Masters Program in Math Finance, Courant Institute, NYU March 14th, 2008 What are Barrier Options?
More informationOn Quantile Hedging and Its Applications to the Pricing of EquityLinked Life Insurance Contracts 1
On Quantile Hedging and Its Applications to the Pricing of EquityLinked Life Insurance Contracts 1 Alexander Melnikov Steklov Mathematical Institute of Russian Academy Sciences and Department of Mathematical
More informationA Summary on Pricing American Call Options Under the Assumption of a Lognormal Framework in the KornRogers Model
BULLEIN of the MALAYSIAN MAHEMAICAL SCIENCES SOCIEY http:/math.usm.my/bulletin Bull. Malays. Math. Sci. Soc. 2 352A 2012, 573 581 A Summary on Pricing American Call Options Under the Assumption of a Lognormal
More informationarxiv:1502.06681v2 [qfin.mf] 26 Feb 2015
ARBITRAGE, HEDGING AND UTILITY MAXIMIZATION USING SEMISTATIC TRADING STRATEGIES WITH AMERICAN OPTIONS ERHAN BAYRAKTAR AND ZHOU ZHOU arxiv:1502.06681v2 [qfin.mf] 26 Feb 2015 Abstract. We consider a financial
More informationAn axiomatic approach to capital allocation
An axiomatic approach to capital allocation Michael Kalkbrener Deutsche Bank AG Abstract Capital allocation techniques are of central importance in portfolio management and riskbased performance measurement.
More informationOption Pricing. Chapter 4 Including dividends in the BS model. Stefan Ankirchner. University of Bonn. last update: 6th November 2013
Option Pricing Chapter 4 Including dividends in the BS model Stefan Ankirchner University of Bonn last update: 6th November 2013 Stefan Ankirchner Option Pricing 1 Dividend payments So far: we assumed
More informationOption Pricing in ARCHtype Models: with Detailed Proofs. Nr. 10 März 1995
Option Pricing in ARCHtype Models: with Detailed Proofs Jan Kallsen Universität Freiburg Murad S. Taqqu Boston University Nr. 1 März 1995 Universität Freiburg i. Br. Institut für Mathematische Stochastik
More informationGrey Brownian motion and local times
Grey Brownian motion and local times José Luís da Silva 1,2 (Joint work with: M. Erraoui 3 ) 2 CCM  Centro de Ciências Matemáticas, University of Madeira, Portugal 3 University Cadi Ayyad, Faculty of
More informationValuing Stock Options: The BlackScholesMerton Model. Chapter 13
Valuing Stock Options: The BlackScholesMerton Model Chapter 13 Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright John C. Hull 2013 1 The BlackScholesMerton Random Walk Assumption
More informationEstimating the Degree of Activity of jumps in High Frequency Financial Data. joint with Yacine AïtSahalia
Estimating the Degree of Activity of jumps in High Frequency Financial Data joint with Yacine AïtSahalia Aim and setting An underlying process X = (X t ) t 0, observed at equally spaced discrete times
More information6.4 The Infinitely Many Alleles Model
6.4. THE INFINITELY MANY ALLELES MODEL 93 NE µ [F (X N 1 ) F (µ)] = + 1 i
More informationValuation of the Surrender Option Embedded in EquityLinked Life Insurance. Brennan Schwartz (1976,1979) Brennan Schwartz
Valuation of the Surrender Option Embedded in EquityLinked Life Insurance Brennan Schwartz (976,979) Brennan Schwartz 04 2005 6. Introduction Compared to traditional insurance products, one distinguishing
More informationFurther Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1
Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1 J. Zhang Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing
More informationA new FeynmanKacformula for option pricing in Lévy models
A new FeynmanKacformula for option pricing in Lévy models Kathrin Glau Department of Mathematical Stochastics, Universtity of Freiburg (Joint work with E. Eberlein) 6th World Congress of the Bachelier
More information