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Theorem 12.16 (Berry-Esseen) Let \( \\left( {X}_{n}\\right) \) be a sequence of i.i.d. random variables such that \( E\\left\\lbrack {X}_{1}\\right\\rbrack = 0 \) and \( {\\sigma }^{2} = \\operatorname{var}\\left( {X}_{1}\\right) ,\\varrho = E\\left\\lbrack {\\left| {X}_{1}\\right| }^{3}\\right\\rbrack \) are finite. I... | For the proof we refer to, for example, Durrett [109]. | No |
Lemma 13.1 For any \( n \in \mathbb{N} \), the random variable \( {T}_{n} \) has probability density\n\n\[ \n{f}_{{T}_{n}}\left( t\right) = \lambda {e}^{-{\lambda t}}\frac{{\left( \lambda t\right) }^{n - 1}}{\left( {n - 1}\right) !}{\mathbb{1}}_{{\mathbb{R}}_{ \geq 0}}\left( t\right) ,\;t \in \mathbb{R}. \n\] | Proof. We prove the thesis by induction. For \( n = 1 \) it is obvious. Next we assume that \( {T}_{n} \) has probability density given by (13.2): by the independence of the variables \( \left\{ {\tau }_{n}\right\} \) and Corollary A.54, we have\n\n\[ \n{f}_{{T}_{n + 1}}\left( t\right) = {f}_{{T}_{n} + {\tau }_{n + 1}}... | Yes |
Proposition 13.3 Let \( {\left( {N}_{t}\right) }_{t \geq 0} \) be a Poisson process. Then:\ni) the trajectories \( t \mapsto {N}_{t}\left( \omega \right) \) are right continuous with finite left limits, that is \( N \) is a càdlà \( {g}^{1} \) process;\n\nii) for any positive \( t \), almost all trajectories are contin... | Proof. Property \( i \) ) follows from the definition of Poisson process. Secondly, the discontinuities of \( N \) are at the jump times \( {T}_{n}, n \in \mathbb{N} \) : however, by Lemma 13.1 for any \( t > 0 \) we have\n\n\[ \nP\left( {{T}_{n} = t}\right) = 0\n\]\n\nand therefore, with probability one, \( t \) is no... | Yes |
Proposition 13.4 Let \( {\left( {N}_{t}\right) }_{t \geq 0} \) be a Poisson process with intensity \( \lambda \) . Then: i) for any \( t \geq 0,{N}_{t} \) has the distribution\n\n\[ P\left( {{N}_{t} = n}\right) = {e}^{-{\lambda t}}\frac{{\left( \lambda t\right) }^{n}}{n!},\;n \in \mathbb{N}, \] | Proof. We only prove \( i \) ): for the other properties, that are consequence of the absence of memory of the exponential distribution (cf. Example A.29), we refer for instante to [76]. We first observe that by (13.2) we have\n\n\[ P\left( {t \geq {T}_{n + 1}}\right) = {\int }_{0}^{t}\lambda {e}^{-{\lambda s}}\frac{{\... | No |
Lemma 13.12 Let \( f \) be a càdlàg function defined on a compact interval \( \left\lbrack {0, T}\right\rbrack \) . Then, for any \( n \in \mathbb{N} \), the number of jumps of \( f \) of size greater than \( \frac{1}{n} \) is finite:\n\n\[ \left. {\left. {\# \{ t \in \rbrack 0, T}\right\rbrack \left| \right| {\Delta f... | Proof. By contradiction, assume that for some \( n \in \mathbb{N} \) the number of jumps of size greater than \( \frac{1}{n} \) is infinite: then, since the domain is compact, there exists a sequence \( \left( {t}_{k}\right) \) in \( \left\lbrack {0, T}\right\rbrack \), strictly increasing or decreasing, which converge... | Yes |
Lemma 13.17 If \( X \) is Lévy process, then \( {X}_{t} \) is infinitely divisible for each \( t \geq 0 \) and we have\n\n\[{\varphi }_{{X}_{t}}\left( \xi \right) = {\left( {\varphi }_{{X}_{\frac{t}{n}}}\left( \xi \right) \right) }^{n},\;t \geq 0, n \in \mathbb{N}.\] | Proof. The thesis follows by the properties \( L - i \) ) and \( L - {ii} \) ) of Definition 13.10: indeed, for any \( n \geq 2 \) we set\n\n\[{Y}_{i}^{\left( n\right) } \mathrel{\text{:=}} {X}_{\frac{it}{n}} - {X}_{\frac{\left( {i - 1}\right) t}{n}}\overset{d}{ = }{X}_{\frac{t}{n}},\;i = 1,\ldots, n,\]\n\nand we remar... | Yes |
Lemma 13.18 If \( {\left( {X}_{t}\right) }_{t \geq 0} \) is stochastically continuous, then the map \( t \mapsto \) \( {\varphi }_{{X}_{t}}\left( \xi \right) \) is continuous for each \( \xi \in {\mathbb{R}}^{d} \) . | Proof. Let \( \xi \in {\mathbb{R}}^{d} \) be fixed: for any \( \varepsilon > 0 \) we consider \( {\delta }_{\varepsilon } > 0 \) such that\n\n\[ \mathop{\sup }\limits_{{\left| y\right| \leq {\delta }_{\varepsilon }}}\left| {{e}^{{i\xi } \cdot y} - 1}\right| < \frac{\varepsilon }{2} \]\n\nIf \( X \) is stochastically co... | Yes |
Lemma 13.19 If \( {\left( {X}_{t}\right) }_{t \geq 0} \) is a Lévy process, then \( {\varphi }_{{X}_{t}}\left( \xi \right) \neq 0 \) for any \( \xi \in {\mathbb{R}}^{d} \) and \( t \geq 0 \) . | Proof. See Sato [297], Lemma 7.5. | No |
Lemma 13.20 Let \( \varphi \in C\left( {{\mathbb{R}}^{d},\mathbb{C}}\right) \) such that \( \varphi \left( 0\right) = 1 \) and \( \varphi \left( \xi \right) \neq 0 \) for any \( \xi \in {\mathbb{R}}^{d} \) . Then there exists a unique continuous function \( g \in C\left( {{\mathbb{R}}^{d},\mathbb{C}}\right) \) such tha... | Proof. See Sato [297], Lemma 7.6. | No |
Example 13.21 (Brownian motion with drift) Let \( {X}_{t} = {\mu t} + \sigma {W}_{t} \) where \( W \) is a standard real Brownian motion: as a consequence of Example A.60 and Lemma A.70, we have | \[ E\left\lbrack {e}^{{i\xi }{X}_{t}}\right\rbrack = {e}^{i\mu t\xi }E\left\lbrack {e}^{{i\xi \sigma }{W}_{t}}\right\rbrack = {e}^{{i\mu t\xi } + \frac{1}{2}{\left( i\xi \sigma \right) }^{2}t}, \] that is \[ \psi \left( \xi \right) = {i\mu \xi } - \frac{{\sigma }^{2}{\xi }^{2}}{2} \] In the \( d \) -dimensional case wh... | Yes |
Example 13.22 (Poisson process) Let \( {N}_{t} \) denote a Poisson process with intensity \( \lambda \) (cf. Definition 13.2). We have\n\n\[ \n{\varphi }_{{N}_{t}}\left( \xi \right) = E\left\lbrack {e}^{{i\xi }{N}_{t}}\right\rbrack = \mathop{\sum }\limits_{{n \geq 0}}E\left\lbrack {{e}^{i\xi n}{\mathbb{1}}_{\left\{ {N}... | \[ \n{\varphi }_{{N}_{t}}\left( \xi \right) = E\left\lbrack {e}^{{i\xi }{N}_{t}}\right\rbrack = \mathop{\sum }\limits_{{n \geq 0}}E\left\lbrack {{e}^{i\xi n}{\mathbb{1}}_{\left\{ {N}_{t} = n\right\} }}\right\rbrack = \n\]\n\n(by (13.3))\n\n\[ \n= {e}^{-{\lambda t}}\mathop{\sum }\limits_{{n \geq 0}}\frac{{\left( {e}^{i\... | Yes |
Example 13.23 (Compound Poisson process) Let\n\n\[ \n{X}_{t} = \mathop{\sum }\limits_{{n = 1}}^{{N}_{t}}{Z}_{n},\;t \geq 0 \n\] \n\nbe a \( d \) -dimensional compound Poisson process (cf. Definition 13.8) with intensity \( \lambda \) and distribution of jumps \( \eta \) . We denote by \n\n\[ \n\widehat{\eta }\left( \xi... | \[ \n= \mathop{\sum }\limits_{{n \geq 0}}{\left( E\left\lbrack {e}^{{i\xi } \cdot {Z}_{1}}\right\rbrack \right) }^{n}P\left( {{N}_{t} = n}\right) = \n\] \n\n(by (13.3)) \n\n\[ \n= {e}^{-{\lambda t}}\mathop{\sum }\limits_{{n \geq 0}}\frac{{\left( \lambda t\widehat{\eta }\left( \xi \right) \right) }^{n}}{n!} = {e}^{-{\la... | Yes |
Example 13.24 (Compensated compound Poisson process) Let\n\n\[ \n{\\widetilde{X}}_{t} = {X}_{t} - {m\\lambda t},\;m = {\\int }_{{\\mathbb{R}}^{d}}{x\\eta }\\left( {dx}\\right) ,\n\]\n\nbe a compensated compound Poisson process with intensity \( \\lambda \) and distribution of jumps \( \\eta \) (cf. Definition 13.9). Th... | \[ \n\\psi \\left( \\xi \\right) = \\lambda \\left( {\\widehat{\\eta }\\left( \\xi \\right) - 1 - {i\\xi } \\cdot m}\\right) = {\\int }_{{\\mathbb{R}}^{d}}\\left( {{e}^{{i\\xi } \\cdot x} - 1 - {i\\xi } \\cdot x}\\right) {\\lambda \\eta }\\left( {dx}\\right) .\n\] | Yes |
Theorem 13.30 Let \( X \) be the jump-diffusion process in (13.17) with jump measure \( J \) and Lévy measure \( \nu \) . For any function \( f = f\left( {t, x}\right) \) we have\n\n\[ \mathop{\sum }\limits_{\substack{{0 < s \leq t} \\ {\Delta {X}_{s} \neq 0} }}f\left( {s,\Delta {X}_{s}}\right) = {\int }_{0}^{t}{\int }... | Proof. For simplicity, we only consider the case \( f = f\left( x\right) \) . We first remark that, for a jump-diffusion process \( X \), we have\n\n\[ \mathop{\sum }\limits_{\substack{{0 < s \leq t} \\ {\Delta {X}_{s} \neq 0} }}f\left( {\Delta {X}_{s}}\right) = \mathop{\sum }\limits_{{n = 1}}^{{N}_{t}}f\left( {Z}_{n}\... | Yes |
Lemma 13.33 Let \( X \) be a Lévy process with jump measure \( J \) and Lévy measure \( \nu \) . Then\ni) if \( H \in \mathcal{B}\left( {\mathbb{R}}^{d}\right) \) is such that \( 0 \notin \bar{H} \) then the process\n\n\[t \mapsto {J}_{t}\left( H\right) \mathrel{\text{:=}} J\left( {\left\lbrack {0, t}\right\rbrack \tim... | Proof. Part i) can be verified directly using the definition of Poisson process: see, for instance, Protter [287], p. 26. | No |
Theorem 13.34 Let \( {\left( {X}_{t}\right) }_{t > 0} \) be a d-dimensional Levy process with Levy measure \( \nu \) and jump measure \( J \) . For any measurable function \( f \) such that\n\n\[ \n{\int }_{0}^{t}{\int }_{\left| x\right| \leq \varepsilon }\left| {f\left( {s, x}\right) }\right| \nu \left( {dx}\right) {d... | Proof. Formulas (13.43) and (13.44) follows from (13.23)-(13.24) by limit arguments (for further details, see Section 2.4 in Applebaum [11] and Section I-4 in Protter [287]).\n\nNow assume that \( f \) satisfies condition (13.40). The idea is that \ | No |
Example 13.37 Let\n\n\[ \n{X}_{t} = {\mu t} + {B}_{t} + \mathop{\sum }\limits_{{n = 1}}^{{N}_{t}}{Z}_{n} \]\n\nbe a jump-diffusion process: \( \mu \in {\mathbb{R}}^{d}, B \) is \( d \) -dimensional correlated Brownian motion with correlation matrix \( \mathcal{C}, N \) is a Poisson process with intensity \( \lambda \) ... | We remark that the second and fourth terms in (13.47) (i.e. Brownian motion and compensated small jumps) form the martingale part of \( X \), while the first and third terms (i.e. drift term and large jumps) govern the drift of the process. More precisely, it is always possible to split a Lévy process into the sum of a... | No |
Corollary 13.38 Let \( X \) be a Lévy process. Then \( X = M + Z \) where \( M \) and \( Z \) are Lévy processes, \( M \) is a martingale such that \( {M}_{t} \in {L}^{p}\left( \Omega \right) \) for any \( p \geq 1 \) and \( Z \) has (locally in time) bounded variation. | Proof. By the Lévy-Ito decomposition (13.47), it is suffices to set\n\n\[ \n{Z}_{t} = {\mu }_{R}t + {X}_{t}^{R} = {\mu }_{R}t + {\int }_{0}^{t}{\int }_{\left| x\right| \geq R}{xJ}\left( {{ds},{dx}}\right) , \n\] \n\n\[ \n{M}_{t} = {B}_{t} + {M}_{t}^{R} = {B}_{t} + {\int }_{0}^{t}{\int }_{\left| x\right| < R}x\widetilde... | Yes |
Theorem 13.40 (Lévy-Khintchine representation) Let \( X \) be a Lévy process in \( {\mathbb{R}}^{d} \) with characteristic triplet \( \left( {{\mu }_{1},\mathcal{C},\nu }\right) \). Then we have\n\n\[ \n{\varphi }_{{X}_{t}}\left( \xi \right) = E\left\lbrack {e}^{{i\xi } \cdot {X}_{t}}\right\rbrack = {e}^{t{\psi }_{X}\l... | Proof. By the Lévy-Itô decomposition \( {X}_{t} \) can be represented as the independent sum of is the a.s. limit of the sum of \( {\mu }_{1}t + {B}_{t},{X}_{t}^{1} \) and \( {\widetilde{X}}_{t}^{\varepsilon ,1} \) in (13.51), as \( \varepsilon \rightarrow {0}^{ + } \). Since these terms are independent, by Remark 13.2... | Yes |
Corollary 13.42 Let \( X \) be a Lévy process with characteristic triplet \( \left( {{\mu }_{1},\mathcal{C},\nu }\right) \) and Lévy measure \( \nu \) such that\n\n\[ \nu \left( {\mathbb{R}}^{d}\right) < \infty \]\n\nThen \( X \) is a jump-diffusion process with intensity \( \lambda = \nu \left( {\mathbb{R}}^{d}\right)... | Proof. Under condition (13.66), we can let \( R \) go to zero in the Lévy-Khintchine representation (13.62): we get\n\n\[ {\psi }_{X}\left( \xi \right) = i{\mu }_{0} \cdot \xi - \frac{1}{2}\langle \mathcal{C}\xi ,\xi \rangle + {\int }_{{\mathbb{R}}^{d}}\left( {{e}^{{i\xi } \cdot x} - 1}\right) \nu \left( {dx}\right) \]... | Yes |
Proposition 13.43 Let \( X \) be a Lévy process with triplet \( \left( {{\mu }_{1},\mathcal{C},\nu }\right) \) . Then \( X \) has (locally in time) bounded variation if and only if | Proof. We only prove the \ | No |
Corollary 13.44 Let \( X \) be a Lévy process with (locally in time) bounded variation and characteristic triplet \( \left( {{\mu }_{1},0,\nu }\right) \). Then we have the Lévy-Itô decomposition | \[ {X}_{t} = {\mu }_{0}t + {\int }_{{\mathbb{R}}^{d}}{xJ}\left( {{ds},{dx}}\right) \] where \[ {\mu }_{0} = {\mu }_{1} - {\int }_{\left| x\right| \leq 1}{x\nu }\left( {dx}\right) \] Moreover the characteristic exponent takes the form \[ {\psi }_{X}\left( \xi \right) = i{\mu }_{0} \cdot \xi + {\int }_{{\mathbb{R}}^{d}}\... | Yes |
The Lévy measure of a stable distribution is of the form: | \[ \nu \left( {dx}\right) = \left( {\frac{{C}_{1}}{{x}^{1 + \alpha }}{\mathbb{1}}_{\{ x > 0\} } + \frac{{C}_{2}}{{\left( -x\right) }^{1 + \alpha }}{\mathbb{1}}_{\{ x < 0\} }}\right) {dx} \] where \( {C}_{1},{C}_{2} > 0 \) . By conditions (13.45) and (13.46), we necessarily have \( \alpha \in \rbrack 0,2\lbrack \) . Mor... | Yes |
Proposition 13.49 Let \( X \) be a Lévy process on \( \mathbb{R} \) with characteristic triplet \( \left( {{\mu }_{1},{\sigma }^{2},\nu }\right) \). The exponential moment \( E\left\lbrack {e}^{\xi {X}_{t}}\right\rbrack ,\xi \in \mathbb{R} \), is finite if and only if \[ {\int }_{\left| x\right| \geq 1}{e}^{\xi x}\nu \... | In this case \[ E\left\lbrack {e}^{\xi {X}_{t}}\right\rbrack = {e}^{{t\psi }\left( {-{i\xi }}\right) } \] where \( \psi \) is the characteristic exponent of \( X \) . | No |
Theorem 13.50 Let \( X \) be a real valued Lévy process with characteristic triplet \( \left( {{\mu }_{1},{\sigma }^{2},\nu }\right) \) . We have:\n\ni) if \( E\left\lbrack \left| {X}_{1}\right| \right\rbrack < \infty \) then \( {\left( {X}_{t} - E\left\lbrack {X}_{t}\right\rbrack \right) }_{t > 0} \) is a martingale; | Proof. i) Since \( E\left\lbrack {X}_{t}\right\rbrack = {tE}\left\lbrack {X}_{1}\right\rbrack \) (cf. (13.72)), if \( E\left\lbrack \left| {X}_{1}\right| \right\rbrack < \infty \) then \( {X}_{t} - E\left\lbrack {X}_{t}\right\rbrack \) is integrable. Moreover, by the independence of increments we have\n\n\[ E\left\lbra... | Yes |
Example 13.51 (Merton model) In the Merton model [251], the log-price is modeled by a process of the form (13.76) where \( \eta = {\mathcal{N}}_{m,{\delta }^{2}} \) . Thus the 0 -triplet is \( \left( {\mu ,{\sigma }^{2},\nu }\right) \) with Lévy measure\n\n\[ \nu \left( {dx}\right) = \frac{\lambda }{\sqrt{{2\pi }{\delt... | As already proved in Example 13.26, the characteristic exponent is\n\n\[ \psi \left( \xi \right) = {i\mu \xi } - \frac{1}{2}{\sigma }^{2}{\xi }^{2} + \lambda \left( {{e}^{{im\xi } - \frac{1}{2}{\delta }^{2}{\xi }^{2}} - 1}\right) . \] | Yes |
In the Kou model [217], the distribution of jumps is defined in terms of an asymmetric double exponential density: more precisely, we have\n\n\[ \eta \left( {dx}\right) = \left( {p{\lambda }_{1}{e}^{-{\lambda }_{1}x}{\mathbb{1}}_{\{ x > 0\} } + \left( {1 - p}\right) {\lambda }_{2}{e}^{{\lambda }_{2}x}{\mathbb{1}}_{\{ x... | \[ {\psi }_{X}\left( \xi \right) = {i\mu \xi } - \frac{{\sigma }^{2}{\xi }^{2}}{2} + {\int }_{\mathbb{R}}\left( {{e}^{i\xi x} - 1}\right) \nu \left( {dx}\right) \] \n\n\[ = {i\mu \xi } - \frac{{\sigma }^{2}{\xi }^{2}}{2} + {i\lambda \xi }\left( {\frac{p}{{\lambda }_{1} - {i\xi }} - \frac{1 - p}{{\lambda }_{2} + {i\xi }... | Yes |
A Poisson processes is a subordinator. A compound Poisson process is a subordinator if and only if all the \( {Z}_{n} \) (cf. Definition 13.8) take only non-negative values. | Since a subordinator \( S \) takes only non-negative values, it is convenient to characterize it by the Laplace transform instead of the Fourier transform: if \( {\psi }_{S} \) denotes as usual the characteristic exponent of \( S \), by Proposition 13.49 the Laplace exponent\n\n\[ \n{\ell }_{S}\left( \xi \right) \mathr... | No |
Example 13.60 (Gamma subordinator) We use slightly different notations and consider a Lévy process \( S \) with 0 -triplet \( \left( {0,0,\varrho }\right) \) and Lévy measure\n\n\[ \varrho \left( {dx}\right) = \frac{a{e}^{-{bx}}}{x}{\mathbb{1}}_{\{ x > 0\} }{dx} \]\n\nwhere \( a, b \) are positive parameters: we call \... | The density of \( {S}_{t} \) can be recovered from the characteristic function by Fourier inversion:\n\n\[ {f}_{{S}_{t}}\left( x\right) = \frac{1}{2\pi }{\int }_{\mathbb{R}}{e}^{-{ix\xi }}{\varphi }_{{S}_{t}}\left( \xi \right) {d\xi } = \frac{{e}^{-{bx}}{\left( bx\right) }^{at}}{{x\Gamma }\left( {at}\right) },\;x > 0, ... | Yes |
Example 13.61 (Variance-Gamma process) By subordinating a Brownian motion with drift \( \mu \) and volatility \( \sigma \) by a Gamma process \( S \) with variance \( v \) (and unitary mean), we obtain the so-called Variance-Gamma (VG) process | \[ {X}_{t} = \mu {S}_{t} + \sigma {W}_{{S}_{t}} \] This is a three-parameter process: the variance \( v \) of the subordinator, the drift \( \mu \) and the volatility \( \sigma \) of the Brownian motion. By Theorem 13.58, the characteristic exponent of \( X \) is \[ {\psi }_{X}\left( \xi \right) = - \frac{1}{v}\log \le... | Yes |
Example 13.62 (Inverse Gaussian subordinator) The Inverse Gaussian (IG) subordinator \( S \) is a tempered stable subordinator with \( \alpha = \frac{1}{2} \) : thus \( S \) has 0-triplet \( \left( {0,0,\varrho }\right) \) and Lévy measure\n\n\[ \varrho \left( {dx}\right) = \frac{a{e}^{-{bx}}}{{x}^{\frac{3}{2}}}{\mathb... | The density of \( {S}_{t} \) can be recovered from the characteristic function by Fourier inversion:\n\n\[ {f}_{{S}_{t}}\left( x\right) = \frac{1}{2\pi }{\int }_{\mathbb{R}}{e}^{-{ix\xi }}{\varphi }_{{S}_{t}}\left( \xi \right) {d\xi } = \frac{at}{{x}^{3/2}}\exp \left( {-\frac{{\left( at\sqrt{\pi } - x\sqrt{b}\right) }^... | Yes |
Proposition 13.64 Assume that \( X \) is a Lévy process with characteristic exponent \( {\psi }_{Q} \) under \( Q \) . The discounted price process \( {\widetilde{S}}_{t} = {S}_{0}{e}^{{X}_{t} - {rt}} \) is a \( Q \) - martingale if and only if\n\n\[ \n{E}^{Q}\left\lbrack {S}_{t}\right\rbrack = {E}^{Q}\left\lbrack {{S}... | Proof. It suffices to recall that, by Theorem 13.50-iii), \( {\widetilde{S}}_{t} = {S}_{0}{e}^{{X}_{t} - {rt}} \) is a martingale if and only if \( 1 = {e}^{-{rt}}{E}^{Q}\left\lbrack {e}^{{X}_{t}}\right\rbrack = {e}^{t\left( {{\psi }_{Q}\left( {-i}\right) - r}\right) } \) . | Yes |
Theorem 13.66 Let \( X = {\left( {X}_{t}\right) }_{t \in \left\lbrack {0, T}\right\rbrack } \) be a Lévy process. Then \( X \) is a Lévy process also with respect to \( {P}_{T}^{\theta } \) in (13.100) and its Laplace exponent \( {\ell }_{\theta } \) under \( {P}_{T}^{\theta } \) is given by \[ {\ell }_{\theta }\left( ... | Proof. In view of Bayes’s formula (Theorem A.113), for all \( 0 \leq s \leq t \leq T \) we have \[ {E}^{{P}_{T}^{\theta }}\left\lbrack {{e}^{z\left( {{X}_{t} - {X}_{s}}\right) } \mid {\mathcal{F}}_{s}}\right\rbrack = \frac{{E}^{P}\left\lbrack {{e}^{z\left( {{X}_{t} - {X}_{s}}\right) }{Z}_{T}^{\theta } \mid {\mathcal{F}... | Yes |
Theorem 13.67 Let \( \left( {\mu ,{\sigma }^{2},\nu }\right) \) be the triplet of a Lévy process \( X \) with respect to the measure \( P \) . Then the triplet \( \left( {{\mu }_{\theta },{\sigma }_{\theta }^{2},{\nu }_{\theta }}\right) \) of \( X \) with respect to the measure \( {P}_{T}^{\theta } \) in (13.100) is de... | Proof. In view of (13.101) we have:\n\n\[ \n{\ell }_{\theta }\left( z\right) = {\mu z} + \frac{{\sigma }^{2}}{2}\left( {{\left( z + \theta \right) }^{2} - {\theta }^{2}}\right) + {\int }_{\mathbb{R}}\left( {\left( {{e}^{zx} - 1}\right) {e}^{\theta x} - {zx}{\mathbb{1}}_{\{ \left| x\right| < 1\} }}\right) \nu \left( {dx... | Yes |
Theorem 13.68 Suppose that \( {\theta }^{ * } \) is a solution to\n\n\[ \ell \left( {1 + {\theta }^{ * }}\right) - \ell \left( {\theta }^{ * }\right) = r \]\n\n(13.103)\n\nIf \( {E}^{P}\left\lbrack {e}^{{\theta }^{ * }{X}_{T}}\right\rbrack < \infty \) (to assure that \( {P}_{T}^{{\theta }^{ * }} \) exists) and \( {E}^{... | Under the assumptions of Theorem 13.68, \( {P}_{T}^{{\theta }^{ * }} \) is called the Esscher martingale transform of the objective measure \( P \) . Let us denote \( {P}_{T}^{{\theta }^{ * }} \) by \( Q \) : in view of (13.101), with obvious notation we have\n\n\[ {\ell }_{Q}\left( z\right) = {\ell }_{P}\left( {z + {\... | Yes |
Example 13.69 (Brownian motion with drift) Let \( {\ell }_{P}\left( z\right) = {\mu z} + \frac{{\sigma }^{2}{z}^{2}}{2} \) be the Laplace exponent under the historic measure \( P \) . The solution to (13.103) | is\n\[ \n{\theta }^{ * } = \frac{1}{{\sigma }^{2}}\left( {r - \mu - \frac{{\sigma }^{2}}{2}}\right) \]\n\nand by (13.102) the characteristic exponent is\n\n\[ \n{\psi }_{Q}\left( \xi \right) = i\left( {r - \frac{{\sigma }^{2}}{2}}\right) \xi - \frac{{\sigma }^{2}{\xi }^{2}}{2}. \]\n\nIt is clear that in this case the E... | Yes |
Let \( {N}_{t} \) be a Poisson process with intensity parameter \( \lambda \) and let \( {X}_{t} = \alpha {N}_{t} - {\beta t} \) with \( \alpha ,\beta > 0 \) . Since\n\n\[ \n{\ell }_{P}\left( z\right) = - {\beta z} + \lambda \left( {{e}^{\alpha z} - 1}\right) \n\] | the solution to (13.103) is \( {\theta }^{ * } = \frac{1}{\alpha }\log \frac{r + \beta }{\lambda \left( {{e}^{\alpha } - 1}\right) } \) . Thus\n\n\[ \n{\ell }_{Q}\left( z\right) = - {\beta z} + {\lambda }^{ * }\left( {{e}^{\alpha z} - 1}\right) \n\]\n\nwith \( {\lambda }^{ * } = \lambda {e}^{{\theta }^{ * }y} = \frac{r... | Yes |
Example 13.71 (NIG process) Let\n\n\\[ \n{\\ell }_{P}\\left( z\\right) = {\\mu z} + \\delta \\left( {{\\left( {\\alpha }^{2} - {\\beta }^{2}\\right) }^{\\frac{1}{2}} - {\\left( {\\alpha }^{2} - {\\left( \\beta + z\\right) }^{2}\\right) }^{\\frac{1}{2}}}\\right) \n\\]\n\nwith \\( - \\alpha - \\beta \\leq \\operatorname{... | Then (13.104) yields\n\n\\[ \n{\\ell }_{Q}\\left( z\\right) = {\\mu z} + \\delta \\left( {{\\left( {\\alpha }^{2} - {\\beta }^{*2}\\right) }^{1/2} - {\\left( {\\alpha }^{2} - {\\left( {\\beta }^{ * } + z\\right) }^{2}\\right) }^{1/2}}\\right) \n\\]\n\nwith \\( {\\beta }^{ * } = - \\frac{1}{2} - \\frac{\\mu - r}{2\\delt... | Yes |
Theorem 13.74 Let \( X \) be a Lévy process with triplet \( \left( {\mu ,{\sigma }^{2},\nu }\right) \) under some probability measure \( P \) . Then the following two conditions are equivalent:\n\ni) there is a probability measure \( Q \), equivalent to \( P \), such that \( X \) is a Lévy process with triplet \( \left... | Proof. See Sato [297], Theorem 33.1. | Yes |
Example 13.75 (Brownian motion) In this case \( \nu \equiv 0 \) and \( \sigma > 0 \) . The unique solution is \( \eta = \frac{r - \mu }{\sigma } - \frac{\sigma }{2} \) . | This implies that the new drift and volatility parameters under \( Q \) are: \( \widetilde{\mu } = r - \frac{{\sigma }^{2}}{2} \) and \( \widetilde{\sigma } = \sigma \) . | Yes |
Let \( {X}_{t} = \alpha {N}_{t} + {\mu t} \) with \( \mu < r \) and \( \alpha > 0 \) . For simplicity, we take \( \alpha > 1 \) . In this case \( \sigma = 0 \) and \( \nu = \lambda {\delta }_{\alpha } \) where \( \lambda \) is the intensity parameter. Then \( H \) is constant | \[ H = \frac{r - \mu }{\lambda \left( {{e}^{\alpha } - 1}\right) } \] and the new Lévy measure under \( Q \) is \( \widetilde{\nu } = {\lambda }^{ * }{\delta }_{\alpha } \) with \( {\lambda }^{ * } = \frac{r - \mu }{{e}^{\alpha } - 1} \) ; moreover \( \widetilde{\mu } = \mu \) . | Yes |
Let \( {T}_{1} > 0 \) and consider the deterministic function\n\n\[ t \mapsto {S}_{t} = {\mathbb{1}}_{\left\lbrack {T}_{1},\infty \lbrack \left( t\right) .\right. } \]\n\nThen, as in Example 3.68, for any continuous function \( u \), the Riemann-Stieltjes integral is well-defined and | \[ {\int }_{0}^{t}{u}_{s}d{S}_{s} = \left\{ \begin{array}{ll} 0 & \text{ if }t < {T}_{1} \\ {u}_{{T}_{1}} & \text{ if }t \geq {T}_{1} \end{array}\right. \] | Yes |
We now examine the convergence of Riemann-Stieltjes sums in case \( u \) is discontinuous and \( S \) as in Example 14.1. We consider \( t \geq {T}_{1} \) and a sequence of partitions \( {\varsigma }_{n} = \left( {{t}_{0}^{n},\ldots ,{t}_{{N}_{n}}^{n}}\right) \) of \( \left\lbrack {0, t}\right\rbrack \) such that \( {T... | \[ \mathop{\sum }\limits_{{k = 1}}^{N}{u}_{{\tau }_{k}^{n}}\left( {{S}_{{t}_{k}^{n}} - {S}_{{t}_{k - 1}^{n}}}\right) = {u}_{{\tau }_{{k}_{n}}^{n}} \] be the Riemann-Stieltjes sum where as usual \( {\tau }_{k}^{n} \in \left\lbrack {{t}_{k - 1}^{n},{t}_{k}^{n}}\right\rbrack \) for \( k = 1,\ldots ,{N}_{n} \) and in parti... | Yes |
Let \( {S}_{t} = {\lambda t} - {N}_{t} \) where \( N \) is a Poisson process with intensity \( \lambda \) . Then \( S \) is a martingale because \( - S \) is a compensated Poisson process: intuitively, \( S \) is a fair investment giving zero gain in average, because the deterministic increase \( {\lambda t} \) is comp... | We denote by \( {T}_{n} \) the jump times of \( S \) and consider the strategy\n\n\[ \n{u}_{t} = {\mathbb{1}}_{\left\lbrack 0,{T}_{1}\right\rbrack }\left( t\right) \]\n\n(14.2)\n\nwhich consists in buying (at zero price) the asset at \( t = 0 \) and selling it at the time of the first jump. Note that \( u \) is left-co... | Yes |
Example 14.8 Let \( N \) be a Poisson process with jump times \( {\left( {T}_{n}\right) }_{n \geq 1} \) (we also set \( {T}_{0} = 0 \) ). Then\n\n\[ \n{N}_{t} = n\;\text{ for }\;t \in \left\lbrack {{T}_{n},{T}_{n + 1}\lbrack, n \geq 0,}\right. \]\n\nand \( N \in \mathbb{D} \) . We already noted that\n\n\[ \n\left. {{N}... | \[ \n{\int }_{0}^{t}{N}_{s - }d{N}_{s} = \frac{{N}_{t}\left( {{N}_{t} - 1}\right) }{2},\;t \geq 0. \]\n\n(14.9) | Yes |
If a process \( S \in \mathbb{D} \) has bounded variation a.s. then it is a semimartingale. | Indeed, for any simple predictable process \( u \), we have\n\n\[ \left| {{\int }_{0}^{T}{u}_{t}d{S}_{t}}\right| \leq {V}_{\left\lbrack 0, T\right\rbrack }\left( S\right) \mathop{\sup }\limits_{{\left\lbrack {0, T}\right\rbrack \times \Omega }}\left| u\right| \]\n\nwhere \( {V}_{\left\lbrack 0, T\right\rbrack }\left( S... | Yes |
Every square integrable martingale \( S \in \mathbb{D} \) is a semimartingale. | Indeed, for any simple predictable process \( u \), we have\n\n\[ E\left\lbrack {\left( {\int }_{0}^{T}{u}_{t}d{S}_{t}\right) }^{2}\right\rbrack = E\left\lbrack {\left( \mathop{\sum }\limits_{{k = 1}}^{N}{e}_{k}\left( {S}_{{T}_{k}} - {S}_{{T}_{k - 1}}\right) \right) }^{2}\right\rbrack = \]\n\n(by the orthogonality of t... | Yes |
A Lévy process is a semimartingale. | Indeed, by the decomposition in Corollary 13.38, every Lévy process is the sum of a càdlàg \( {L}^{2} \) - martingale with a BV process. Since the set of semimartingales forms a vector space, the thesis is a consequence of the two examples above. | Yes |
The space of simple predictable processes is dense in \( \mathbb{L} \) under the uniform convergence in probability: for any \( u \in \mathbb{L} \) there exists a sequence \( \left( {u}^{n}\right) \) of simple predictable processes such that (14.10) holds. The stochastic integral with respect to a semimartingale \( S \... | Proof. See Protter [287], Theorems II-10 and II-11. | No |
Theorem 14.15 Let \( S \) be a semimartingale and \( u \) be a process in \( \mathbb{D} \) or in \( \mathbb{L} \) . Then the Riemann-Stieltjes sum \( {}^{5} \n\n\[ \n\mathop{\sum }\limits_{{k = 1}}^{{N}_{n}}{u}_{{T}_{k - 1}^{n}}\left( {{S}_{{T}_{k}^{n}}^{t} - {S}_{{T}_{k - 1}^{n}}^{t}}\right) \n\] \n\nconverges uniform... | Proof. See Theorem II-21 in Protter [287]. | No |
Proposition 14.19 For any \( \varphi \in {\mathbb{L}}_{\nu }^{2} \), the process \( M \) in (14.13) is a square-integrable martingale such that\n\n\[ E\left\lbrack {M}_{t}\right\rbrack = 0,\;\operatorname{var}\left( {M}_{t}\right) = E\left\lbrack {{\int }_{0}^{t}{\int }_{\mathbb{R}}{\varphi }^{2}\left( {t, x}\right) \n... | Proof. The thesis follows from Proposition 14.17 by a limit argument: for full details see, for instance, Theorem 4.2.3 in Applebaum [11]. | No |
Proposition 14.20 For any \( \varphi \in {\mathbb{L}}_{\nu ,\text{ loc }}^{2} \), the integral process \( M \) in (14.13) is a local martingale. | Proof. See, for instance, Theorem 4.2.12 in Applebaum [11]. | No |
Proposition 14.24 Let \( S \) be a one dimensional Lévy process with Lévy-Itô decomposition\n\n\[ \n{S}_{t} = {\mu }_{R}t + \sigma {W}_{t} + {S}_{t}^{R} + {M}_{t}^{R} \]\n\n(14.18)\n\nwhere \( W \) is a standard Brownian motion and\n\n\[ \n{S}_{t}^{R} = {\int }_{0}^{t}{\int }_{\left| y\right| \geq R}{yJ}\left( {{ds},{d... | Proof. Any \( u \in \mathbb{L} \) is progressively measurable and such that\n\n\[ \n{\int }_{0}^{T}{u}_{t}^{2}{dt} < \infty \;\text{ a.s. } \]\n\nor in other terms \( u \in {\mathbb{L}}_{\text{loc }}^{2} \) (cf. Definition 4.1) so that the Brownian integral in (14.21) is well-defined and it is a local martingale. Moreo... | Yes |
Theorem 14.26 (Deterministic Itô formula) Let \( f \in {C}^{1}\left( \mathbb{R}\right) \) and \( X \) be a càdlàg (deterministic) function with bounded variation. Then we have\n\n\[ f\left( {X}_{t}\right) - f\left( {X}_{0}\right) = {\int }_{0}^{t}{f}^{\prime }\left( {X}_{s - }\right) d{X}_{s} \]\n\n\[ + \mathop{\sum }\... | Proof. We show that the series in (14.27) is convergent: first of all, it has a countable number of terms because \( X \in \mathrm{{BV}} \) . For the same reason, \( \parallel X{\parallel }_{\infty } = \) sup \( \left| {X}_{t}\right| < \infty \) and therefore we have \( t \in \left\lbrack {0, T}\right\rbrack \)\n\n\[ \... | Yes |
Theorem 14.28 If \( S \) is a semimartingale, then \( \langle S\rangle \) is a càdlàg, adapted and increasing process such that \( \langle S{\rangle }_{0} = {S}_{0}^{2} \) and \( \Delta \langle S{\rangle }_{t} = {\left( \Delta {S}_{t}\right) }^{2},\;t \in \left\lbrack {0, T}\right\rbrack \) | Proof. We use repeatedly the elementary equality \[ {\left( b - a\right) }^{2} = {b}^{2} - {a}^{2} - {2a}\left( {b - a}\right) ,\;a, b \in \mathbb{R}. \] It is clear that, by definition of stochastic integral, \( \langle S\rangle \in \mathbb{D} \). Moreover, by (14.32) we have \[ {\left( \Delta {S}_{t}\right) }^{2} = {... | Yes |
Example 14.30 By (14.31) and Theorem 3.74, for a real Brownian motion \( W \) we have\n\n\[ \langle W{\rangle }_{t} = \langle W{\rangle }_{t}^{c} = t \] | By (14.31) and Proposition 4.24, the quadratic variation of a Brownian integral\n\n\[ {X}_{t} = {\int }_{0}^{t}{u}_{s}d{W}_{s} \]\n\nwith \( u \in {\mathbb{L}}_{\text{loc }}^{2} \), is given by\n\n\[ \langle X{\rangle }_{t} = \langle X{\rangle }_{t}^{c} = {\int }_{0}^{t}{u}_{s}^{2}{ds} \] | No |
If \( N \) is a Poisson process, then by definition of quadratic variation and Example 14.8, we have\n\n\[\n\langle N{\rangle }_{t} = {N}_{t}\n\] | Moreover\n\n\[\n\langle N{\rangle }_{t}^{c} = {N}_{t} - \mathop{\sum }\limits_{{0 < s \leq t}}{\left( \Delta {N}_{s}\right) }^{2} = 0.\n\] | Yes |
If \( S \) is a continuous semimartingale with bounded variation then \( \langle S{\rangle }_{t} = {S}_{0}^{2}, t \in \left\lbrack {0, T}\right\rbrack \) . | The proof is the same as in the deterministic case (cf. Proposition 3.73) and uses the characterization (14.31) of the quadratic variation process. | No |
If \( S \) is a one-dimensional Lévy process with characteristic triplet \( \left( {{\mu }_{1},\sigma ,\nu }\right) \) then | \[ \langle S{\rangle }_{t} = {\sigma }^{2}t + \mathop{\sum }\limits_{{0 < s \leq t}}{\left( \Delta {S}_{s}\right) }^{2} = {\sigma }^{2}t + {\int }_{\mathbb{R}}{x}^{2}J\left( {{ds},{dx}}\right) ,\] where \( {J}_{t} \) denotes the jump measure of \( S \) (cf. (13.59)). Moreover \[ \langle S{\rangle }_{t}^{c} = {\sigma }^... | Yes |
Example 14.35 We apply the Itô formula with \( f\left( x\right) = {x}^{2} \) and \( {X}_{t} = {N}_{t} \) , Poisson process. We get | \[ {N}_{t}^{2} = 2{\int }_{0}^{t}{N}_{s - }d{N}_{s} + \mathop{\sum }\limits_{{0 < s \leq t}}\left( {{N}_{s}^{2} - {N}_{s - }^{2} - 2{N}_{s - }\Delta {N}_{s}}\right) \] \[ = 2{\int }_{0}^{t}{N}_{s - }d{N}_{s} + \mathop{\sum }\limits_{{k = 1}}^{{N}_{t}}\left( {{k}^{2} - {\left( k - 1\right) }^{2} - 2\left( {k - 1}\right)... | Yes |
Lemma 14.36 Let \( X \) be a one dimensional Lévy process with R-triplet \( \left( {{\mu }_{R},{\sigma }^{2},\nu }\right) \) and \( f = f\left( {t, x}\right) \in {C}^{1,2}\left( {\left\lbrack {0, T}\right\rbrack \times \mathbb{R}}\right) \) . Then we have\n\n\[ \n{df}\left( {t,{X}_{t}}\right) = \left( {{\mathcal{A}}_{R... | Proof. First of all we show that each term in (14.39) and (14.40) is well defined. Let us recall that (cf. (13.46))\n\n\[ \n{\int }_{\left| y\right| < 1}{y}^{2}\nu \left( {dy}\right) < \infty \n\]\n\n(14.41)\n\nand let us denote by\n\n\[ \nD\left( {z, R}\right) = \rbrack z - R, z + R\lbrack \n\]\n\nthe interval centere... | Yes |
Theorem 14.37 (Itô formula) Let \( X \) be a one dimensional Lévy process with \( R \) -triplet \( \left( {{\mu }_{R},{\sigma }^{2},\nu }\right) \) and \( f = f\left( {t, x}\right) \in {C}^{1,2}\left( {\left\lbrack {0, T}\right\rbrack \times \mathbb{R}}\right) \) . If \( f \) is bounded then we have\n\n\[ \n{df}\left( ... | Proof. Formulas (14.44)-(14.45) follow directly by adding to (14.40), and subtracting from (14.39), the integral term\n\n\[ \n{\int }_{\left| y\right| \geq R}\left( {f\left( {t, y + {X}_{t - }}\right) - f\left( {t,{X}_{t - }}\right) }\right) \nu \left( {dy}\right) {dt} \]\n\nthat is well defined because \( f \) is boun... | Yes |
Let \( X \) be a one dimensional Lévy process with \( R \) -triplet \( \left( {{\mu }_{R},{\sigma }^{2},\nu }\right) \) and characteristic exponent \( {\psi }_{X} \). We assume that \[ {\int }_{\left| y\right| \geq 1}{e}^{2y}\nu \left( {dy}\right) < \infty \] | By Proposition 13.49, condition (14.47) is equivalent to the existence of the second moment of \( X \): \[ E\left\lbrack {e}^{2{X}_{t}}\right\rbrack = {e}^{t{\psi }_{X}\left( {-{2i}}\right) }.\] | Yes |
By the Lévy-Itô decomposition, a Lévy process \( X \) with triplet \( \left( {{\mu }_{1},{\sigma }^{2},\nu }\right) \) can be written as\n\n\[ d{X}_{t} = {\mu }_{1}{dt} + {\sigma d}{W}_{t} + {\int }_{\left| y\right| < 1}y\widetilde{J}\left( {{dt},{dy}}\right) + {\int }_{\left| y\right| \geq 1}{yJ}\left( {{dt},{dy}}\rig... | and therefore it can be considered as a solution of a SDE of type (14.55) where\n\n\[ \bar{b}\left( {t, x}\right) = {\mu }_{1},\;\sigma \left( {t, x}\right) = \sigma ,\;\widetilde{a}\left( {t, x, y}\right) = \bar{a}\left( {t, x, y}\right) = y. \] | Yes |
Under the hypotheses of Theorem 14.43, let \( X \) be a strong solution of the SDE (14.55) and \( f = f\left( {t, x}\right) \in {C}^{1,2}\left( {\left\lbrack {0, T}\right\rbrack \times \mathbb{R}}\right) \) . Then we have\n\n\[ {df}\left( {t,{X}_{t}}\right) = \left( {\mathcal{A} + {\partial }_{t}}\right) f\left( {t,{X}... | where \( \mathcal{A} \) is the integro-differential operator with variable coefficients\n\n\[ \mathcal{A}f\left( {t, x}\right) = \bar{b}\left( {t, x}\right) {\partial }_{x}f\left( {t, x}\right) + \frac{{\sigma }^{2}\left( {t, x}\right) }{2}{\partial }_{xx}f\left( {t, x}\right) \]\n\n\[ + {\int }_{\left| y\right| < 1}\l... | Yes |
Theorem 14.45 Assume the following Lipschitz and growth conditions:\n\ni) for every \( n \in \mathbb{N} \) there exists a constant \( {K}_{n} \) such that\n\n\[ \n{\left| b\left( t,{x}_{1}\right) - b\left( t,{x}_{2}\right) \right| }^{2} + {\left| \sigma \left( t,{x}_{1}\right) - \sigma \left( t,{x}_{2}\right) \right| }... | Moreover, there exists a positive constant \( C \), depending on \( K,{K}_{n} \) and \( T \) only, such that\n\n\[ \nE\left\lbrack {X}_{t}^{2}\right\rbrack \leq C\left( {1 + {X}_{0}^{2}}\right) ,\;t \in \left\lbrack {0, T}\right\rbrack . \n\]\n\n(14.61) | Yes |
The SDE (14.49) for an exponential Lévy process \( {S}_{t} = {e}^{{X}_{t}} \) is of the form (14.58), that is\n\n\[ d{S}_{t} = b\left( {t,{S}_{t - }}\right) {dt} + \sigma \left( {t,{S}_{t - }}\right) d{W}_{t} + {\int }_{\mathbb{R}}a\left( {t,{S}_{t - }, y}\right) \widetilde{J}\left( {{dt},{dy}}\right) \]\n\nwhere\n\n\[... | Under condition (14.47) on the Lévy measure \( \nu \), the Lipschitz and growth conditions (14.59)-(14.60) are satisfied. | No |
Example 14.51 (Cauchy distribution) We consider a \( \alpha \) -stable process with \( \alpha = 1 \) : this is a pure jump process with no diffusion component, see Section 13.4.2. By (13.80), the characteristic exponent takes the form\n\n\[ \n\psi \left( \xi \right) = {i\mu \xi } - \sigma \left| \xi \right| \left( {1 +... | Even if the expression of \( {\psi }_{X} \) is known, in most cases it is not possible to compute explicitly \( \Gamma \) by Fourier inversion. However, formula (14.66) is used \( {}^{8} \) in the practical applications because it can be inverted numerically in various efficient ways: numerical methods in option pricin... | No |
Proposition 14.52 Let \( Z \) be a Lévy process with characteristic exponent \( \psi \) . Then the characteristic function of \( X \) in (14.69) is equal to\n\n\[ \n{\varphi }_{{X}_{t}}\left( \xi \right) = E\left\lbrack {e}^{{i\xi }{X}_{t}}\right\rbrack = \exp \left( {{i\xi }{X}_{0}{e}^{-{Bt}} + {\int }_{0}^{t}\psi \le... | Proof. The thesis is a consequence of the identity\n\n\[ \nE\left\lbrack {e}^{i{\int }_{0}^{t}f\left( s\right) d{Z}_{s}}\right\rbrack = {e}^{{\int }_{0}^{t}\psi \left( {f\left( s\right) }\right) {ds}} \]\n\n(14.71)\n\nwhich holds for any continuous function \( f : \left\lbrack {0, T}\right\rbrack \rightarrow \mathbb{R}... | Yes |
If \( {Z}_{t} = {\mu t} + \sigma {W}_{t} \) is a Brownian motion with drift and characteristic exponent\n\n\[ \n{\psi }_{Z}\left( \xi \right) = {i\mu \xi } - \frac{{\sigma }^{2}{\xi }^{2}}{2} \n\]\n\nthen, by (14.70), the characteristic function of the solution \( X \) in (14.68) is equal to\n\n\[ \n{\varphi }_{{X}_{t}... | In this case, as we already showed in Section 9.5, \( {X}_{t} \) is a Gaussian random variable. | No |
In an exponential Lévy model, the underlying asset is of the form (14.72) where \( X \) is a Lévy process satisfying the SDE\n\n\[ d{X}_{t} = {\mu }_{\infty }{dt} + {\sigma d}{W}_{t} + {\int }_{\mathbb{R}}y\widetilde{J}\left( {{dt},{dy}}\right) \]\n\nunder condition\n\n\[ {\int }_{\left| y\right| \geq 1}{e}^{2y}\nu \le... | More precisely, as in Example 14.40, we have that \( \widetilde{S} \) is a martingale if and only if (14.78) is satisfied. Thus, under an EMM the drift parameter \( {\mu }_{\infty } \) is determined by (14.78):\n\n\[ {\mu }_{\infty } = r - \frac{{\sigma }^{2}}{2} - {\int }_{\mathbb{R}}\left( {{e}^{y} - 1 - y}\right) \n... | Yes |
In the Black-Scholes model, we have\n\n\[ \n{S}_{T} = {S}_{0}{e}^{{X}_{T}},\;{X}_{T} = \left( {r - \frac{{\sigma }^{2}}{2}}\right) T + \sigma {W}_{T}, \n\] | and\n\n\[ \n{\varphi }_{{X}_{T}}\left( \xi \right) = {e}^{i\left( {r - \frac{{\sigma }^{2}}{2}}\right) {T\xi } - \frac{{\sigma }^{2}{\xi }^{2}}{2}T}. \n\] | Yes |
Example 15.3 We have\n\n\[ \n{f}_{\alpha }^{\mathrm{{Call}}}\left( x\right) = {e}^{-{\alpha x}}{\left( {e}^{x} - K\right) }^{ + },\;{f}_{\alpha }^{\mathrm{{Put}}}\left( x\right) = {e}^{-{\alpha x}}{\left( K - {e}^{x}\right) }^{ + }, \n\]\n\nand therefore \( {f}_{\alpha }^{\text{Call }} \in {L}^{1}\left( \mathbb{R}\righ... | ▱ | No |
Lemma 15.4 If \( g \in {W}^{1,2}\left( \mathbb{R}\right) \), that is \( g \in {L}^{2}\left( \mathbb{R}\right) \) and the weak derivative \( {Dg} \in {L}^{2}\left( \mathbb{R}\right) \), then \( \widehat{g} \in {L}^{1}\left( \mathbb{R}\right) \) . | Proof. It is well known that if \( g \in {W}^{1,2}\left( \mathbb{R}\right) \) then \( g,\widehat{g} \in {L}^{2}\left( \mathbb{R}\right) \) and\n\n\[ \n\widehat{Dg}\left( \xi \right) = - {i\xi }\widehat{g}\left( \xi \right) \n\]\n\nThen we have\n\n\[ \n{\int }_{\mathbb{R}}\left( {{\left| \widehat{g}\left( \xi \right) \r... | Yes |
Example 15.5 By Lemma 15.4, \( {f}_{\alpha }^{\text{Call }},\widehat{{f}_{\alpha }^{\text{Call }}} \in {L}^{1}\left( \mathbb{R}\right) \) for any \( \alpha > 1 \) . Indeed \( {f}_{\alpha }^{\text{Call }} \in {W}^{1,2}\left( \mathbb{R}\right) \) because \( {f}_{\alpha }^{\text{Call }} \in {L}^{2} \) and | \[ D{f}_{\alpha }^{\text{Call }}\left( x\right) = \left\{ \begin{array}{ll} 0 & \text{ if }x < \log K, \\ \left( {1 - \alpha }\right) {e}^{\left( {1 - \alpha }\right) x} + {\alpha K}{e}^{-{\alpha x}} & \text{ if }x > \log K, \end{array}\right. \] is a square integrable function for \( \alpha > 1 \) . | Yes |
Proposition 15.6 Assume that there exists \( \alpha \in \mathbb{R} \) such that\ni) \( {f}_{\alpha },{\widehat{f}}_{\alpha } \in {L}^{1}\left( \mathbb{R}\right) \) ;\nii) \( {E}^{Q}\left\lbrack {S}_{T}^{\alpha }\right\rbrack \) is finite.\n\nThen the following pricing formula holds:\n\n\[ H\left( {{S}_{0}, T}\right) = ... | Proof. First of all we show that conditions \( i \) ) and \( {ii} \) ) guarantee that the integral in (15.8) converges. Indeed, by \( i \) ) we have that \( \xi \mapsto \widehat{f}\left( {\xi + {i\alpha }}\right) \) is integrable because\n\n\[ \widehat{f}\left( {\xi + {i\alpha }}\right) = {\int }_{\mathbb{R}}{e}^{i\lef... | Yes |
Corollary 15.8 (Delta) Under the assumptions of Proposition 15.6, if in addition one of the functions \( \xi \mapsto \left( {1 + \left| \xi \right| }\right) {\varphi }_{{X}_{T}}\left( {-\left( {\xi + {i\alpha }}\right) }\right) \) or \( \widehat{D{f}_{\alpha }} \) is integrable, then we have\n\n\[ \operatorname{Delta}\... | Proof. The thesis follows by differentiating formula (15.8) with respect to \( {S}_{0} \) : the additional assumptions guarantee that we can exchange the integral and differential signs. | No |
Theorem 15.9 (Call option) For any \( \alpha > 1 \) such that \( {E}^{Q}\left\lbrack {S}_{T}^{\alpha }\right\rbrack \) is finite, we have the following pricing formula for a Call option with strike \( K \) and maturity \( T \) :\n\n\[ \operatorname{Call}\left( {{S}_{0}, K, T}\right) = \frac{{e}^{-{rT}}{S}_{0}^{\alpha }... | Proof. We recall Example 15.5 and use the pricing formula (15.8) of Proposition 15.6. To this end, we compute \( \widehat{{f}^{\text{Call }}}\left( {\xi + {i\alpha }}\right) \) : by (15.9), we have\n\n\[ \widehat{{f}^{\text{Call }}}\left( {\xi + {i\alpha }}\right) = \widehat{{f}_{\alpha }^{\text{Call }}}\left( \xi \rig... | Yes |
Example 15.15 (Heston model) We consider another example where the option prices are sensitive to the choice of the damping parameter \( \alpha \) . In the Heston stochastic volatility model (cf. Example 10.33) the risk-neutral dynamics of the asset and its variance is given by (15.4), that is\n\n\[ d{S}_{t} = r{S}_{t}... | To avoid complex discontinuities (cf. Paragraph 15.1), we use the risk-neutral characteristic function given by Bakshi, Cao and Chen [17], which takes the form\n\n\[ {\varphi }_{{X}_{T}}\left( \xi \right) = \exp \left( {{i\xi rT} + \frac{{\nu }_{0}}{{\eta }^{2}}\left( \frac{1 - {e}^{-D\left( \xi \right) T}}{1 - G\left(... | Yes |
Example 15.17 (Call price and Delta) For a Call option we have\n\n\[ \n{B}_{0}^{\mathrm{{Call}}}\left( {S}_{0}\right) = {\int }_{a}^{b}{\left( {e}^{x + \log {S}_{0}} - K\right) }^{ + }{dx} = \n\]\n\n(assuming \( a < - \log \frac{{S}_{0}}{K} \) )\n\n\[ \n= {\int }_{-\log \frac{{S}_{0}}{K}}^{b}\left( {{e}^{x + \log {S}_{... | The coefficients of the Delta of a Call option can be obtained by differentiating with respect to \( {S}_{0} \) :\n\n\[ \n{B}_{0}^{\text{Delta }}\left( {S}_{0}\right) = \frac{d}{d{S}_{0}}{B}_{0}^{\text{Call }}\left( {S}_{0}\right) = {e}^{b} - \frac{K}{{S}_{0}}, \n\]\n\nand, for \( k \geq 1 \) ,\n\n\[ \n{B}_{k}^{\text{D... | Yes |
Example 15.20 (Heston) We consider the Heston model (cf. Example 15.15) with\n\n\[ r = 0, k = {1.5768},{\nu }_{0} = {0.0175},{\nu }_{\infty } = {0.0398},\eta = {0.5751},\varrho = - {0.5711}\text{.} \]\n\nWe analyze the Fourier-cosine approximation with \( a, b \) as in (15.36) and different choices of \( L \) and \( N ... | In particular, we compute the price of a Call with \( {S}_{0} = K = {100} \) and maturity \( T = 1 \) : for \( L = {10},{20},{30},{40} \) and \( N = {1000} \), we obtain the same price that is our reference value \( \mathrm{{RV}} = {5.785155434} \) . Figure 15.17 shows the percentage difference (15.37) between RV and t... | Yes |
Example 15.22 (CGMY) The CGMY model is based on a particular tempered stable process (cf. Section 13.4.3) and encompasses the VG and Black-Scholes models: as usual, the underlying asset is in the form \( {S}_{T} = {S}_{0}{e}^{{X}_{T}} \) and by (13.98)-(13.99) the risk-neutral characteristic exponent given by | \[ \psi \left( \xi \right) = i{\mu }_{\infty }^{Q}\xi + C\left( {{\left( M - i\xi \right) }^{Y} - {M}^{Y} + {\left( G + i\xi \right) }^{Y} - {G}^{Y}}\right. \]\n\[ \left. {+{i\xi Y}\left( {{M}^{Y - 1} - {G}^{Y - 1}}\right) }\right) \Gamma \left( {-Y}\right) \]\n\[ {\mu }_{\infty }^{Q} = r + {C\Gamma }\left( {-Y}\right)... | Yes |
Example 16.10 Let \( u \in {L}^{2}\left( {0, T}\right) \) be a (deterministic) function and\n\n\[ \nX = {\int }_{0}^{t}u\left( r\right) d{W}_{r} \n\]\n\nThen \( X \in {\mathbb{D}}^{1,2} \) and\n\n\[ \n{D}_{s}X = \left\{ \begin{array}{ll} u\left( s\right) & \text{ for }s \leq t \\ 0 & \text{ for }s > t \end{array}\right... | Indeed the sequence defined by\n\n\[ \n{X}_{n} = \mathop{\sum }\limits_{{k = 1}}^{{{k}_{n}\left( t\right) }}u\left( {t}_{n}^{k - 1}\right) {\Delta }_{n}^{k} \n\]\n\n is such that\n\n\[ \n{D}_{s}{X}_{n} = \varphi \left( {t}_{n}^{{k}_{n}\left( s\right) }\right) \n\]\n\n if \( s \leq {t}_{n}^{{k}_{n}\left( t\right) } \) a... | Yes |
Proposition 16.12 (Chain rule) Let \( {}^{5}\varphi \in {C}_{\text{pol }}^{\infty }\left( \mathbb{R}\right) \). Then:\ni) if \( X \in {\mathbb{D}}^{1,\infty } \), then \( \varphi \left( X\right) \in {\mathbb{D}}^{1,\infty } \) and\n\n\[{D\varphi }\left( X\right) = {\varphi }^{\prime }\left( X\right) {DX}\]\n\n(16.3)\n\... | Proof. We prove only \( {ii} \) ) since the other parts can be proved essentially in an analogous way. If \( X \in \mathcal{S},\varphi \in {C}^{1} \) and both \( \varphi \) and its first-order derivative are bounded, then \( \varphi \left( X\right) \in \mathcal{S} \) and the claim is obvious.\n\nIf \( X \in {\mathbb{D}... | Yes |
Example 16.13 By the chain rule, \( {\left( {W}_{t}\right) }^{2} \in {\mathbb{D}}^{1,\infty } \) and | \[ {D}_{s}{W}_{t}^{2} = 2{W}_{t}{\mathbb{1}}_{\left\lbrack 0, t\right\rbrack }\left( s\right) \] | Yes |
Let \( u \in {\mathbb{L}}^{2} \) such that \( {u}_{t} \in {\mathbb{D}}^{1,2} \) for every \( t \) . Then \[ X \mathrel{\text{:=}} {\int }_{0}^{t}{u}_{r}d{W}_{r} \in {\mathbb{D}}^{1,2} \] and for \( s \leq t \) \[ {D}_{s}{\int }_{0}^{t}{u}_{r}d{W}_{r} = {u}_{s} + {\int }_{s}^{t}{D}_{s}{u}_{r}d{W}_{r} \] | Indeed, for fixed \( t \), we consider the sequence defined by \[ {X}_{n} \mathrel{\text{:=}} \mathop{\sum }\limits_{{k = 1}}^{{{k}_{n}\left( t\right) }}{u}_{{t}_{n}^{k - 1}}{\Delta }_{n}^{k},\;n \in \mathbb{N}, \] approximating \( X \) in \( {L}^{2}\left( \Omega \right) \) . Then \( {X}_{n} \in {\mathbb{D}}^{1,2} \) a... | Yes |
Let us consider the solution \( \left( {X}_{t}\right) \) of the SDE\n\n\[ \n{X}_{t} = x + {\int }_{0}^{t}b\left( {r,{X}_{r}}\right) {dr} + {\int }_{0}^{t}\sigma \left( {r,{X}_{r}}\right) d{W}_{r},\n\]\n\nwith \( x \in \mathbb{R} \) and the coefficients \( b,\sigma \in {C}_{b}^{1} \) . Then \( {X}_{t} \in {\mathbb{D}}^{... | We do not go into the details of the proof of the first claim. The idea is to use an approximation argument based on the Euler scheme (cf. Paragraph 12.2): more precisely, the claim follows from the fact that \( \left( {X}_{t}\right) \) is the limit of the sequence of piecewise constant processes defined by\n\n\[ \n{X}... | No |
Lemma 16.17 Let \( Y \) be as in (16.7) and \( Z \) be solution of the SDE\n\n\[ \n{Z}_{t} = 1 + {\int }_{0}^{t}\left( {{\left( {\partial }_{x}\sigma \right) }^{2} - {\partial }_{x}b}\right) \left( {r,{X}_{r}}\right) {Z}_{r}{dr} - {\int }_{0}^{t}{\partial }_{x}\sigma \left( {r,{X}_{r}}\right) {Z}_{r}d{W}_{r}.\n\]\n\n(1... | Proof. We have \( {Y}_{0}{Z}_{0} = 1 \) and, omitting the arguments, by the Itô formula we have\n\n\[ \nd\left( {{Y}_{t}{Z}_{t}}\right) = {Y}_{t}d{Z}_{t} + {Z}_{t}d{Y}_{t} + d\langle Y, Z{\rangle }_{t}\n\]\n\n\[ \n= {Y}_{t}{Z}_{t}\left( {\left( {{\left( {\partial }_{x}\sigma \right) }^{2} - \left( {{\partial }_{x}b}\ri... | Yes |
Proposition 16.18 Let \( X, Y, Z \) be the solutions of the SDEs (16.4),(16.7) and (16.8), respectively. Then\n\n\[ \n{D}_{s}{X}_{t} = {Y}_{t}{Z}_{s}\sigma \left( {s,{X}_{s}}\right) \n\]\n\n(16.9) | Proof. We recall that, for fixed \( s \), the process \( {D}_{s}{X}_{t} \) verifies the SDE (16.5) over \( \left\lbrack {s, T}\right\rbrack \) and we prove that \( {A}_{t} \mathrel{\text{:=}} {Y}_{t}{Z}_{s}\sigma \left( {s,{X}_{s}}\right) \) verifies the same equation: the claim will then follow from the uniqueness res... | Yes |
Theorem 16.26 (Clark-Ocone formula) If \( X \in {\mathbb{D}}^{1,2} \), then\n\n\[ X = E\left\lbrack X\right\rbrack + {\int }_{0}^{T}E\left\lbrack {{D}_{t}X \mid {\mathcal{F}}_{t}^{W}}\right\rbrack d{W}_{t} \] | Proof. It is not restrictive to suppose \( E\left\lbrack X\right\rbrack = 0 \) . For every simple adapted process \( U \in \mathcal{P} \) we have, by the duality relation of Theorem 16.23,\n\n\[ E\left\lbrack {X{D}^{ * }U}\right\rbrack = E\left\lbrack {{\int }_{0}^{T}\left( {{D}_{t}X}\right) {U}_{t}{dt}}\right\rbrack =... | Yes |
Theorem 16.28 (Stochastic integration by parts) Let \( F \in {C}_{b}^{1} \) and let \( X \in {\mathbb{D}}^{1,2} \) . Then the following integration by parts holds:\n\n\[ E\left\lbrack {{F}^{\prime }\left( X\right) Y}\right\rbrack = E\left\lbrack {F\left( X\right) {\int }_{0}^{T}\frac{{u}_{t}Y}{{\int }_{0}^{T}{u}_{s}{D}... | Sketch of the proof. By the chain rule we have\n\n\[ {D}_{t}F\left( X\right) = {F}^{\prime }\left( X\right) {D}_{t}X \]\n\nmultiplying by \( {u}_{t}Y \) and integrating from 0 to \( T \) we get\n\n\[ {\int }_{0}^{T}{u}_{t}Y{D}_{t}F\left( X\right) {dt} = {F}^{\prime }\left( X\right) Y{\int }_{0}^{T}{u}_{t}{D}_{t}{Xdt} \... | No |
Example 16.30 (Delta) We observe that \( {D}_{s}{S}_{T} = \sigma {S}_{T} \) and \( {\partial }_{x}{S}_{T} = \frac{{S}_{T}}{x} \) . Then, by (16.17) we have the following expression for the Black-Scholes Delta | \[ \Delta = {e}^{-{rT}}{\partial }_{x}E\left\lbrack {F\left( {S}_{T}\right) }\right\rbrack \] \[ = {e}^{-{rT}}E\left\lbrack {F\left( {S}_{T}\right) {\int }_{0}^{T}\frac{{\partial }_{x}{S}_{T}}{{\int }_{0}^{T}{D}_{s}{S}_{T}{ds}}\diamond d{W}_{t}}\right\rbrack \] \[ = {e}^{-{rT}}E\left\lbrack {F\left( {S}_{T}\right) {\in... | Yes |
Proposition 16.31 Let \( X \in {\mathbb{D}}^{1,2} \) and let \( U \) be a second-order Skorohod-integrable process. Then\n\n\[ \n{\int }_{0}^{T}X{U}_{t}\diamond d{W}_{t} = X{\int }_{0}^{T}{U}_{t}\diamond d{W}_{t} - {\int }_{0}^{T}\left( {{D}_{t}X}\right) {U}_{t}{dt}.\n\] | Proof. For every \( Y \in \mathcal{S} \), by the duality relation, we have\n\n\[ \nE\left\lbrack {Y{D}^{ * }\left( {XU}\right) }\right\rbrack = E\left\lbrack {{\int }_{0}^{T}\left( {{D}_{t}Y}\right) X{U}_{t}{dt}}\right\rbrack = \n\]\n\n(by the chain rule)\n\n\[ \n= E\left\lbrack {{\int }_{0}^{T}\left( {{D}_{t}\left( {Y... | Yes |
[{\int }_{0}^{T}{W}_{T}\diamond d{W}_{t} = {W}_{T}^{2} - T] | By a direct application of (16.20), we have\n\n\[ \n{\int }_{0}^{T}{W}_{T}\diamond d{W}_{t} = {W}_{T}^{2} - T \n\] | Yes |
Example 16.33 (Vega) Let us compute the Vega of a European option with payoff function \( F \) in the Black-Scholes model: we first notice that\n\n\[ \n{\partial }_{\sigma }{S}_{T} = \left( {{W}_{T} - {2\sigma T}}\right) {S}_{T},\;{D}_{s}{S}_{T}\sigma {S}_{T}.\n\]\n\nThen\n\n\[ \n\mathcal{V} = {e}^{-{rT}}{\partial }_{\... | \n(by the integration-by-parts formula (16.17))\n\n\[ \n= {e}^{-{rT}}E\left\lbrack {F\left( {S}_{T}\right) {\int }_{0}^{T}\frac{{W}_{T} - {\sigma T}}{\sigma T}\diamond d{W}_{t}}\right\rbrack =\n\]\n\n(by \( \left( {16.20}\right) \) )\n\n\[ \n= {e}^{-{rT}}E\left\lbrack {F\left( {S}_{T}\right) \left( {\frac{{W}_{T} - {\s... | No |
Example 16.34 (Gamma) We compute the Gamma of a European option with payoff function \( F \) in the Black-Scholes model: | \[ \Gamma = {e}^{-{rT}}{\partial }_{xx}E\left\lbrack {F\left( {S}_{T}\right) }\right\rbrack = \] \[ = \frac{{e}^{-{rT}}}{\sigma T}E\left\lbrack {{\partial }_{x}\left( \frac{F\left( {S}_{T}\right) }{x}\right) {W}_{T}}\right\rbrack = - \frac{{e}^{-{rT}}}{{\sigma T}{x}^{2}}E\left\lbrack {F\left( {S}_{T}\right) {W}_{T}}\ri... | Yes |
We give the expression of the Delta of an arithmetic Asian option with Black-Scholes dynamics (16.18) for the underlying asset. We denote the average by\n\n\\[ \nX = \frac{1}{T}{\\int }_{0}^{T}{S}_{t}{dt} \n\\]\n\nand we observe that \\( {\\partial }_{x}X = \frac{X}{x} \\) and\n\n\\[ \n{\\int }_{0}^{T}{D}_{s}{Xds} = {\... | Then we have\n\n\\[ \n\\Delta = {e}^{-{rT}}{\\partial }_{x}E\\left\\lbrack {F\\left( X\\right) }\\right\\rbrack = \\frac{{e}^{-{rT}}}{x}E\\left\\lbrack {{F}^{\\prime }\\left( X\\right) X}\\right\\rbrack = \n\\]\n\n(by (16.17) and (16.21))\n\n\\[ \n= \\frac{{e}^{-{rT}}}{\\sigma x}E\\left\\lbrack {F\\left( X\\right) {\\i... | Yes |
We extend Example 16.30 to the case of a model with local volatility\n\n\[ \n{S}_{t} = x + {\int }_{0}^{t}b\left( {s,{S}_{s}}\right) {ds} + {\int }_{0}^{t}\sigma \left( {s,{S}_{s}}\right) d{W}_{s}.\n\]\n\nUnder suitable assumptions on the coefficients, we prove the following Bismut-Elworthy formula:\n\n\[ \nE\left\lbra... | We recall that, by Proposition 16.18, we have\n\n\[ \n{D}_{s}{S}_{T} = {Y}_{T}{Z}_{s}\sigma \left( {s,{S}_{s}}\right)\n\]\n\n(16.23)\n\nsince\n\n\[ \n{Y}_{t} \mathrel{\text{:=}} {\partial }_{x}{S}_{t} = : {Z}_{t}^{-1}.\n\]\n\nLet us apply (16.16) after choosing\n\n\[ \nX = {S}_{T},\;Y = G{Y}_{T},\;{u}_{t} = \frac{{Y}_{... | Yes |
Corollary 6.8 (Sufficiency in Theorem 6.6): If \( N = D \) and \( \sigma \left( t\right) \) is nonsingular for Lebesgue-almost-every \( t \in \left\lbrack {0, T}\right\rbrack \) almost surely, then the financial market is complete. | Proof. We verify the condition of Proposition 6.2. Let \( B \) be an \( \mathcal{F}\left( T\right) \) -measurable random variable satisfying (6.3), and define the Lévy \( {P}_{0} \) -martingale\n\n\[ \n{M}_{0}\left( t\right) = {E}_{0}\left\lbrack {\left. \frac{B}{{S}_{0}\left( T\right) }\right| \;\mathcal{F}\left( t\ri... | Yes |
Example 3.1 (Forward contract to purchase a stock that pays no dividends): Suppose the contract is to purchase one share of the first stock, i.e., \( B = {S}_{1}\left( T\right) \) . If the first stock pays no dividends and \( \sigma \left( \cdot \right) \) satisfies the Novikov condition (1.5.17) with \( n = 1 \), then... | \[ {V}^{FC}\left( {t;q}\right) = {S}_{1}\left( t\right) - q{S}_{0}\left( t\right) \cdot {E}_{0}\left\lbrack {1/{S}_{0}\left( T\right) \mid \mathcal{F}\left( t\right) }\right\rbrack ,\;0 \leq t \leq T. \] (3.3) If in addition \( {S}_{0}\left( T\right) \) is nonrandom, the hedging portfolio is particularly simple. The ag... | Yes |
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