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Proposition 7.3.2 Assume that the weights \( {p}_{1},\ldots ,{p}_{m} \) are given and \( r = 0 \) . Let the function \( \widetilde{\sigma }\left( {s, t}\right) \) satisfy\n\n\[ \n{\widetilde{\sigma }}^{2}\left( {s, t}\right) = \frac{\mathop{\sum }\limits_{{i = 1}}^{m}{p}_{i}{g}_{i}^{2}\left( {s, t}\right) {f}^{i}\left(... | Proof. Let us sketch the proof. For any maturity \( T > 0 \) and strike \( K > 0 \), we define\n\n\[ \n{C}_{0}^{i}\left( {T, K}\right) = {B}_{0}{\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {{B}_{T}^{-1}{\left( {S}_{T}^{i} - K\right) }^{ + }}\right) ,\n\]\n\nwhere the process \( {S}^{i} \) satisfies (7.34) with \( {S}_{0}^{i... | Yes |
Proposition 7.4.2 The process \( {S}^{ * } \) given by, for \( t \in \left\lbrack {0,{T}^{ * }}\right\rbrack \) , \[ {S}_{t}^{ * } = {B}_{t}^{-1}{S}_{t} = {S}_{0}\exp \left( {{\int }_{0}^{t}{\sigma }_{u}d{W}_{u}^{ * } - \frac{1}{2}{\int }_{0}^{t}{\sigma }_{u}^{2}{du}}\right) \] is a martingale with respect to \( \mathb... | Proof. First, by virtue of continuity of sample paths of \( \sigma \), it is clear that \[ {\mathbb{P}}^{ * }\left\{ {{\int }_{0}^{{T}^{ * }}{\sigma }_{t}^{2}{dt} < \infty }\right\} = 1 \] Thus, the process \( {S}^{ * } \) is well defined and it is a local martingale under \( {\mathbb{P}}^{ * } \) . Since, in addition,... | Yes |
Proposition 7.4.3 The price at time 0 of any contingent claim \( X = h\left( {S}_{T}\right) \) settling at time \( T \), satisfies\n\n\[ \n{\pi }_{0}\left( X\right) = {\int }_{0}^{\infty }h\left( s\right) d{F}_{S}\left( s\right) = {\int }_{0}^{\infty }h\left( s\right) {f}_{S \mid A}\left( {s \mid w}\right) {dsd}{F}_{A}... | From the proposition above, it is clear that valuation of any contingent claim \( X = h\left( {S}_{T}\right) \) is rather straightforward, provided that we know how to value this claim analytically in the Black-Scholes set-up, and the cumulative distribution function \( {F}_{A} \) is given explicitly. Indeed, we can th... | Yes |
Lemma 8.1.1 A probability measure \( {\mathbb{P}}^{ * } \) is a martingale measure for \( {Z}^{ * } \) if and only if it is a martingale measure for \( {\mathcal{M}}^{k} \) . | Proof. For the inclusion \( \mathcal{P}\left( {Z}^{ * }\right) \subseteq \mathcal{P}\left( {\mathcal{M}}^{k}\right) \), note that using Itô’s integration by parts formula, we obtain\n\n\[ d{V}_{t}^{ * }\left( \phi \right) = \left( {1/{Z}_{t}^{k}}\right) d{V}_{t}\left( \phi \right) + {V}_{t}\left( \phi \right) d\left( {... | No |
Proposition 8.1.1 For any martingale measure \( {\mathbb{P}}^{ * } \in \mathcal{P}\left( {\mathcal{M}}^{k}\right) \), the spot market model \( {\mathcal{M}}^{k}\left( {\mathbb{P}}^{ * }\right) \) is arbitrage-free. Any contingent claim \( X \) attainable in \( {\mathcal{M}}^{k}\left( {\mathbb{P}}^{ * }\right) \) admits... | Proof. To prove the first statement, it is enough to verify that the class \( {\Phi }^{k}\left( {\mathbb{P}}^{ * }\right) \) of trading strategies does not contain arbitrage opportunities. For any \( \phi \in {\Phi }^{k}\left( {\mathbb{P}}^{ * }\right) \), with \( {V}_{0}\left( \phi \right) = 0 \), we have \( {V}^{ * }... | Yes |
Proposition 8.1.2 For any European contingent claim \( X \) that settles at time \( T \) and is attainable in \( \mathcal{M}\left( {\mathbb{P}}^{ * }\right) \), we have\n\n\[{\pi }_{t}^{k}\left( {X \mid {\mathbb{P}}^{ * }}\right) = {v}_{t}^{k}\left( {X \mid {\mathbb{P}}^{ * }}\right) ,\;\forall t \in \left\lbrack {0, T... | Proof. Let us define the relative price of \( X \) by setting\n\n\[{\pi }_{t}^{k, * }\left( {X \mid {\mathbb{P}}^{ * }}\right) \overset{\text{ def }}{ = }{\pi }_{t}^{k}\left( {X \mid {\mathbb{P}}^{ * }}\right) /{Z}_{t}^{k}.\](8.6)\n\nFor any replicating strategy \( \phi \in {\Phi }^{k}\left( {\mathbb{P}}^{ * }\right) \... | Yes |
Proposition 8.1.4 Let \( X \) be a contingent claim that settles at time \( T \) and is attainable in the market model \( {\mathcal{M}}^{k} \). The arbitrage price of \( X \) satisfies\n\n\[ \n{\pi }_{0}^{k}\left( X\right) = \mathop{\sup }\limits_{{{\mathbb{P}}^{ * } \in \mathcal{P}\left( {\mathcal{M}}^{k}\right) }}{Z}... | Proof. Since a claim \( X \) is \( {\Phi }_{0}^{k}\left( {\mathbb{P}}_{1}^{ * }\right) \) -attainable for some martingale measure \( {\mathbb{P}}_{1}^{ * } \in \) \( \mathcal{P}\left( {\mathcal{M}}^{k}\right) \), there exists a strategy \( \phi \in {\Phi }_{0}^{k}\left( {\mathbb{P}}_{1}^{ * }\right) \) such that\n\n\[ ... | Yes |
Corollary 8.1.1 Let \( X \) be a contingent claim such that the random variable \( \left| X\right| /{Z}_{T}^{k} \) is bounded by a constant. Then \( X \) is \( {\mathbb{P}}^{ * } \) -attainable for some \( {\mathbb{P}}^{ * } \in \mathcal{P}\left( {\mathcal{M}}^{k}\right) \) if and only if \( X \) is \( {\mathbb{P}}^{ *... | Proof. The first assertion is clear. The \ | No |
Proposition 8.1.5 Assume that for every \( i = 1,\ldots, k - 1 \) we have\n\n\[ \n{\left\langle {f}^{i},1/{Z}^{k}\right\rangle }_{t} = 0,\;\forall t \in \left\lbrack {0,{T}^{ * }}\right\rbrack .\n\]\n\n(8.14)\n\nThen any probability measure \( \widetilde{\mathbb{P}} \) from \( \mathcal{P}\left( f\right) \) is a marting... | Proof. From Itô's formula, we have\n\n\[ \nd\left( {1/{Z}_{t}^{k}}\right) = - {\left( 1/{Z}_{t}^{k}\right) }^{2}d{Z}_{t}^{k} + {\left( 1/{Z}_{t}^{k}\right) }^{3}d{\left\langle {Z}^{k},{Z}^{k}\right\rangle }_{t}.\n\]\n\nThis implies that \( d{\left\langle {Z}^{k},1/{Z}^{k}\right\rangle }_{t} = - {\left( {Z}_{t}^{k}\righ... | Yes |
Lemma 8.1.2 A trading strategy \( \psi \) is a self-financing forward strategy; that is, the forward wealth \( \widetilde{V}\left( \psi \right) \) satisfies\n\n\[ d{\widetilde{V}}_{t}\left( \psi \right) = \mathop{\sum }\limits_{{i = 1}}^{k}{\psi }_{t}^{i}d{F}^{i}\left( {t, T}\right) = {\psi }_{t} \cdot d{F}_{Z}\left( {... | Proof. We shall prove the \ | No |
Proposition 8.1.6 Let the class \( \mathcal{P}\left( {\mathcal{M}}^{i}\right) \) be non-empty for some \( i \leq k \) . More precisely, we assume that relative prices \( {Z}^{j}/{Z}^{i} \) are \( {\mathbb{P}}^{ * } \) -martingales for some \( {\mathbb{P}}^{ * } \in \) \( \mathcal{P}\left( {\mathcal{M}}^{i}\right) \) . ... | Proof. Since buy-and-hold strategies are manifestly self-financing, the first statement follows from the second. To prove (ii), we may in fact take an arbitrary strictly positive process \( N \) such that \( N/{Z}^{i} \) is a \( {\mathbb{P}}^{ * } \) -martingale. We define a measure \( {\mathbb{Q}}^{ * } \) on \( \left... | Yes |
Proposition 8.1.7 Let \( X \) be a contingent claim that can be priced by arbitrage in standard market models \( {\mathcal{M}}^{i} \) and \( {\mathcal{M}}^{j} \) . More specifically, we assume that \( X \) is \( {\mathbb{P}}_{i}^{ * } \) - attainable and \( {\mathbb{P}}_{j}^{ * } \) -attainable, where \( {\mathbb{P}}_{... | Proof. The proof is quite standard, and thus it is left to the reader. | No |
To illustrate the way in which Proposition 8.1.7 can be applied to facilitate the valuation of derivative securities, we place ourselves once again within the standard Black-Scholes framework. Suppose that we wish to price a European call option maturing at time \( T \) . For convenience, as the second security we take... | \[ {C}_{T} = {\left( {S}_{T} - K\right) }^{ + } = {S}_{T}{\mathbb{1}}_{A} - {KB}\left( {T, T}\right) {\mathbb{1}}_{A}, \] where \( A = \left\{ {{S}_{T} > K}\right\} \) . Consequently, \[ {\pi }_{t}\left( {C}_{T}\right) = {\pi }_{t}\left( {{S}_{T}{\mathbb{1}}_{A}}\right) - {\pi }_{t}\left( {K{\mathbb{1}}_{A}}\right) = {... | Yes |
Proposition 8.1.8 Let \( Z \) be a real-valued special \( {}^{12} \) semimartingale under \( \mathbb{P} \), with the canonical decomposition \( Z = {Z}_{0} + M + A \) . Then a local martingale \( U \), satisfying (8.20), defines a martingale measure for \( Z \) if and only if\n\n\[ \n{A}_{t} + \langle M, U{\rangle }_{t... | Proof. The first assertion follows from Girsanov's theorem. The second can be proved using the Kunita-Watanabe inequality (for the latter, see Protter (2003)). | No |
Proposition 8.2.1 The following are equivalent: (i) the multidimensional Black-Scholes model is complete; (ii) inequality \( d \leq k \) holds and the volatility matrix \( \sigma \) has full rank for Lebesgue a.e. \( t \in \left\lbrack {0,{T}^{ * }}\right\rbrack \), with probability 1; (iii) there exists a unique marti... | Proof. We shall merely outline the proof. Let us first examine the implication (ii) \( \Rightarrow \) (i). Essentially, it is a consequence of the representation theorem for the filtration of a multidimensional Brownian motion, \( {}^{16} \) combined with the possibility of expressing the underlying Brownian motions \(... | No |
Proposition 8.2.2 Let \( X \) be a European claim, settling at time \( T \), such that \( {X}^{ * } \) is square-integrable under \( \mathbb{P} \) . The following are equivalent: (i) \( X \) admits a locally risk-minimizing replicating strategy; and (ii) \( {X}^{ * } \) admits a Föllmer-Schweizer decomposition with res... | \[ {\int }_{0}^{t}{\phi }_{u} \cdot d{S}_{u}^{ * } = {\int }_{0}^{t}{\xi }_{u} \cdot d{S}_{u}^{ * },\;\forall t \in \left\lbrack {0,{T}^{ * }}\right\rbrack . \] | Yes |
Proposition 9.5.1 Assume the short-term interest rate \( r \) follows an Itô process under the actual probability \( \mathbb{P} \), as specified by (9.17). Let \( B\left( {t, T}\right) \) be an arbitrage-free family of bond prices relative to \( r \) . For any martingale measure \( {\mathbb{P}}^{ * } = {\mathbb{P}}^{\l... | Proof. To show (i), it is enough to combine (9.17) with (9.19). For (ii), it suffices to observe that the process \( M = {Z}^{ * }\eta \) follows a (local) martingale under \( \mathbb{P} \) (recall that \( {Z}^{ * } \) is given by (9.12)). In view of Theorem A.11.1, we have\n\n\[ \n{M}_{t} = {Z}^{ * }\left( {t, T}\righ... | Yes |
Corollary 9.5.1 Let \( {\mathbb{P}}^{\lambda } \) and \( {\mathbb{P}}^{\widetilde{\lambda }} \) be two probability measures equivalent to the underlying probability measure \( \mathbb{P} \). Assume that the bond price \( B\left( {t, T}\right) \) is given by formula (9.14), with \( {\mathbb{P}}^{ * } = {\mathbb{P}}^{\la... | Proof. For any process \( \lambda \), let \( {U}^{\lambda } \) stand for the integral\n\n\[ {U}_{t}^{\lambda } = {\int }_{0}^{t}{\lambda }_{u} \cdot d{W}_{u},\;\forall t \in \left\lbrack {0,{T}^{ * }}\right\rbrack .\n\nStraightforward calculations show that\n\n\[ \frac{d{\mathbb{P}}^{\lambda }}{d{\mathbb{P}}^{\widetild... | Yes |
Lemma 9.6.1 The forward price \( {F}_{X}\left( {t, T}\right) \) at time \( t \leq T \), for the settlement date \( T \) , of an attainable contingent claim \( X \) equals\n\n\[ \n{F}_{X}\left( {t, T}\right) = \frac{{\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {X{B}_{T}^{-1} \mid {\mathcal{F}}_{t}}\right) }{{\mathbb{E}}_{{\m... | Proof. It is sufficient to observe that\n\n\[ \n{\pi }_{t}\left( {G}_{T}\right) = {B}_{t}{\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {{G}_{T}{B}_{T}^{-1} \mid {\mathcal{F}}_{t}}\right)\n\]\n\n\[ \n= {B}_{t}\left( {{\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {X{B}_{T}^{-1} \mid {\mathcal{F}}_{t}}\right) - {F}_{X}\left( {t, T}\r... | Yes |
Proposition 9.6.1 The relative prices \( {B}_{t}/B\left( {t, T}\right) \) and \( B\left( {t, U}\right) /B\left( {t, T}\right) \) are martingales under \( {\mathbb{P}}_{T} \) . | Proof. The result is an immediate consequence of the martingale property of \( B\left( {t, T}\right) /{B}_{t} \) and \( B\left( {t, U}\right) /{B}_{t} \) under \( {\mathbb{P}}^{ * } \) and Lemma A.14.1. | No |
Lemma 9.6.2 The forward price at time \( t \) for the delivery date \( T \) of an attainable contingent claim \( X \), settling at time \( T \), equals\n\n\[ \n{F}_{X}\left( {t, T}\right) = {\mathbb{E}}_{{\mathbb{P}}_{T}}\left( {X \mid {\mathcal{F}}_{t}}\right) ,\;\forall t \in \left\lbrack {0, T}\right\rbrack ,\n\]\n\... | Proof. The abstract Bayes rule (see Lemma A.1.4) yields\n\n\[ \n{\mathbb{E}}_{{\mathbb{P}}_{T}}\left( {X \mid {\mathcal{F}}_{t}}\right) = \frac{{\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {{\eta }_{T}X \mid {\mathcal{F}}_{t}}\right) }{{\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {{\eta }_{T} \mid {\mathcal{F}}_{t}}\right) } = {... | Yes |
Proposition 9.6.2 The arbitrage price at time \( t \) of an attainable contingent claim \( X \) settling at time \( T \) is given by the formula\n\n\[ \n{\pi }_{t}\left( X\right) = B\left( {t, T}\right) {\mathbb{E}}_{{\mathbb{P}}_{T}}\left( {X \mid {\mathcal{F}}_{t}}\right) ,\;\forall t \in \left\lbrack {0, T}\right\rb... | Proof. Equality (9.36) is an immediate consequence of (9.28) combined with (9.34). For a more direct proof, note that the price \( {\pi }_{t}\left( X\right) \) can be re-expressed as follows\n\n\[ \n{\pi }_{t}\left( X\right) = {B}_{t}{\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {X{B}_{T}^{-1} \mid {\mathcal{F}}_{t}}\right) ... | Yes |
Consider an attainable contingent claim \( X \) settling at time \( U \) . (i) If \( U \leq T \), then the price of \( X \) at time \( t \leq U \) equals\n\n\[ \n{\pi }_{t}\left( X\right) = B\left( {t, T}\right) {\mathbb{E}}_{{\mathbb{P}}_{T}}\left( {X{B}^{-1}\left( {U, T}\right) \mid {\mathcal{F}}_{t}}\right) .\n\]\n\... | Proof. Both equalities are intuitively clear. In case (i), we invest at time \( U \) an \( {\mathcal{F}}_{U} \) - measurable payoff \( X \) in zero-coupon bonds which mature at time \( T \) . For the second case, observe that in order to replicate an \( {\mathcal{F}}_{T} \) -measurable claim \( X \) at time \( U \), it... | No |
Lemma 9.6.3 For any maturities \( T, U \in \left\lbrack {0,{T}^{ * }}\right\rbrack \), the dynamics under \( \mathbb{P} \) of the forward process are given by the following expression\n\n\[ d{F}_{B}\left( {t, T, U}\right) = {F}_{B}\left( {t, T, U}\right) \gamma \left( {t, T, U}\right) \cdot \left( {d{W}_{t} - \gamma \l... | Combining Lemma 9.6.3 with Girsanov's theorem, we obtain\n\n\[ d{F}_{B}\left( {t, T, U}\right) = {F}_{B}\left( {t, T, U}\right) \gamma \left( {t, T, U}\right) \cdot d{W}_{t}^{U},\] \n\n(9.41)\n\nwhere for every \( t \in \left\lbrack {0, U}\right\rbrack \)\n\n\[ {W}_{t}^{U} = {W}_{t} - {\int }_{0}^{t}\gamma \left( {u, U... | Yes |
Proposition 10.1.1 The arbitrage price \( B\left( {t, T}\right) \) of a \( T \) -maturity zero-coupon bond in Merton's model equals\n\n\[ B\left( {t, T}\right) = {e}^{-{r}_{t}\left( {T - t}\right) - \frac{1}{2}a{\left( T - t\right) }^{2} + \frac{1}{6}{\sigma }^{2}{\left( T - t\right) }^{3}}. \] | Proof. Let us evaluate the price \( B\left( {0, T}\right) \) using the formula\n\n\[ B\left( {0, T}\right) = {\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {e}^{-{\int }_{0}^{T}{r}_{u}{du}}\right) ,\;\forall T \in \left\lbrack {0,{T}^{ * }}\right\rbrack . \]\n\nTo this end, we need to find the probability distribution of the ... | Yes |
Lemma 10.1.2 The unique solution to the \( {SDE}\left( {10.5}\right) \) is given by the formula\n\n\[ \n{r}_{t} = {r}_{s}{e}^{-b\left( {t - s}\right) } + \frac{a}{b}\left( {1 - {e}^{-b\left( {t - s}\right) }}\right) + \sigma {\int }_{s}^{t}{e}^{-b\left( {t - u}\right) }d{W}_{u}^{ * }.\n\] | Proof. Let us fix \( s > 0 \) and let us consider the process \( {Y}_{t} = {r}_{t}{e}^{b\left( {t - s}\right) } \) where \( t \geq s \) . Using Itô's formula and (10.5), we obtain\n\n\[ \nd{Y}_{t} = {e}^{b\left( {t - s}\right) }d{r}_{t} + b{e}^{b\left( {t - s}\right) }{r}_{t}{dt} \]\n\n\[ \n= {e}^{b\left( {t - s}\right... | Yes |
Proposition 10.1.2 The price at time \( t \) of a zero-coupon bond in Vasicek’s model equals:\n\n\[ B\left( {t, T}\right) = {e}^{m\left( {t, T}\right) - n\left( {t, T}\right) {r}_{t}}, \]\n\nwhere\n\n\[ n\left( {t, T}\right) = \frac{1}{b}\left( {1 - {e}^{-b\left( {T - t}\right) }}\right) \]\n\nand\n\n\[ m\left( {t, T}\... | Proof. We shall first evaluate \( B\left( {0, T}\right) \) using the formula:\n\n\[ B\left( {0, T}\right) = {\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {B}_{T}^{-1}\right) = {\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {e}^{-{\int }_{0}^{T}{r}_{t}{dt}}\right) . \]\n\nWe know already that\n\n\[ {r}_{t} = {r}_{0}{e}^{-{bt}} + \fr... | Yes |
Proposition 10.1.3 The price at time \( t \) of a zero-coupon bond in Vasicek’s model equals\n\n\[ B\left( {t, T}\right) = v\left( {{r}_{t}, t, T}\right) = {e}^{m\left( {t, T}\right) - n\left( {t, T}\right) {r}_{t}}, \]\n\nwhere\n\n\[ n\left( {t, T}\right) = \frac{1}{b}\left( {1 - {e}^{-b\left( {T - t}\right) }}\right)... | Proof. To establish the bond pricing formula through the PDE approach, it is enough to assume that the bond price is given by (10.10), with the functions \( m \) and \( n \) satisfying \( m\left( {T, T}\right) = n\left( {T, T}\right) = 0 \), and to make use of the valuation PDE. By separating terms that do not depend o... | Yes |
Consider the special case of the Black and Karasinski (1991) model, specifically,\n\n\[ d\ln {r}_{t} = \left( {a\left( t\right) - b\ln {r}_{t}}\right) {dt} + {\sigma d}{W}_{t}^{ * }, \] | As was mentioned already, the process \( {Y}_{t} = \ln {r}_{t} \) satisfies Vasicek’s (1977) dynamics. Consequently, the unique solution to (10.32) is given by the following expression\n\n\[ {r}_{t} = \exp \left( {{e}^{-{bt}}\ln {r}_{0} + {\int }_{0}^{t}{e}^{-b\left( {t - u}\right) }a\left( u\right) {du} + \sigma {\int... | Yes |
Lemma 10.3.1 Let \( X \) be a \( \delta \) -dimensional squared Bessel process under \( \mathbb{P} \) . Then\n\n\[ \n{\mathbb{E}}_{\mathbb{P}}\left\{ {\exp \left\lbrack {\frac{1}{2}\left( {\beta \left( t\right) {X}_{t} - {\int }_{0}^{t}\left( {{\beta }^{2}\left( u\right) + {\beta }^{\prime }\left( u\right) }\right) {X}... | More generally, for any \( t < T \) we have\n\n\[ \n{\mathbb{E}}_{\mathbb{P}}\left\{ \left. {\exp \left\lbrack {\frac{1}{2}\left( {\beta \left( T\right) {X}_{T} - {\int }_{t}^{T}\left( {{\beta }^{2}\left( u\right) + {\beta }^{\prime }\left( u\right) }\right) {X}_{u}{du}}\right) }\right\rbrack \;{\mathcal{F}}_{t}}\right... | No |
Proposition 10.3.1 Assume that \( \beta < 0 \) . Then for any fixed \( T > 0 \) we have\n\n\[ B\left( {0, T}\right) = \exp \left\{ {\frac{1}{2}\left( {\left( {{F}_{T}\left( 0\right) - \beta \left( 0\right) }\right) {X}_{0} + \delta {\int }_{0}^{T}\left( {{F}_{T}\left( u\right) - \beta \left( u\right) }\right) {du}}\rig... | The most convenient approach to the Riccati equation (10.46) is through the corresponding Sturm-Liouville equation. Let us write \( F\left( u\right) = \frac{{\varphi }^{\prime }\left( u\right) }{\varphi \left( u\right) } \) for \( u \in \lbrack 0, T) \) , where \( \varphi \left( 0\right) = 1 \) . Then for \( \varphi \l... | Yes |
Corollary 10.3.1 The price at time 0 of a zero-coupon bond with maturity date \( T \) satisfies\n\n\[ B\left( {0, T}\right) = \exp \left\{ {\frac{1}{2}\left( {\left( {{\varphi }_{T}^{\prime }\left( 0\right) - \beta \left( 0\right) }\right) {X}_{0} + \delta \ln {\varphi }_{T}\left( T\right) - \delta {\int }_{0}^{T}\beta... | We write here \( {\varphi }_{T} \) rather than \( \varphi \) since we wish to stress the dependence on the maturity date \( T \) of the solution \( \varphi \) in \( \left\lbrack {0, T}\right\rbrack \) . From (10.47), we obtain the intuitively obvious fact that the bond price is a decreasing function of the short-term r... | No |
Proposition 10.3.2 For any dates \( 0 \leq t \leq T \), we have\n\n\[ B\left( {t, T}\right) = \exp \left\{ {\frac{1}{2}\left( {\left( {{F}_{T}\left( t\right) - \beta \left( t\right) }\right) {X}_{t} + \delta {\int }_{t}^{T}\left( {{F}_{T}\left( u\right) - \beta \left( u\right) }\right) {du}}\right) }\right\} ,\] | where the function \( {F}_{T} : \left\lbrack {0, T}\right\rbrack \rightarrow \mathbb{R} \) solves (10.46) with the terminal condition \( {F}_{T}\left( T\right) = \beta \left( T\right) \) . Put another way, the bond price equals\n\n\[ B\left( {t, T}\right) = {e}^{m\left( {t, T}\right) - n\left( {t, T}\right) {r}_{t}}, \... | Yes |
Lemma 10.3.2 The dynamics of the process \( X \) under the forward martingale measure \( {\mathbb{P}}_{T} \) are\n\n\[ d{X}_{t} = \left( {\delta + 2{F}_{T}\left( t\right) {X}_{t}}\right) {dt} + 2\sqrt{{X}_{t}}d{W}_{t}^{T}. \] | For a fixed \( t < T \), let \( \widetilde{\varphi } : \left\lbrack {t, T}\right\rbrack \rightarrow \mathbb{R} \) be the unique solution to the following ordinary differential equation:\n\n\[ \frac{{\widetilde{\varphi }}^{\prime \prime }\left( u\right) }{\widetilde{\varphi }\left( u\right) } = {\beta }^{2}\left( u\righ... | No |
Proposition 10.3.4 The arbitrage price in the extended CIR model of a call option written on the \( U \)-maturity zero-coupon bond equals\n\n\[ \n{C}_{0} = B\left( {0, U}\right) {\mathbb{P}}_{U}\left\{ {{X}_{T} < \bar{K}}\right\} - {KB}\left( {0, T}\right) {\mathbb{P}}_{T}\left\{ {{X}_{T} < \bar{K}}\right\} ,\n\]\n\nwh... | Proof. The statement follows from results of Sect. 10.3.8. | No |
Proposition 10.3.5 Consider a contingent claim \( Y = h\left( {r}_{T}\right) \) at time \( T \) . Its arbitrage price at time \( t \in \left\lbrack {0, T}\right\rbrack \) equals | \[ {\pi }_{0}\left( X\right) = {B}_{0}{\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {{B}_{T}^{-1}h\left( {r}_{T}\right) }\right) = B\left( {0, T}\right) {\mathbb{E}}_{{\mathbb{P}}_{T}}\left( {h\left( {s\left( T\right) {X}_{T}}\right) }\right) ,\] where the random variable \( {X}_{T} \) has under the forward measure \( {\math... | Yes |
Lemma 11.1.1 The dynamics of the bond price \( B\left( {t, T}\right) \) are determined by the expression\n\n\[ \n{dB}\left( {t, T}\right) = B\left( {t, T}\right) \left( {a\left( {t, T}\right) {dt} + b\left( {t, T}\right) \cdot d{W}_{t}}\right) ,\n\]\n\nwhere \( a \) and \( b \) are given by the following formulas\n\n\[... | Proof. Let us denote \( {I}_{t} = \ln B\left( {t, T}\right) \) . According to formulas (11.2) and (11.4), we have that\n\n\[ \n{I}_{t} = - {\int }_{t}^{T}f\left( {0, u}\right) {du} - {\int }_{t}^{T}{\int }_{0}^{t}\alpha \left( {v, u}\right) {dvdu} - {\int }_{t}^{T}{\int }_{0}^{t}\sigma \left( {v, u}\right) \cdot d{W}_{... | Yes |
For any fixed maturity \( T \leq {T}^{ * } \), the dynamics of the bond price \( B\left( {t, T}\right) \) under the spot martingale measure \( {\mathbb{P}}^{ * } \) are | \[ {dB}\left( {t, T}\right) = B\left( {t, T}\right) \left( {{r}_{t}{dt} - {\sigma }^{ * }\left( {t, T}\right) \cdot d{W}_{t}^{ * }}\right) ,\] | Yes |
Lemma 11.1.2 Conditions (M.1) and (M.2) are equivalent. | Proof. Let us sketch the proof. Suppose first that condition (M.1) holds. Let us define \( \lambda \) by setting\n\n\[ \lambda_t \overset{\text{ def }}{=} h_t - b(t, T^*) = h_t + \int_t^{T^*} \sigma(t, u) \, du, \quad \forall t \in \left[ 0, T^* \right], \]\n\nand check that for such a process \( \lambda \) condition (... | Yes |
Proposition 11.1.1 Suppose that the coefficients \( \alpha \left( {t, T}\right) \) and \( \sigma \left( {t, T}\right) \) and the initial forward curve \( f\left( {0, T}\right) \) are differentiable with respect to maturity \( T \), with bounded partial derivatives \( {\alpha }_{T}\left( {t, T}\right) ,{\sigma }_{T}\lef... | Proof. Observe first that \( r \) satisfies\n\n\[ \n{r}_{t} = f\left( {t, t}\right) = f\left( {0, t}\right) + {\int }_{0}^{t}\alpha \left( {u, t}\right) {du} + {\int }_{0}^{t}\sigma \left( {u, t}\right) \cdot d{W}_{u}.\n\]\n\nApplying the stochastic Fubini theorem to the Itô integral, we obtain\n\n\[ \n{\int }_{0}^{t}\... | Yes |
Proposition 11.2.1 Suppose that the coefficient \( \sigma \) is deterministic. Then for any choice \( \mathcal{T} \) of maturity dates and of a spot martingale measure \( {\mathbb{P}}^{ * } \), the probability distribution of the process \( {Z}^{ * }\left( {t,\mathcal{T}}\right), t \in \left\lbrack {0,{T}^{ * }}\right\... | Proof. The assertion follows easily by Girsanov's theorem. Indeed, for any fixed \( 0 < T \leq {T}^{ * } \) the dynamics of \( {Z}^{ * }\left( {t, T}\right) \) under a spot martingale measure \( {\mathbb{P}}^{ * } = {\mathbb{P}}^{\lambda } \) are\n\n\[ d{Z}^{ * }\left( {t, T}\right) = - {Z}^{ * }\left( {t, T}\right) {\... | Yes |
Proposition 11.2.2 Suppose that the short-term rate \( r \) is Markovian. Assume, in addition, that for any maturity \( T \leq {T}^{ * } \) we have \( \sigma \left( {t, T}\right) \neq 0 \) for every \( t \in \left\lbrack {0, T}\right\rbrack \) . Then there exist functions \( g : \left\lbrack {0,{T}^{ * }}\right\rbrack ... | Proof. In view of (11.27), it is clear that the short-term rate \( r \) is Markovian if and only if the process \( {D}_{t} = {\int }_{0}^{t}\sigma \left( {u, t}\right) \cdot d{W}_{u}^{ * } \) is Markovian - that is, if\n\n\[ {\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {h\left( {D}_{s}\right) \mid {\mathcal{F}}_{t}^{D}}\rig... | Yes |
Let us assume that the volatility of each forward rate is constant, i.e., independent of the maturity date and the level of the forward interest rate. Taking \( d = 1 \), we thus have \( \sigma \left( {t, T}\right) = \sigma \) for a strictly positive constant \( \sigma > 0 \) . By virtue of (11.15), the dynamics of the... | \[ {df}\left( {t, T}\right) = {\sigma }^{2}\left( {T - t}\right) {dt} + {\sigma d}{W}_{t}^{ * }, \] (11.32) so that the dynamics of the bond price \( B\left( {t, T}\right) \) are \[ {dB}\left( {t, T}\right) = B\left( {t, T}\right) \left( {{r}_{t}{dt} - \sigma \left( {T - t}\right) d{W}_{t}^{ * }}\right) , \] where the ... | Yes |
It is a conventional wisdom that forward rates of longer maturity fluctuate less than rates of shorter maturity. To account for this feature in the HJM framework, we assume now that the volatility of a forward rate is a decreasing function of the time to its effective date. For instance, we may assume that the volatili... | Then \( {\sigma }^{ * }\left( {t, T}\right) \) equals\n\n\[ \n{\sigma }^{ * }\left( {t, T}\right) = {\int }_{t}^{T}\sigma {e}^{-b\left( {u - t}\right) }{du} = - \sigma {b}^{-1}\left( {{e}^{-b\left( {T - t}\right) } - 1}\right) , \n\]\n\n(11.34)\n\nand consequently\n\n\[ \n{df}\left( {t, T}\right) = - {\sigma }^{2}{b}^{... | Yes |
Corollary 11.3.1 For any fixed \( T \in \left\lbrack {0,{T}^{ * }}\right\rbrack \), the instantaneous forward rate \( f\left( {0, T}\right) \) is equal to the expected value of the spot rate \( {r}_{T} \) under the forward probability measure \( {\mathbb{P}}_{T} \) . | Proof. Observe that in view of (11.16), we have\n\n\[ \n{r}_{T} = f\left( {0, T}\right) + {\int }_{0}^{T}\sigma \left( {t, T}\right) \cdot \left( {{\sigma }^{ * }\left( {t, T}\right) {dt} + d{W}_{t}^{ * }}\right) = f\left( {0, T}\right) + {\int }_{0}^{T}\sigma \left( {t, T}\right) \cdot d{W}_{t}^{T}, \n\] \n\nsince \( ... | Yes |
Proposition 11.3.1 Assume that the bond price volatilities \( b\left( {\cdot, T}\right) \) and \( b\left( {\cdot, U}\right) \) are \( {\mathbb{R}}^{d} \) -valued, bounded, deterministic functions. The arbitrage price at time \( t \in \left\lbrack {0, T}\right\rbrack \) of a European call option with expiry date \( T \)... | Proof. In view of the general valuation formula (11.38), it is clear that we have to evaluate the conditional expectations\n\n\[ \n{C}_{t} = B\left( {t, T}\right) {\mathbb{E}}_{{\mathbb{P}}_{T}}\left( {{F}_{B}\left( {T, U, T}\right) {\mathbb{1}}_{D} \mid {\mathcal{F}}_{t}}\right) - {KB}\left( {t, T}\right) {\mathbb{P}}... | Yes |
Let \( {W}^{ * } = \left( {{W}^{1 * },{W}^{2 * }}\right) \) be a two-dimensional standard Brownian motion given on a probability space \( \left( {\Omega ,\mathbb{F},{\mathbb{P}}^{ * }}\right) \). We assume that the bond price \( B\left( {t, T}\right) \) satisfies, under \( {\mathbb{P}}^{ * } \)\n\n\[ \n{dB}\left( {t, T... | An application of Proposition 11.3.2 yields the following result, first established in Merton (1973).\n\nCorollary 11.3.2 Assume | No |
Assume that the dynamics of a bond and a stock price are given by (11.53) and (11.54) respectively. If the volatility coefficients \( \widehat{b} \) and \( \widehat{\sigma } \) are deterministic functions, then the arbitrage price of a European call option written on a stock \( S \) is given by (11.50)-(11.51), with | \[ {v}^{2}\left( {t, T}\right) = {\int }_{t}^{T}\left( {{\widehat{\sigma }}^{2}\left( u\right) - {2\rho }\widehat{\sigma }\left( u\right) \widehat{b}\left( {u, T}\right) + {\widehat{b}}^{2}\left( {u, T}\right) }\right) {du}. \] | Yes |
Corollary 11.3.3 The following put-call parity relationship is valid\n\n\\[ \n{C}_{t} - {P}_{t} = {Z}_{t} - B\\left( {t, T}\\right) K,\\;\\forall t \\in \\left\\lbrack {0, T}\\right\\rbrack .\n\\]\n\n(11.57) | Proof. We make use of the forward measure method. We have\n\n\\[ \n{C}_{t} - {P}_{t} = B\\left( {t, T}\\right) {\\mathbb{E}}_{{\\mathbb{P}}_{T}}\\left( {{F}_{Z}\\left( {T, U, T}\\right) - K \\mid {\\mathcal{F}}_{t}}\\right) ,\n\\]\n\nand thus we obtain\n\n\\[ \n{C}_{t} - {P}_{t} = B\\left( {t, T}\\right) {F}_{Z}\\left(... | Yes |
Lemma 11.3.2 Let us denote \( D = \left\{ {{Z}_{T} > K}\right\} \) . Then the arbitrage price of a European call option written on a coupon bond satisfies\n\n\[ \n{C}_{t} = \mathop{\sum }\limits_{{j = 1}}^{m}{c}_{j}B\left( {t,{T}_{j}}\right) {\mathbb{P}}_{{T}_{j}}\left( {D \mid {\mathcal{F}}_{t}}\right) - {KB}\left( {t... | Proof. We have \( {Z}_{T} = \mathop{\sum }\limits_{{j = 1}}^{m}{c}_{j}{B}_{T}{\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {{B}_{{T}_{j}}^{-1} \mid {\mathcal{F}}_{T}}\right) \), and thus\n\n\[ \n{C}_{t} = {B}_{t}{\mathbb{E}}_{{\mathbb{P}}^{ * }}\left\{ {{\mathbb{1}}_{D}{B}_{T}^{-1}\left( {\mathop{\sum }\limits_{{j = 1}}^{m}{... | Yes |
Proposition 11.3.5 Assume that \( {\gamma }_{i} \) is a deterministic function for \( i = 1,\ldots, n \) . Then the arbitrage price at time \( t \in \left\lbrack {0, T}\right\rbrack \) of a European contingent claim \( X = \) \( g\left( {{Z}_{T}^{1},\ldots ,{Z}_{T}^{n}}\right) \) at time \( T \) equals\n\n\[ \n{\pi }_{... | Proof. We have\n\n\[ \n{\pi }_{t}\left( X\right) = {B}_{t}{\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {{B}_{T}^{-1}g\left( {{Z}_{T}^{1},\ldots ,{Z}_{T}^{n}}\right) \mid {\mathcal{F}}_{t}}\right) \n\]\n\n\[ \n= B\left( {t, T}\right) {\mathbb{E}}_{{\mathbb{P}}_{T}}\left( {g\left( {{F}_{{Z}^{1}}\left( {T, T}\right) ,\ldots ,{... | Yes |
Lemma 11.5.1 The forward and futures prices of a claim \( X \) satisfy\n\n\[ \n{F}_{X}\left( {t, T}\right) - {f}_{X}\left( {t, T}\right) = \frac{{\operatorname{Cov}}_{{\mathbb{P}}^{ * }}\left( {X,{B}_{t}{B}_{T}^{-1} \mid {\mathcal{F}}_{t}}\right) }{B\left( {t, T}\right) },\;\forall t \in \left\lbrack {0, T}\right\rbrac... | Proof. Indeed, the conditional covariance is defined as follows\n\n\[ \n{\operatorname{Cov}}_{{\mathbb{P}}^{ * }}\left( {X,{B}_{t}{B}_{T}^{-1} \mid {\mathcal{F}}_{t}}\right) = {\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {X{B}_{t}{B}_{T}^{-1} \mid {\mathcal{F}}_{t}}\right) - {\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {X \mid {... | Yes |
Assume that the volatility \( {\gamma }_{Z}\left( {\cdot, T}\right) = \xi - b\left( {\cdot, T}\right) \) of the forward price process \( {F}_{Z}\left( {t, T}\right) \) and the bond volatility \( b\left( {\cdot, T}\right) \) are deterministic. Then the futures price \( {f}_{Z}\left( {t, T}\right) \) equals\n\n\[ \n{f}_{... | Proof. It is clear that\n\n\[ \n{F}_{Z}\left( {T, T}\right) = {F}_{Z}\left( {t, T}\right) {\zeta }_{t}\exp \left( {{\int }_{t}^{T}\left( {b\left( {u, T}\right) - {\xi }_{u}}\right) \cdot b\left( {u, T}\right) {du}}\right) ,\n\]\n\nwhere \( {\zeta }_{t} \) stands for the following random variable\n\n\[ \n{\zeta }_{t} = ... | Yes |
Proposition 11.5.2 Assume that \( U \geq T \) . The arbitrage price at time \( t \in \left\lbrack {0, T}\right\rbrack \) of a European call option with expiry date \( T \) and exercise price \( K \), written on the futures contract for a \( U \) -maturity zero-coupon bond with delivery date \( T \), equals\n\n\[ \n{C}_... | Proof. We need to evaluate\n\n\[ \n{C}_{t}^{f} = B\left( {t, T}\right) {\mathbb{E}}_{{\mathbb{P}}_{T}}\left( {{\left( {f}_{B}\left( T, U, T\right) - K\right) }^{ + } \mid {\mathcal{F}}_{t}}\right) \n\]\n\n\[ \n= B\left( {t, T}\right) {\mathbb{E}}_{{\mathbb{P}}_{T}}\left( {{\left( {F}_{B}\left( T, U, T\right) - K\right)... | Yes |
Proposition 11.5.4 Under the assumptions of Proposition 11.5.3, the following put-call parity relationship is valid\n\n\[ \n{C}_{t}^{f} - {P}_{t}^{f} = B\left( {t, T}\right) \left( {{f}_{Z}\left( {t, T}\right) \exp \left( {{\int }_{t}^{T}{\gamma }_{Z}\left( {u, T}\right) \cdot b\left( {u, T}\right) {du}}\right) - K}\ri... | The put-call parity relationship (11.57) for European spot options can be easily established by a direct construction of two portfolios. Let us consider the following two portfolios: long call and short put option; and one unit of the underlying asset and \( K \) units of \( T \) -maturity zero-coupon bonds. The portfo... | No |
Proposition 11.6.1 Assume that the price processes \( Z \) and \( B\left( {t, D}\right) \) follow (11.78) and (11.79) respectively, and the volatility \( {\xi }_{t} - b\left( {t, D}\right) \) of the forward price is deterministic. Consider a European contingent claim \( X \) of the form \( X = B\left( {T, D}\right) g\l... | \[ {\pi }_{t}\left( X\right) = v\left( {{Z}_{t}, B\left( {t, D}\right), t}\right) = B\left( {t, D}\right) H\left( {{Z}_{t}{B}^{-1}\left( {t, D}\right), t}\right) \] where the function \( H : {\mathbb{R}}_{ + } \times \left\lbrack {0, T}\right\rbrack \rightarrow \mathbb{R} \) solves the following PDE \[ \frac{\partial H... | Yes |
Proposition 11.6.2 Suppose that the futures price \( {f}_{t} = {F}_{Z}\left( {t, R}\right) \) of an asset \( Z \) satisfies (11.87), where the volatility \( {\zeta }_{t} = {\xi }_{t} - b\left( {t, R}\right) \) is such that \( {\zeta }_{t} \cdot b\left( {t, D}\right) \) is a deterministic function. Let \( X \) be a Euro... | \[ {\pi }_{t}^{f}\left( X\right) = v\left( {{f}_{t}, B\left( {t, D}\right), t}\right) = B\left( {t, D}\right) L\left( {{f}_{t}{e}^{\eta \left( {t, T}\right) }, t}\right) \] for every \( t \in \left\lbrack {0, T}\right\rbrack \), where \[ \eta \left( {t, T}\right) = {\int }_{t}^{T}\left( {{\xi }_{u} - b\left( {u, R}\rig... | Yes |
Proposition 12.1.1 Assume the Gaussian HJM model of the term structure of interest rates. Then the following relationships are valid\n\n\[ 1 + {\delta }_{j + 1}L\left( {t,{T}_{j}}\right) = {F}_{B}\left( {t,{T}_{j},{T}_{j + 1}}\right) \]\n\n(12.10)\n\n\[ 1 + {\delta }_{j + 1}\widetilde{L}\left( {t,{T}_{j}}\right) = {F}_... | Proof. For brevity, we shall write \( {F}_{B}\left( t\right) = {F}_{B}\left( {t,{T}_{j},{T}_{j + 1}}\right) \) . The first formula is in fact universal (see (12.1)). For the second, note that (cf. (12.2))\n\n\[ d{F}_{B}\left( t\right) = {F}_{B}\left( t\right) \gamma \left( {t,{T}_{j},{T}_{j + 1}}\right) \cdot \left( {d... | Yes |
Corollary 12.1.1 We have\n\n\[ \n{dL}\left( {t,{T}_{j}}\right) = {\delta }_{j + 1}^{-1}\left( {1 + {\delta }_{j + 1}L\left( {t,{T}_{j}}\right) }\right) \gamma \left( {t,{T}_{j},{T}_{j + 1}}\right) \cdot d{W}_{t}^{{T}_{j + 1}}, \]\n\n(12.13)\n\n\[ \nd\widetilde{L}\left( {t,{T}_{j}}\right) = {\delta }_{j + 1}^{-1}\left( ... | Proof. Formula (12.13) is an immediate consequence of (12.1) combined with (12.2). Expressions (12.14) and (12.15) can be derived by applying Itô's rule to equalities (12.11) and (12.12) respectively. | Yes |
Lemma 12.3.1 Consider a caplet with settlement date \( T \), accrual period \( \delta \), and strike level \( \kappa \), that pays at time \( T + \delta \) the amount \( {\left( L\left( T, T\right) - \kappa \right) }^{ + }\delta \) . Its arbitrage price at time \( t \in \left\lbrack {0, T}\right\rbrack \) in the Gaussi... | \[ {\mathbf{{Cpl}}}_{t} = B\left( {t, T}\right) \left( {N\left( {{e}_{1}\left( {t, T}\right) }\right) - \widetilde{\delta }{F}_{B}\left( {t, T + \delta, T}\right) N\left( {{e}_{2}\left( {t, T}\right) }\right) }\right) ,\] where \( \widetilde{\delta } = 1 + {\kappa \delta } \) and \[ {e}_{1,2}\left( {t, T}\right) = \fra... | Yes |
Proposition 12.3.1 Assume the Gaussian HJM framework, so that the volatilities \( \gamma \left( {t,{T}_{j - 1},{T}_{j}}\right) \) are deterministic. Then the arbitrage price at time \( t \leq {T}_{0} \) of an interest rate cap with strike level \( \kappa \), settled in arrears at times \( {T}_{j}, j = 1,\ldots, n \), e... | Proof. We represent the price of a forward cap in the following way\n\n\[ \n{\mathbf{{FC}}}_{t} = \mathop{\sum }\limits_{{j = 1}}^{n}{\mathbb{E}}_{{\mathbb{P}}^{ * }}\left\{ {\left. {\frac{{B}_{t}}{{B}_{{T}_{j}}}{\left( L\left( {T}_{j - 1},{T}_{j - 1}\right) - \kappa \right) }^{ + }{\delta }_{j}}\right| \;{\mathcal{F}}... | Yes |
Lemma 12.4.1 Let \( G \) and \( H \) be real-valued adapted processes, such that\n\n\[ d{G}_{t} = {\alpha }_{t} \cdot d{W}_{t},\;d{H}_{t} = {\beta }_{t} \cdot d{W}_{t}. \]\n\nAssume, in addition, that \( {H}_{t} > - 1 \) for every \( t \) and write \( {Y}_{t} = {\left( 1 + {H}_{t}\right) }^{-1} \) . Then\n\n\[ d\left( ... | It follows immediately from Lemma 12.4.1 that\n\n\[ d{U}_{2}\left( {t,{T}_{k}^{ * }}\right) = {\eta }_{t}^{k} \cdot \left( {d{W}_{t} - \frac{{\delta }_{n}L\left( {t,{T}_{1}^{ * }}\right) }{1 + {\delta }_{n}L\left( {t,{T}_{1}^{ * }}\right) }\lambda \left( {t,{T}_{1}^{ * }}\right) {dt}}\right) \]\n\nfor a certain process... | No |
Proposition 12.4.1 The drift term \( {\widehat{\mu }}_{i}\left( t\right) \) in the dynamics of \( {L}_{i}\left( t\right) = L\left( {t,{T}_{i}}\right) \) under the forward measure \( {\mathbb{P}}_{{T}_{n}} \) equals\n\n\[ \n{\widehat{\mu }}_{i}\left( t\right) = - \mathop{\sum }\limits_{{j = i + 1}}^{{n - 1}}\frac{{\delt... | Proof. Using Girsanov’s theorem, we get from (12.30), for \( i = 0,\ldots, n - 1 \) ,\n\n\[ \nd{L}_{i}\left( t\right) = {L}_{i}\left( t\right) \left( {{\widehat{\mu }}_{i}\left( t\right) {dt} + {\lambda }_{i}\left( {{L}_{i}\left( t\right), t}\right) d{\widehat{W}}_{t}^{i}}\right) , \n\]\n\nwhere \( {\widehat{W}}^{0},{\... | Yes |
Proposition 12.4.2 For any \( j = 0,\ldots, n - 1 \), the process \( L\left( {t,{T}_{j}}\right) \) satisfies\n\n\[ \n{dL}\left( {t,{T}_{j}}\right) = \mathop{\sum }\limits_{{k = m\left( t\right) }}^{j}\frac{{\delta }_{k + 1}}{1 + {\delta }_{k + 1}L\left( {t,{T}_{k}}\right) }\zeta \left( {t,{T}_{k}}\right) \cdot \zeta \l... | To further specify the model, we postulate that the processes \( \zeta \left( {t,{T}_{j}}\right), j = \) \( 0,\ldots, n - 1 \) are exogenously given. Specifically, let\n\n\[ \n\zeta \left( {t,{T}_{j}}\right) = {\lambda }_{j}\left( {t, L\left( {t,{T}_{j}}\right), L\left( {t,{T}_{j + 1}}\right) ,\ldots, L\left( {t,{T}_{n... | Yes |
Proposition 12.4.3 Assume that the forward LIBORs \( {L}_{0},{L}_{1},\ldots ,{L}_{n - 1} \) satisfy (12.30) under the actual probability \( \mathbb{P} \) . Then they satisfy the following stochastic differential equations under the spot LIBOR measure \( {\mathbb{P}}^{L} \)\n\n\[ d{L}_{i}\left( t\right) = {L}_{i}\left( ... | Proof. Since the relative bond price \( {\widetilde{U}}_{i} \) should follow a (local) martingale under the spot LIBOR measure \( {\mathbb{P}}^{L} \), we require that the drift term in the dynamics of \( {\widetilde{U}}_{i} \) under \( {\mathbb{P}}^{L} \) equals zero. As expected, the proof is based on an application o... | No |
Proposition 12.5.1 Let \( t \leq u \leq {T}_{j} \) . Then we have\n\n\[ \n{\mathbb{E}}_{{\mathbb{P}}_{{T}_{j}}}\left( {L\left( {u,{T}_{j}}\right) \mid {\mathcal{F}}_{t}}\right) = L\left( {t,{T}_{j}}\right) + \frac{{\delta }_{j + 1}{\operatorname{Var}}_{{\mathbb{P}}_{{T}_{j + 1}}}\left( {L\left( {u,{T}_{j}}\right) \mid ... | Proof. Combining (12.5) with the martingale property of \( L\left( {t,{T}_{j}}\right) \) under \( {\mathbb{P}}_{{T}_{j + 1}} \), we obtain\n\n\[ \n{\mathbb{E}}_{{\mathbb{P}}_{{T}_{j}}}\left( {L\left( {u,{T}_{j}}\right) \mid {\mathcal{F}}_{t}}\right) = \frac{{\mathbb{E}}_{{\mathbb{P}}_{{T}_{j + 1}}}\left( {\left( {1 + {... | Yes |
Lemma 12.5.1 Let \( \zeta \) be a nonnegative random variable on a probability space \( \left( {\Omega ,\mathcal{F},\mathbb{P}}\right) \) with the probability density function \( {f}_{\mathbb{P}} \) . Let \( \mathbb{Q} \) be a probability measure equivalent to \( \mathbb{P} \) . Suppose that for any bounded Borel measu... | Proof. The assertion is in fact trivial since, by assumption,\n\n\[ \n{\int }_{-\infty }^{\infty }g\left( y\right) {f}_{\mathbb{P}}\left( y\right) {dy} = {\int }_{-\infty }^{\infty }g\left( y\right) \left( {1 + y}\right) {f}_{\mathbb{Q}}\left( y\right) {dy} \n\]\n\nfor any bounded Borel measurable function \( g : \math... | Yes |
Corollary 12.5.2 The transition probability density function under \( {\mathbb{P}}_{{T}_{j}} \) of the forward bond price \( {F}_{B}\left( {t,{T}_{j + 1},{T}_{j}}\right) \) equals\n\n\[ \n{p}_{B}\left( {t, x, u, y}\right) = \frac{x}{\sqrt{2\pi }{v}_{j}\left( {t, u}\right) {y}^{2}\left( {1 - y}\right) }\exp \left\{ {-\f... | Proof. Let us fix \( x \in \left( {0,1}\right) \) . Using (12.50), it is easy to show that\n\n\[ \n{p}_{B}\left( {t, x, u, y}\right) = {y}^{-2}{\widetilde{p}}_{L}\left( {t,\frac{1 - x}{\delta x}, u,\frac{1 - y}{\delta y}}\right) , \n\] \n\nwhere \( \delta = {\delta }_{j + 1} \) . The formula now follows from Corollary ... | Yes |
For any \( x \in \left( {0,1}\right) \), the process\n\n\[ \n{X}_{t} = {\left( 1 + {e}^{-{Z}_{t}}\right) }^{-1},\;\forall t \in \left\lbrack {0,{T}^{ * }}\right\rbrack ,\n\]\n\nwhere \( z \) satisfies\n\n\[ \n{Z}_{0} = z = \ln \frac{x}{1 - x}\n\]\n\nis the unique strong and non-exploding solution to equation (12.57). M... | Proof. It is easy to see that equation (12.58) can be rewritten in the form\n\n\[ \nd{Z}_{t} = \left( {{X}_{t} - \frac{1}{2}}\right) {\left| \lambda \left( t\right) \right| }^{2}{dt} - \lambda \left( t\right) \cdot d{W}_{t}.\n\]\n\nHence, applying the Itô formula to the process \( X \) given by formula (12.59), we find... | Yes |
Corollary 12.6.1 Let \( g : \mathbb{R} \rightarrow \mathbb{R} \) be a nonnegative Borel function. Then for every \( x \in \left( {0,1}\right) \) and any \( 0 < t < T \) the conditional expectation \( {\mathbb{E}}_{\mathbb{P}}\left( {g\left( {X}_{T}\right) \mid {\mathcal{F}}_{t}}\right) \) equals \( {\mathbb{E}}_{\mathb... | where the function \( k : \left( {0,1}\right) \rightarrow \mathbb{R} \) is given by the formula\n\n\[ k\left( x\right) = \sqrt{x\left( {1 - x}\right) }{e}^{-\frac{{v}^{2}\left( {t, T}\right) }{8}}{\mathbb{E}}_{\mathbb{Q}}\left\{ {g\left( \frac{1}{1 + {e}^{-\left( {z + \zeta }\right) }}\right) \left( {{e}^{\left( {z + \... | Yes |
Proposition 12.6.3 The price \( {C}_{t} \) at time \( t \leq {T}_{j - 1} \) of a European call option, with expiration date \( {T}_{j - 1} \) and strike price \( 0 < K < 1 \), written on a zero-coupon bond maturing at \( {T}_{j} = {T}_{j - 1} + {\delta }_{j} \), equals\n\n\[ \n{C}_{t} = \left( {1 - K}\right) B\left( {t... | Proof. Let us write \( T = {T}_{j - 1} \) and \( T + \delta = {T}_{j} \) . In view of Corollary 12.6.1, it is clear that\n\n\[ \n{C}_{t} = B\left( {t, T}\right) {\mathbb{E}}_{{\mathbb{P}}_{T}}\left( {{\left( F\left( T, T + \delta, T\right) - K\right) }^{ + } \mid {\mathcal{F}}_{t}}\right) = B\left( {t, T}\right) k\left... | Yes |
Lemma 13.1.1 We have, for every \( t \in \left\lbrack {0,{T}_{0}}\right\rbrack \) ,\n\n\[ \n{\mathbf{{FS}}}_{t}\left( \kappa \right) = B\left( {t,{T}_{0}}\right) - \mathop{\sum }\limits_{{j = 1}}^{n}{c}_{j}B\left( {t,{T}_{j}}\right) ,\n\]\n\n(13.2)\n\nwhere \( {c}_{j} = \kappa {\delta }_{j} \) for \( j = 1,\ldots, n - ... | Alternatively, formula (13.2) can be established using the forward measure approach. Indeed, since we have\n\n\[ \nL\left( {t,{T}_{j - 1}}\right) = \frac{B\left( {t,{T}_{j - 1}}\right) - B\left( {t,{T}_{j}}\right) }{{\delta }_{j}B\left( {t,{T}_{j}}\right) },\n\]\n\nthe process \( L\left( {t,{T}_{j - 1}}\right) \) is a ... | Yes |
Lemma 13.1.2 The payoff \( {\mathbf{{PS}}}_{T} \) at expiration date \( T \leq {T}_{0} \) of a forward swaption equals\n\n\[ \n{\mathbf{{PS}}}_{T} = \mathop{\sum }\limits_{{j = 1}}^{n}{\delta }_{j}B\left( {T,{T}_{j}}\right) {\left( \kappa \left( T,{T}_{0}, n\right) - \kappa \right) }^{ + }.\n\] | Formula (13.8) shows that, if exercised, the forward swaption gives rise to a sequence of payments \( {\delta }_{j}\left( {\kappa \left( {T,{T}_{0}, n}\right) - \kappa }\right) \) at each settlement date \( {T}_{j}, j = 1,\ldots, n \) . By setting \( T = {T}_{0} \), we recover, in a more general framework, the previous... | No |
Proposition 13.2.2 Assume the Gaussian HJM model of the term structure of interest rates. Then the arbitrage price at time \( t \) of a CMS spread call option equals\n\n\[ \n{\mathbf{{CMSC}}}_{t} = B\left( {t, T}\right) {\int }_{{\mathbb{R}}^{k}}{\left( {f}_{1}\left( x\right) - {f}_{2}\left( x\right) - \kappa \right) }... | The corresponding put option pays at time \( T \) the amount\n\n\[ \n{P}_{T}\left( {\kappa ,{m}_{1},{m}_{2}}\right) = {\left( \kappa - \kappa \left( T, T,{m}_{1}\right) + \kappa \left( T, T,{m}_{2}\right) \right) }^{ + }. \n\]\n\nConsequently, the arbitrage price at time \( t \in \left\lbrack {0, T}\right\rbrack \) of ... | Yes |
Proposition 13.2.3 The value of a yield curve swap at time \( t \) is given by the risk-neutral valuation formula\n\n\[ \n{\mathbf{{YCS}}}_{t} = {\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {\mathop{\sum }\limits_{{i = 0}}^{{n - 1}}\frac{{B}_{t}}{{B}_{{T}_{i}}}\left( {\kappa \left( {{T}_{i},{T}_{i},{m}_{1}}\right) - \kappa ... | Proof. The proof of the proposition relies on a straightforward application of the valuation formula established in Proposition 11.3.5. | No |
Lemma 13.3.1 Equality \( {G}_{t}\left( {n - j}\right) = B\left( {t,{T}_{n}}\right) {g}_{t}^{j} \) holds for \( j = 1,\ldots, n - 1 \) . | Proof. We shall proceed by induction with respect to \( j \) . Let us first check that the equality holds for \( j = n - 1 \) . We have\n\n\[ \n{G}_{t}\left( {n - \left( {n - 1}\right) }\right) = {G}_{t}\left( 1\right) = {\delta }_{n}B\left( {t,{T}_{n}}\right) = B\left( {t,{T}_{n}}\right) {g}_{t}^{n - 1}, \n\] \n\nsinc... | Yes |
Lemma 13.3.2 Let us denote \( \widetilde{\kappa }\left( {t,{T}_{i}}\right) = {\widetilde{\kappa }}_{t}^{i} \) and\n\n\[ \n{z}_{t}^{j} = \ln \frac{B\left( {t,{T}_{n}}\right) }{{G}_{t}\left( {n - j}\right) } = - \ln {g}_{t}^{j}.\n\]\n\nThen we have for \( j = 0,\ldots, n - 1 \) and \( t \in \left\lbrack {0,{T}_{j}}\right... | Proof. Observe that\n\n\[ \n\ln {g}_{t}^{j} = \ln \left( {\mathop{\sum }\limits_{{i = j}}^{{n - 1}}{\delta }_{i + 1}\mathop{\prod }\limits_{{k = j + 1}}^{i}\left( {1 + {\delta }_{k}{\widetilde{\kappa }}_{t}^{k}}\right) }\right) .\n\]\n\nThe result thus follows by an application of Itô's formula to the right-hand side o... | No |
Proposition 13.3.1 The following recursive relationship is valid for every \( j = 0,\ldots, n - 1 \) and \( t \in \left\lbrack {0,{T}_{j}}\right\rbrack \)\n\n\[ d\widetilde{\kappa }\left( {t,{T}_{j}}\right) = - \mathop{\sum }\limits_{{k = j + 1}}^{{n - 1}}\frac{{\delta }_{k}{g}_{t}^{jk}{\phi }_{t}^{j} \cdot {\phi }_{t}... | Proof. The process \( \widetilde{\kappa }\left( {t,{T}_{j}}\right) \) is a martingale under the forward swap measure, and thus\n\n\[ d\widetilde{\kappa }\left( {t,{T}_{j}}\right) = {\phi }_{t}^{j} \cdot d{\widetilde{W}}_{t}^{{T}_{j + 1}}, \]\n\n(13.25)\n\nBut from equality\n\n\[ {\left\langle {\widetilde{\kappa }}^{j},... | Yes |
Proposition 13.3.2 Assume the lognormal market model of co-terminal forward swap rates. For any \( j = 0,\ldots, n - 1 \), the arbitrage price \( {\mathbf{{PS}}}_{t}^{j} \) of the \( {j}^{\text{th }} \) co-terminal swaption equals, for every \( t \in \left\lbrack {0,{T}_{j}}\right\rbrack \) , \[ {\mathbf{{PS}}}_{t}^{j}... | Proof. The proof of the proposition is similar to that of Proposition 12.6.1, and thus it is omitted. | No |
Proposition 13.5.2 Assume the lognormal market model of co-sliding forward swap rates. For any \( j = 1,\ldots, n - K \), the arbitrage price at time \( t \in \left\lbrack {0, T - j}\right\rbrack \) of the \( {j}^{\text{th }} \) co-sliding swaption equals\n\n\[ \n{\widehat{\mathbf{{PS}}}}_{t}^{j} = \mathop{\sum }\limit... | Note that the swaption price at time 0 depends only on the initial term structure and the volatility of the underlying forward swap rate. In particular, the volatilities \( {\zeta }_{m + 1}\left( {t,{T}_{m + K}}\right), m = 1,\ldots, n - K \) of relative bond prices that were used as auxiliary inputs in the model's con... | No |
Lemma 13.6.1 The following equality holds for every \( t \in \left\lbrack {0, T}\right\rbrack \)\n\n\[{\mathbf{{PS}}}_{t} = \mathop{\sum }\limits_{{j = 1}}^{n}{\delta }_{j}B\left( {t,{T}_{j}}\right) {\mathbb{E}}_{{\mathbb{P}}_{{T}_{j}}}\left( {\left( {L\left( {T,{T}_{j - 1}}\right) - \kappa }\right) {\mathbb{1}}_{D} \m... | Proof. Since\n\n\[{\mathbf{{PS}}}_{t} = {\mathbb{E}}_{{\mathbb{P}}^{ * }}\left\{ {\left. {\frac{{B}_{t}}{{B}_{T}}{\mathbb{1}}_{D}{\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {\mathop{\sum }\limits_{{j = 1}}^{n}\frac{{B}_{T}}{{B}_{{T}_{j}}}\left( {L\left( {T}_{j - 1}\right) - \kappa }\right) {\delta }_{j} \mid {\mathcal{F}}_... | Yes |
Proposition 13.6.1 Assume the lognormal LIBOR model. The price at time 0 of a payer swaption with expiry date \( T = {T}_{0} \) and strike level \( \kappa \) equals\n\n\[ \n{\mathbf{{PS}}}_{0} = \mathop{\sum }\limits_{{j = 1}}^{n}{\delta }_{j}B\left( {0,{T}_{j}}\right) {\int }_{{\mathbb{R}}^{n}}\left( {L\left( {0,{T}_{... | Proof. Let us start by considering arbitrary \( t \in \left\lbrack {0, T}\right\rbrack \) . Notice that\n\n\[ \n\frac{B\left( {t,{T}_{j}}\right) }{B\left( {t, T}\right) } = \mathop{\prod }\limits_{{k = 1}}^{j}\frac{B\left( {t,{T}_{k}}\right) }{B\left( {t,{T}_{k - 1}}\right) } = \mathop{\prod }\limits_{{k = 1}}^{j}{\lef... | Yes |
Lemma 13.8.1 Let \( {\mathbb{P}}^{ * } \) be an arbitrary probability measure equivalent to \( \mathbb{P} \) . Define the process \( B \) by setting \( {B}_{t} = {\eta }_{t}{A}_{t}^{-1} \) for every \( t \in \left\lbrack {0,{T}^{ * }}\right\rbrack \), where \( \eta \) is the Radon-Nikodým derivative of \( {\mathbb{P}}^... | Proof. We have\n\n\[ \n{B}_{t}{\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {{B}_{T}^{-1} \mid {\mathcal{F}}_{t}}\right) = \frac{{\mathbb{E}}_{\mathbb{P}}\left( {{\eta }_{T}{B}_{T}^{-1} \mid {\mathcal{F}}_{t}}\right) }{{\eta }_{t}{B}_{t}^{-1}} = \frac{{\mathbb{E}}_{\mathbb{P}}\left( {{A}_{T} \mid {\mathcal{F}}_{t}}\right) }{... | Yes |
Proposition 13.8.1 Let \( A \) be a strictly positive supermartingale. Then there exists a unique strictly positive martingale \( \eta \), with \( {\eta }_{0} = 1 \), such that the process \( {B}_{t} = {\eta }_{t}{A}_{t}^{-1} \) is an increasing process. The bond price model \( B\left( {t, T}\right) \) is arbitrage-fre... | Proof. The first statement follows directly from the multiplicative decomposition of a strictly positive supermartingale \( A \) . The first equality in (13.61) is standard; the second is an immediate consequence of Bayes's rule. | No |
Proposition 13.8.2 Assume that the coefficients \( {a}_{j} \) and \( {b}_{j} \) are strictly positive. Then the price of a caplet maturing at time \( {T}_{j} \) equals, for \( t \in \left\lbrack {0,{T}_{j}}\right\rbrack \) , \[ {\mathbf{{Cpl}}}_{t}^{j} = {\left( f\left( t\right) + g\left( t\right) {M}_{t}\right) }^{-1}... | Proof. We have \[ {\mathbb{E}}_{\mathbb{P}}\left( {{\left( {a}_{j} - {b}_{j}{M}_{{T}_{j}}\right) }^{ + } \mid {\mathcal{F}}_{t}}\right) = {\mathbb{E}}_{\mathbb{P}}\left( {{\left( {a}_{j} - {b}_{j}{M}_{t}\zeta \right) }^{ + } \mid {\mathcal{F}}_{t}}\right) , \] where the random variable \( \zeta \), which is given by th... | Yes |
Lemma 14.1.1 The domestic forward price \( {\widetilde{F}}_{{Z}^{i}}\left( {t, T}\right) \) for the settlement at time \( T \) of the foreign market security \( {Z}^{i} \) (which pays no dividends) satisfies\n\n\[ \n{\widetilde{F}}_{{Z}^{i}}\left( {t, T}\right) = \frac{{Z}_{t}^{i}{Q}_{t}^{i}}{B\left( {t, T}\right) } = ... | Proof. The first equality follows by standard no-arbitrage arguments. For the second, notice that\n\n\[ \n\frac{{Z}_{t}^{i}{Q}_{t}^{i}}{B\left( {t, T}\right) } = \frac{{Z}_{t}^{i}}{{B}^{i}\left( {t, T}\right) }\frac{{B}^{i}\left( {t, T}\right) }{B\left( {t, T}\right) }{Q}_{t}^{i} = {F}_{{Z}^{i}}\left( {t, T}\right) {F}... | Yes |
Lemma 14.1.2 The Radon-Nikodým derivative on \( \left( {\Omega ,{\mathcal{F}}_{T}}\right) \) of the forward measure \( {\mathbb{P}}_{T}^{i} \) of the \( {i}^{\text{th }} \) foreign market with respect to the domestic spot martingale measure \( {\mathbb{P}}^{ * } \) equals\n\n\[ \frac{d{\mathbb{P}}_{T}^{i}}{d{\mathbb{P}... | Proof. For any two continuous semimartingales \( X, Y \) defined on a probability space \( \left( {\Omega ,\mathbb{F},\mathbb{Q}}\right) \), with \( {X}_{0} = {Y}_{0} = 0 \), we have (see Theorem II. 37 in Protter (2003))\n\n\[ {\mathcal{E}}_{t}\left( X\right) {\mathcal{E}}_{t}\left( Y\right) = {\mathcal{E}}_{t}\left( ... | Yes |
Lemma 14.2.1 Let \( {V}^{k}\left( {T, U}\right) \) stand for the following process\n\n\[ \n{V}_{t}^{k}\left( {T, U}\right) = \frac{B\left( {t, U}\right) {B}^{k}\left( {t, T}\right) {G}_{t}^{k}\left( {T, U}\right) }{{B}^{k}\left( {t, U}\right) },\;\forall t \in \left\lbrack {0, T}\right\rbrack ,\n\]\n\n(14.38)\n\nwhere\... | Proof. Since \( {G}^{k}\left( {T, U}\right) \) is an adapted process of finite variation, Itô’s formula yields\n\n\[ \nd{V}_{t}^{k}\left( {T, U}\right) = {G}_{t}^{k}\left( {T, U}\right) \left( {{F}_{t}{dB}\left( {t, U}\right) + B\left( {t, U}\right) d{F}_{t} + d\langle B\left( {\cdot, U}\right), F{\rangle }_{t}}\right)... | Yes |
Proposition 14.2.1 Let us consider the portfolio \( \phi = \left( {{\phi }^{1},{\phi }^{2},{\phi }^{3}}\right) \) that equals\n\n\[{\phi }_{t}^{1} = \frac{{V}_{t}^{k}\left( {T, U}\right) }{B\left( {t, U}\right) },\;{\phi }_{t}^{2} = - \frac{{V}_{t}^{k}\left( {T, U}\right) }{{\widetilde{B}}^{k}\left( {t, U}\right) },\;{... | Proof. For the last claim, it is enough to check that\n\n\[{V}_{t}\left( \phi \right) = {\phi }_{t}^{1}B\left( {t, U}\right) + {Q}_{t}^{k}\left( {{\phi }_{t}^{2}{B}^{k}\left( {t, U}\right) + {\phi }_{t}^{3}{B}^{k}\left( {t, T}\right) }\right) = {V}_{t}^{k}\left( {T, U}\right)\]\n\nfor every \( t \in \left\lbrack {0, T}... | Yes |
Proposition 14.2.2 The arbitrage price of the floating-for-floating cross-currency \( \left( {k,0;0}\right) \) swap at time \( t \in \left\lbrack {0,{T}_{0}}\right\rbrack \) equals\n\n\[ \n{\operatorname{CCFS}}_{t}\left( {k,0;0}\right) = \mathop{\sum }\limits_{{j = 1}}^{n}\left( {\frac{B\left( {t,{T}_{j}}\right) {B}^{k... | Proof. It is enough to observe that\n\n\[ \n{\operatorname{CCFS}}_{t}\left( {k,0;0}\right) = {\mathbb{E}}_{{\mathbb{P}}^{ * }}\left\{ {\left. {\mathop{\sum }\limits_{{j = 1}}^{n}\frac{{B}_{t}}{{B}_{{T}_{j}}}\left( {\frac{1}{{B}^{k}\left( {{T}_{j - 1},{T}_{j}}\right) } - \frac{1}{B\left( {{T}_{j - 1},{T}_{j}}\right) }}\... | Yes |
Lemma 14.2.2 The following equalities hold for every \( t \in \left\lbrack {0,{T}_{0}}\right\rbrack \)\n\n\[ \n{J}_{t} = \mathop{\sum }\limits_{{j = 1}}^{n}{\widetilde{\delta }}_{j}{Q}_{t}^{m}{B}^{m}\left( {t,{T}_{j}}\right) \n\]\n\n(14.49)\n\nand\n\n\[ \n{I}_{t} = \mathop{\sum }\limits_{{j = 1}}^{n}\frac{{Q}_{t}^{m}{B... | Proof. For the first formula, it is enough to observe that the equality\n\n\[ \n{\mathbb{E}}_{{\mathbb{P}}^{ * }}\left\{ {\left. {\frac{{B}_{t}}{{B}_{{T}_{j}}}{Q}_{{T}_{j}}^{m}}\right| \;{\mathcal{F}}_{t}}\right\} = {Q}_{t}^{m}{B}^{m}\left( {t,{T}_{j}}\right) \n\]\n\nis valid for every \( t \in \left\lbrack {0,{T}_{j}}... | Yes |
Proposition 14.2.3 The arbitrage price at time \( t \in \left\lbrack {0,{T}_{0}}\right\rbrack \) of a fixed-for-floating cross-currency \( \left( {k;m}\right) \) swap with the underlying fixed interest rate equal to \( \kappa \) is given by the expression (recall that \( {\widetilde{\delta }}_{j} = 1 + \kappa {\delta }... | \[ {\operatorname{CCFS}}_{t}^{\kappa }\left( {k;m}\right) = \mathop{\sum }\limits_{{j = 1}}^{n}{Q}_{t}^{m}{B}^{m}\left( {t,{T}_{j}}\right) \left( {\frac{{B}^{k}\left( {t,{T}_{j - 1}}\right) {G}_{t}^{km}\left( {{T}_{j - 1},{T}_{j}}\right) }{{B}^{k}\left( {t,{T}_{j}}\right) } - {\widetilde{\delta }}_{j}}\right) . \] | Yes |
Proposition 14.2.4 The price \( {\operatorname{CCFS}}_{t} = {\operatorname{CCFS}}_{t}\left( {k, l;m}\right) \) at time \( t \in \left\lbrack {0,{T}_{0}}\right\rbrack \) of the floating-for-floating cross-currency forward swap of type \( \left( {k, l;m}\right) \) in a Gaussian HJM model equals | \[ {\mathbf{{CCFS}}}_{t} = \mathop{\sum }\limits_{{j = 1}}^{n}{Q}_{t}^{m}{B}^{m}\left( {t,{T}_{j}}\right) \left( {\frac{{B}^{k}\left( {t,{T}_{j - 1}}\right) }{{B}^{k}\left( {t,{T}_{j}}\right) }{e}_{t}^{kmj} - \frac{{B}^{l}\left( {t,{T}_{j - 1}}\right) }{{B}^{l}\left( {t,{T}_{j}}\right) }{e}_{t}^{lmj}}\right) ,\] where ... | Yes |
Proposition 14.3.1 The arbitrage price of a quanto caplet written on a LIBOR rate of the \( {i}^{\text{th }} \) market satisfies\n\n\[ \n{\mathbf{{Cplq}}}_{t} \approx \bar{Q}{\delta B}\left( {t, T + \delta }\right) \left( {{L}^{i}\left( {t, T}\right) {e}^{-{\widetilde{\eta }}^{i}\left( {t, T}\right) }N\left( {{\widetil... | Proof. It is enough to observe that \n\n\[ \n{\mathbf{{Cplq}}}_{t} \approx \bar{Q}{\delta B}\left( {t, T + \delta }\right) {e}^{-{\widetilde{\eta }}^{i}\left( {t, T}\right) }{\mathbb{E}}_{{\mathbb{P}}_{T + \delta }}\left\{ {\left. {\left( {L}^{i}\left( t, T\right) \Lambda \left( t, T\right) - \widetilde{\kappa }\right)... | Yes |
Corollary 1.1 (Put-Call parity) Under the previous assumptions, we have\n\n\[ \n{c}_{t} = {p}_{t} + {S}_{t} - K{e}^{-r\left( {T - t}\right) },\;t \in \left\lbrack {0, T}\right\rbrack .\n\] | Proof. It suffices to note that the investments\n\n\[ \n{X}_{t} = {c}_{t} + \frac{K}{{B}_{T}}{B}_{t}\;\text{ and }\;{Y}_{t} = {p}_{t} + {S}_{t},\n\]\n\nhave the same final value\n\n\[ \n{X}_{T} = {Y}_{T} = \max \left\{ {K,{S}_{T}}\right\}\n\]\n\nThe claim follows from (1.3). | Yes |
Corollary 1.2 (Estimates from above and below for European options) For every \( t \in \left\lbrack {0, T}\right\rbrack \) we have\n\n\[ \n{\left( {S}_{t} - K{e}^{-r\left( {T - t}\right) }\right) }^{ + } < {c}_{t} < {S}_{t} \]\n\n(1.5)\n\n\[ \n{\left( K{e}^{-r\left( {T - t}\right) } - {S}_{t}\right) }^{ + } < {p}_{t} <... | Proof. By (1.2)\n\n\[ \n{c}_{t},{p}_{t} > 0 \]\n\n(1.6)\n\nConsequently by (1.4) we get\n\n\[ \n{c}_{t} > {S}_{t} - K{e}^{-r\left( {T - t}\right) }.\n\]\n\nMoreover, since \( {c}_{t} > 0 \), we get the first estimate from below. Finally \( {c}_{T} < {S}_{T} \) and so by (1.2) we get the first estimate from above. The s... | No |
In the case of one risky asset (i.e. \( d = 1 \) ) (2.6) is equivalent to\n\n\[ \n{\beta }_{n} = {\beta }_{n - 1} - \left( {{\alpha }_{n} - {\alpha }_{n - 1}}\right) \frac{{S}_{n - 1}}{{B}_{n - 1}}.\n\] | The variation, from time \( {t}_{n - 1} \) to \( {t}_{n} \), of the value of a self-financing strategy \( \left( {\alpha ,\beta }\right) \) is given by\n\n\[ \n{V}_{n}^{\left( \alpha ,\beta \right) } - {V}_{n - 1}^{\left( \alpha ,\beta \right) } = {\alpha }_{n}\left( {{S}_{n} - {S}_{n - 1}}\right) + {\beta }_{n}\left( ... | No |
Lemma 2.6 The value of a self-financing strategy \( \left( {\alpha ,\beta }\right) \) is determined by its initial value \( {V}_{0} \) and recursively by\n\n\[ \n{V}_{n} = {V}_{n - 1}\left( {1 + {r}_{n}}\right) + \mathop{\sum }\limits_{{i = 1}}^{d}{\alpha }_{n}^{i}{S}_{n - 1}^{i}\left( {{\mu }_{n}^{i} - {r}_{n}}\right)... | Proof. By (2.7), the variation of a self-financing portfolio in the period \( \left\lbrack {{t}_{n - 1},{t}_{n}}\right\rbrack \) is equal to\n\n\[ \n{V}_{n} - {V}_{n - 1} = {\alpha }_{n}\left( {{S}_{n} - {S}_{n - 1}}\right) + {\beta }_{n}\left( {{B}_{n} - {B}_{n - 1}}\right) \n\]\n\n\[ \n= \mathop{\sum }\limits_{{i = 1... | Yes |
Proposition 2.7 Given \( {V}_{0} \in \mathbb{R} \) and a predictable process \( \alpha \), there exists a unique predictable process \( \beta \) such that \( \left( {\alpha ,\beta }\right) \in \mathcal{A} \) and \( {V}_{0}^{\left( \alpha ,\beta \right) } = {V}_{0} \) . | Proof. Given \( {V}_{0} \in \mathbb{R} \) and a predictable process \( \alpha \), we define the process\n\n\[ \n{\beta }_{n} = \frac{{V}_{n - 1} - {\alpha }_{n}{S}_{n - 1}}{{B}_{n - 1}},\;n = 1,\ldots, N, \n\] \n\nwhere \( \left( {V}_{n}\right) \) is recursively defined by (2.8). Then by construction \( \left( {\beta }... | Yes |
Lemma 2.9 The discounted value of a self-financing strategy \( \left( {\alpha ,\beta }\right) \) is uniquely determined by its initial value \( {V}_{0} \) and recursively by\n\n\[ \n{\widetilde{V}}_{n}^{\left( \alpha ,\beta \right) } = {\widetilde{V}}_{n - 1}^{\left( \alpha ,\beta \right) } + \mathop{\sum }\limits_{{i ... | The following formula, analogous to (2.10), holds:\n\n\[ \n{\widetilde{V}}_{n}^{\left( \alpha ,\beta \right) } = {V}_{0} + {G}_{n}^{\left( \alpha \right) }\n\]\n\n\( \left( {2.13}\right) \)\n\nwhere\n\n\[ \n{G}_{n}^{\left( \alpha \right) } = \mathop{\sum }\limits_{{k = 1}}^{n}{\alpha }_{k}\left( {{\widetilde{S}}_{k} - ... | Yes |
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