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Example 1.6.1 Let \( {f}_{t} = {f}_{S}\left( {t,{T}^{ * }}\right) \) be a one-period process that models the futures price of a certain asset \( S \), for the settlement date \( {T}^{ * } \geq T \). We assume that \( {f}_{0} = {280} \), and \[ {f}_{T}\left( \omega \right) = \left\{ \begin{array}{ll} {f}^{u} = {320}, & ... | For Example 1.6.1, this gives the following system of linear equations \[ \begin{cases} {40}{\alpha }_{0} + {1.05}{\beta }_{0} & = {40} \\ - {20}{\alpha }_{0} + {1.05}{\beta }_{0} & = 0 \end{cases} \] yielding \( {\alpha }_{0} = 2/3 \) and \( {\beta }_{0} = {12.70} \). The manufacturing cost of a futures call option is... | Yes |
Proposition 1.6.1 The futures market \( {\mathcal{M}}^{f} = \left( {f, B,{\Phi }^{f}}\right) \) is arbitrage-free if and only if the process \( f \) that models the futures price admits a (unique) martingale measure \( \widetilde{\mathbb{P}} \) equivalent to \( \mathbb{P} \) . In this case, the arbitrage price at time ... | Proof. If there is no martingale measure for \( f \), equivalent to \( \mathbb{P} \), we have either \( \widetilde{p} \geq 1 \) or \( \widetilde{p} \leq 0 \) . In the first case, we have \( {f}_{0} - {f}^{d} \geq {f}^{u} - {f}^{d} \) and thus \( {f}_{0} \geq {f}^{u} > {f}^{d} \) . Consequently, a portfolio \( \phi = \l... | Yes |
Corollary 1.6.1 The futures price at time 0 for the delivery date \( T \) of the underlying asset \( S \) that makes the spot/futures market arbitrage-free equals \( {f}_{0} = \left( {1 + r}\right) {S}_{0} \) . | Proof. Suppose an investor enters at time 0 into one futures contract. The payoff of his position at time \( T \) corresponds to a time \( T \) contingent claim \( X = {f}_{T} - {f}_{0} = \) \( {S}_{T} - {f}_{0} \) . Since it costs nothing to enter a futures contract we should have\n\n\[ \n{\pi }_{0}\left( X\right) = {... | Yes |
Proposition 1.7.1 Assume that the risk-free interest rate \( r \) is a nonnegative real number. Then the arbitrage price \( {C}_{0}^{a} \) of an American call option in the arbitrage-free market model \( \mathcal{M} = \left( {S, B,\Phi }\right) \) coincides with the price \( {C}_{0} \) of the European call option with ... | Proof. Assume, on the contrary, that \( {C}_{0}^{a} \neq {C}_{0} \) . Suppose first that \( {C}_{0}^{a} > {C}_{0} \) . Note that the arbitrage price \( {C}_{0} \) satisfies\n\n\[ \n{C}_{0} = {p}_{ * }\frac{{S}^{u} - K}{1 + r} = \frac{\left( {1 + r}\right) {S}_{0} - {S}^{d}}{{S}^{u} - {S}^{d}}\frac{{S}^{u} - K}{1 + r} >... | Yes |
Lemma 1.7.1 Let \( h : \mathbb{R} \rightarrow \mathbb{R} \) be a convex function, and \( \xi \) a random variable on a finite probability space \( \left( {\Omega ,\mathcal{F},\mathbb{P}}\right) \) . Then Jensen’s inequality holds, that is, \( g\left( {{\mathbb{E}}_{\mathbb{P}}\left( \xi \right) }\right) \leq {\mathbb{E... | Proof. Under the present assumptions we have \( \Omega = \left\{ {{\omega }_{1},\ldots ,{\omega }_{n}}\right\} \), and thus\n\n\[ g\left( {{\mathbb{E}}_{\mathbb{P}}\left( \xi \right) }\right) = g\left( {\mathop{\sum }\limits_{{i = 1}}^{n}{p}_{i}\xi \left( {\omega }_{i}\right) }\right) \]\n\nwhere \( {p}_{i} = \mathbb{P... | Yes |
Proposition 1.7.2 Assume that \( r > 0 \) . Then \( {P}_{0}^{a} = {P}_{0} \) if and only if the inequality\n\n\[ K - {S}_{0} \leq \frac{{S}^{u} - \left( {1 + r}\right) {S}_{0}}{{S}^{u} - {S}^{d}}\frac{K - {S}^{d}}{1 + r} \]\n\n(1.35)\n\nis valid. Otherwise, \( {P}_{0}^{a} = K - {S}_{0} > {P}_{0} \) . If \( r = 0 \), th... | Proof. In view of (1.21), it is clear that inequality (1.35) is equivalent to \( {P}_{0} \geq K - {S}_{0} \) . Suppose first that the last inequality holds. If, in addition, \( {P}_{0}^{a} > {P}_{0} \) (respectively, \( {P}_{0}^{a} < {P}_{0} \) ), by selling the American put and buying the European put (respectively, b... | Yes |
Proposition 1.8.1 Let \( {C}_{t} \) and \( {P}_{t} \) (respectively, \( {C}_{t}^{a} \) and \( {P}_{t}^{a} \) ) stand for the arbitrage prices at time \( t \) of European (respectively, American) call and put options, with strike price \( K \) and expiry date \( T \) . Then the following inequalities are valid for every... | Proof. All inequalities may be derived by constructing appropriate portfolios at time \( t \) and holding them to the terminal date. Let us derive, for instance, the first one. Consider the following (static) portfolios, A and B, established at time \( t \) . Portfolio A consists of one European call option and \( K{e}... | Yes |
Proposition 1.8.2 Assume that \( {K}_{1} < {K}_{2} \) . The following inequalities are valid\n\n\[ \n{e}^{-{rT}}\left( {{K}_{1} - {K}_{2}}\right) \leq C\left( {{S}_{0}, T,{K}_{2}}\right) - C\left( {{S}_{0}, T,{K}_{1}}\right) \leq 0 \n\] \n\nand \n\n\[ \n{K}_{1} - {K}_{2} \leq {C}^{a}\left( {{S}_{0}, T,{K}_{2}}\right) -... | Proof. Let us consider, for instance, the case of European call options. Take the two following portfolios at time 0: portfolio A - one European call with exercise price \( {K}_{2} \) and \( {e}^{-{rT}}\left( {{K}_{2} - {K}_{1}}\right) \) units of cash; and portfolio B - one European call with exercise price \( {K}_{1}... | No |
Proposition 1.8.3 The price of a European (or American) call (or put) option is a convex function of the exercise price \( K \) . | Proof. Let us consider, for instance, the case of a European put option. We denote its price at time 0 by \( P\left( {{S}_{0}, T, K}\right) \) . Assume that \( {K}_{1} < {K}_{2} \) and put \( {K}_{3} = \gamma {K}_{1} + (1 - \gamma ){K}_{2} \), where \( \gamma \in \left\lbrack {0,1}\right\rbrack \) is a constant. We con... | Yes |
Proposition 2.1.2 For every \( m = 1,2,\ldots, T \), the arbitrage price of a European call option at time \( t = T - m \) is given by the Cox-Ross-Rubinstein valuation formula\n\n\[ \n{C}_{T - m} = {S}_{T - m}\mathop{\sum }\limits_{{j = a}}^{m}\left( \begin{matrix} m \\ j \end{matrix}\right) {\bar{p}}^{j}{\left( 1 - \... | Proof. Straightforward calculations yield \( 1 - \bar{p} = d\left( {1 - {p}_{ * }}\right) /\widehat{r} \), and thus\n\n\[ \n{\bar{p}}^{j}{\left( 1 - \bar{p}\right) }^{m - j} = {p}_{ * }^{j}{\left( 1 - {p}_{ * }\right) }^{m - j}{u}^{j}{d}^{m - j}/{\widehat{r}}^{m}. \n\]\n\nFormula (2.17) is therefore equivalent to the f... | Yes |
A martingale measure \( {\mathbb{P}}^{ * } \) for the discounted stock price \( {S}^{ * } \) exists if and only if \( d < 1 + r < u \) . In this case, the martingale measure \( {\mathbb{P}}^{ * } \) is the unique element from the class \( \mathcal{P} \) that corresponds to \( p = {p}_{ * } = \left( {1 + r - d}\right) /... | Using (2.23)-(2.24), we may re-express equality (2.25) in the following way\n\n\[ \n{\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {{\widehat{r}}^{-\left( {t + 1}\right) }{\xi }_{t + 1}{S}_{t} \mid {\mathcal{F}}_{t}^{S}}\right) = {\widehat{r}}^{-t}{S}_{t},\;\forall t \leq {T}^{ * } - 1, \n\] \n\n(2.26) \n\nor equivalently \n\... | Yes |
Lemma 2.2.1 Let \( \xi \) be a random variable on a probability space \( \left( {\Omega ,\mathcal{F},\mathbb{P}}\right) \) such that \( \mathbb{P}\{ \xi = a\} + \mathbb{P}\{ \xi = b\} = 1 \) for some real numbers \( a, b \) . Let \( \mathcal{G} \) be a sub- \( \sigma \) -field of \( \mathcal{F} \) . If \( {\mathbb{E}}_... | Proof. Equality \( {\mathbb{E}}_{\mathbb{P}}\left( {\xi \mid \mathcal{G}}\right) = {\mathbb{E}}_{\mathbb{P}}\xi \) implies that for any event \( B \in \mathcal{G} \) we have\n\n\[ \n{\int }_{B}{\xi d}\mathbb{P} = {\int }_{B}{\mathbb{E}}_{\mathbb{P}}{\xi d}\mathbb{P} \n\]\n\nLet us write \( A = \{ \xi = a\} \) . Then we... | Yes |
Corollary 2.2.2 The CRR binomial model \( \left( {S, B,\Phi }\right) \) is arbitrage-free if and only if \( d < 1 + r < u \) . | Proof. Suppose first that the condition \( d < 1 + r < u \) does not hold. Then is it possible to produce an example of an arbitrage opportunity by considering, for instance, the first time-period and proceeding as in the case of a single-period model (buy stock if \( 1 + r \leq d \) and short stock if \( 1 + r \geq u ... | No |
Consider a European call option, with expiry date \( T \) and strike price \( K \), written on one share of a common stock whose price \( S \) is assumed to follow the CRR multiplicative binomial process (2.5). Then for any \( m = 0,1,\ldots, T \) the arbitrage price \( {C}_{T - m} \), given by formula (2.17), coincide... | Proof. It is enough to find the conditional expectation in (2.28) explicitly. Recall that we have\n\n\[ \n{S}_{T} = {S}_{T - m}{\xi }_{T - m + 1}{\xi }_{T - m + 2}\ldots {\xi }_{T} = {S}_{T - m}{\eta }_{m},\n\]\n\nwhere we write \( {\eta }_{m} = {\xi }_{T - m + 1}{\xi }_{T - m + 2}\ldots {\xi }_{T} \) . Note that the s... | Yes |
Proposition 2.2.4 The Radon-Nikodým derivative process \( \eta \) of \( \overline{\mathbb{P}} \) with respect to \( {\mathbb{P}}^{ * } \) equals, for every \( t = 0,1,\ldots ,{T}^{ * } \) , \[ {\eta }_{t} = {\left. \frac{d\overline{\mathbb{P}}}{d{\mathbb{P}}^{ * }}\right| }_{{\mathcal{F}}_{t}^{S}} = \frac{{B}_{0}{S}_{t... | Proof. It is clear that \( {\mathbb{P}}^{ * } \) and \( \overline{\mathbb{P}} \) are equivalent on \( \left( {\Omega ,{\mathcal{F}}_{{T}^{ * }}}\right) \) and thus the Radon-Nikodým derivative \( {\eta }_{{T}^{ * }} = d\overline{\mathbb{P}}/d{\mathbb{P}}^{ * } \) exists. Since \( {S}^{ * } = S/B \) is a martingale unde... | Yes |
Corollary 2.2.3 For any European contingent claim \( X \) settling at time \( T \) the arbitrage price satisfies, for every \( t = 0,1,\ldots, T \) , | \[ {\pi }_{t}\left( X\right) = {B}_{t}{\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {{B}_{T}^{-1}X \mid {\mathcal{F}}_{t}^{S}}\right) = {S}_{t}{\mathbb{E}}_{\overline{\mathbb{P}}}\left( {{S}_{T}^{-1}X \mid {\mathcal{F}}_{t}^{S}}\right) . \] | Yes |
Proposition 2.2.5 The arbitrage price at time \( t \) of a European call option settling at time \( T \) equals\n\n\[ \n{C}_{t} = {S}_{t}\overline{\mathbb{P}}\left\{ {{S}_{T} > K \mid {\mathcal{F}}_{t}^{S}}\right\} - K{\widehat{r}}^{-\left( {T - t}\right) }{\mathbb{P}}^{ * }\left\{ {{S}_{T} > K \mid {\mathcal{F}}_{t}^{... | Proof. To derive (2.34), let us denote \( D = \left\{ {{S}_{T} > K}\right\} \) and let us observe that\n\n\[ \n{C}_{T} = {\left( {S}_{T} - K\right) }^{ + } = {S}_{T}{\mathbb{1}}_{D} - K{\mathbb{1}}_{D}.\n\]\n\nConsequently, using the linearity of the arbitrage price and Corollary 2.2.3, we obtain\n\n\[ \n{C}_{t} = {\pi... | Yes |
Proposition 2.3.1 For any dyadic \( t \in \left\lbrack {0, T}\right\rbrack \), the following convergence holds\n\n\[ \mathop{\lim }\limits_{{n \rightarrow + \infty }}\mathop{\sum }\limits_{{j = {b}_{n}\left( t\right) }}^{{{m}_{n}\left( t\right) }}\left( \begin{matrix} {m}_{n}\left( t\right) \\ j \end{matrix}\right) \le... | Proof of Proposition 2.3.1. Let \( {S}_{t} = s \) be the generic value of the stock price at time \( t \) . Our first goal is to check that\n\n\[ \mathop{\lim }\limits_{{n \rightarrow + \infty }}\mathop{\sum }\limits_{{j = {b}_{n}\left( t\right) }}^{{{m}_{n}\left( t\right) }}\left( \begin{matrix} {m}_{n}\left( t\right)... | Yes |
Corollary 2.4.1 Let the nonnegative adapted process \( {U}_{t}, t \leq T \), be defined recursively by setting \( {U}_{T} = {\left( K - {S}_{T}\right) }^{ + } \) and, for \( t \leq T - 1 \) , \[ {U}_{t} = \max \left\{ {K - {S}_{t},{\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {{\widehat{r}}^{-1}{U}_{t + 1} \mid {\mathcal{F}}... | It is also possible to go the other way around - that is, to first show directly that the price \( {P}_{t}^{a} \) needs to satisfy the recursive relation, for \( t \leq T - 1 \) , \[ {P}_{t}^{a} = \max \left\{ {K - {S}_{t},{\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {{\widehat{r}}^{-1}{P}_{t + 1}^{a} \mid {\mathcal{F}}_{t}... | Yes |
For every \( t \leq T \), the arbitrage price \( \pi \left( {X}^{a}\right) \) of an arbitrary American claim \( {X}^{a} \) in the CRR model equals\n\n\[ \n{\pi }_{t}\left( {X}^{a}\right) = \mathop{\max }\limits_{{\tau \in {\mathcal{T}}_{\left\lbrack t, T\right\rbrack }}}{\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {{\wideha... | The price process \( \pi \left( {X}^{a}\right) \) can be determined using the following recurrence relation, for every \( t \leq T - 1 \) ,\n\n\[ \n{\pi }_{t}\left( {X}^{a}\right) = \max \left\{ {{X}_{t},{\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {{\widehat{r}}^{-1}{\pi }_{t + 1}\left( {X}^{a}\right) \mid {\mathcal{F}}_{t... | Yes |
Lemma 2.6.1 A spot trading strategy \( \phi \) is self-financing if and only if we have that, for every \( t = 0,1,\ldots ,{T}^{ * } \) ,\n\n\[ \n{V}_{t}\left( \phi \right) = {V}_{0}\left( \phi \right) + {G}_{t}\left( \phi \right) \n\] | Proof. Assume first that \( \phi \) is self-financing. Then, taking into account formulas (2.58)-(2.59), we obtain\n\n\[ \n{V}_{t}\left( \phi \right) = {\phi }_{0} \cdot {S}_{0} + \mathop{\sum }\limits_{{u = 0}}^{{t - 1}}\left( {{\phi }_{u + 1} \cdot {S}_{u + 1} - {\phi }_{u} \cdot {S}_{u}}\right) \n\]\n\n\[ \n= {\phi ... | Yes |
Proposition 2.6.1 Suppose that the market model \( \mathcal{M} \) is arbitrage-free. Then any attainable contingent claim \( X \) is uniquely replicated in \( \mathcal{M} \) . | Proof. Suppose, on the contrary, that there exists a time \( T \) attainable contingent claim \( X \) which admits two replicating strategies, say \( \phi \) and \( \psi \), such that for some \( t < T \) we have: \( {V}_{u}\left( \phi \right) = {V}_{u}\left( \psi \right) \) for every \( u < t \), and \( {V}_{t}\left( ... | Yes |
Lemma 2.6.2 A trading strategy \( \phi \) is self-financing if and only if the relative wealth process \( {V}^{ * }\left( \phi \right) = V\left( \phi \right) {B}^{-1} \) satisfies, for every \( t = 0,1,\ldots ,{T}^{ * } \) ,\n\n\[ \n{V}_{t}^{ * }\left( \phi \right) = {V}_{0}^{ * }\left( \phi \right) + \mathop{\sum }\li... | Proof. Let us write \( V = V\left( \phi \right) \) and \( {V}^{ * } = {V}^{ * }\left( \phi \right) \) . Then (2.61) is equivalent to\n\n\[ \n{\Delta }_{t}{V}^{ * } = {V}_{t + 1}^{ * } - {V}_{t}^{ * } = {\phi }_{t} \cdot {\Delta }_{t}{S}^{ * },\;\forall t \leq {T}^{ * } - 1. \n\]\n\n(2.62)\n\nBut, on the one hand, we ha... | Yes |
Proposition 2.6.2 Assume that the class \( \mathcal{P}\left( \mathcal{M}\right) \) is non-empty. Then the spot market \( \mathcal{M} \) is arbitrage-free. Moreover, the arbitrage price process of any attainable contingent claim \( X \), which settles at time \( T \), is given by the risk-neutral valuation formula\n\n\[... | Proof. Let \( {\mathbb{P}}^{ * } \) be some martingale measure for \( \mathcal{M} \) . We know already that the relative wealth process \( {V}^{ * }\left( \phi \right) \) of any strategy \( \phi \in \Phi \) follows a \( {\mathbb{P}}^{ * } \) -martingale, and thus\n\n\[ \n{V}_{t}\left( \phi \right) = {B}_{t}{V}_{t}^{ * ... | Yes |
Theorem 2.6.2 Assume that a spot market model \( \mathcal{M} \) is arbitrage-free so that the class \( \mathcal{P}\left( \mathcal{M}\right) \) is non-empty. Then \( \mathcal{M} \) is complete if and only if the uniqueness of a martingale measure for \( \mathcal{M} \) holds. | Proof. \( \Rightarrow \) ) Assume that \( \mathcal{M} \) is an arbitrage-free complete market. Then for every \( {\mathcal{F}}_{{T}^{ * }} \) -measurable random variable \( X \) there exists at least one trading strategy \( \phi \in \Phi \) such that \( {V}_{{T}^{ * }}\left( \phi \right) = X \) . By virtue of Propositi... | Yes |
Corollary 2.6.2 A contingent claim \( X \in \mathcal{X} \) is attainable if and only if the map \( {\mathbb{P}}^{ * } \mapsto {\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {X{B}_{T}^{-1}}\right) \) from \( \mathcal{P}\left( \mathcal{M}\right) \) to \( \mathbb{R} \) is constant. | Proof. We consider the case of \( T = {T}^{ * } \) ; the arguments for any \( T \leq {T}^{ * } \) are analogous. Suppose first that \( X \) is attainable. Then its arbitrage price at time 0 is well defined and it is unique. The risk-neutral valuation formula thus shows that the map \( {\mathbb{P}}^{ * } \mapsto {\mathb... | Yes |
Corollary 2.6.3 Let \( W \) be a linear subspace of \( {\mathbb{R}}^{d} \) and let \( K \) be a compact, convex subset of \( {\mathbb{R}}^{d} \) such that \( K \cap W = \varnothing \) . Then there exists a linear functional \( y \in {\mathbb{R}}^{d} \) and a real number \( \alpha > 0 \) such that \( y \cdot w = 0 \) fo... | Proof. Let \( U \) stand for the following set\n\n\[ U = K - W = \left\{ {x \in {\mathbb{R}}^{d} \mid x = k - w, k \in K, w \in W}\right\} .\n\]\n\nIt is easy to check that \( U \) is a convex, closed set. Since \( K \) and \( W \) are disjoint, we manifestly have that \( 0 \notin U \) . From Theorem 2.6.3, we deduce t... | Yes |
Proposition 2.6.4 The Radon-Nikodým derivative process \( \eta \) of \( \overline{\mathbb{P}} \) with respect to \( {\mathbb{P}}^{ * } \) equals, for every \( t = 0,1,\ldots ,{T}^{ * } \) , | Proof. It is clear that \( {\mathbb{P}}^{ * } \) and \( \overline{\mathbb{P}} \) are equivalent on \( \left( {\Omega ,{\mathcal{F}}_{{T}^{ * }}}\right) \) and thus the Radon-Nikodým derivative \( {\eta }_{{T}^{ * }} = d\overline{\mathbb{P}}/d{\mathbb{P}}^{ * } \) exists. Note that \( {S}^{l}/{S}^{k} \) is a martingale ... | No |
Lemma 2.7.1 A futures trading strategy \( \phi \) is self-financing if and only if we have that, for every \( t = 0,1,\ldots ,{T}^{ * } \) ,\n\n\[ \n{V}_{t}^{f}\left( \phi \right) = {V}_{0}^{f}\left( \phi \right) + {G}_{t}^{f}\left( \phi \right) \n\] | Proof. Taking into account (2.73)-(2.74), for any self-financing futures strategy \( \phi \) we obtain\n\n\[ \n{V}_{t}^{f}\left( \phi \right) = {V}_{0}^{f}\left( \phi \right) + \mathop{\sum }\limits_{{u = 0}}^{{t - 1}}\left( {{V}_{u + 1}^{f} - {V}_{u}^{f}}\right) \n\]\n\n\[ \n= {V}_{0}^{f}\left( \phi \right) + \mathop{... | Yes |
Lemma 2.7.2 A futures trading strategy \( \phi \) is self-financing if and only if the relative wealth process \( {\widetilde{V}}^{f}\left( \phi \right) \) admits the following representation, for every \( t = \) \( 0,1,\ldots ,{T}^{ * } \) , \[ {\widetilde{V}}_{t}^{f}\left( \phi \right) = {\widetilde{V}}_{0}^{f}\left(... | Proof. Let us denote \( {V}^{f} = {V}^{f}\left( \phi \right) \) and \( {\widetilde{V}}^{f} = {\widetilde{V}}^{f}\left( \phi \right) \) . For the first statement, it is sufficient to check that, for every \( t = 0,1,\ldots ,{T}^{ * } - 1 \) , \[ {\widetilde{V}}_{t + 1}^{f} - {\widetilde{V}}_{t}^{f} = {B}_{t + 1}^{-1}{\p... | Yes |
Proposition 2.7.2 Assume that the class \( \mathcal{P}\left( {\mathcal{M}}^{f}\right) \) of futures martingale measures is non-empty. Then the futures market model \( {\mathcal{M}}^{f} \) is arbitrage-free. Moreover, the arbitrage price in \( {\mathcal{M}}^{f} \) of any attainable contingent claim \( X \) which settles... | Proof. It is easy to check the absence of arbitrage opportunities in \( {\mathcal{M}}^{f} \) . Also, equality (2.76) is a straightforward consequence of Lemma 2.7.2. Indeed, it follows immediately from the martingale property of the discounted wealth of a strategy that replicates \( X \) . | No |
Theorem 2.7.1 The following statements are true.\n\n(i) A finite futures market \( {\mathcal{M}}^{f} \) is arbitrage-free if and only if the class \( \mathcal{P}\left( {\mathcal{M}}^{f}\right) \) of martingale measures is non-empty.\n\n(ii) An arbitrage-free futures market \( {\mathcal{M}}^{f} \) is complete if and onl... | Proof. Both statements can be proved by means of the same arguments as those used in the case of a spot market. | No |
Proposition 2.7.3 The forward price at time \( t \leq T \) of a stock \( S \) for the settlement date \( T \) equals\n\n\[ \n{F}_{S}\left( {t, T}\right) = \frac{{S}_{t}}{B\left( {t, T}\right) },\;\forall t \leq T.\n\] | Proof. In view of (2.63), we obtain\n\n\[ \n{\pi }_{t}\left( X\right) = {B}_{t}{\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {{B}_{T}^{-1}\left( {{S}_{T} - {F}_{S}\left( {t, T}\right) }\right) \mid {\mathcal{F}}_{t}}\right) = 0.\n\]\nOn the other hand, since by assumption \( {F}_{S}\left( {t, T}\right) \) is \( {\mathcal{F}}... | Yes |
Proposition 2.7.4 Let the bond price \( B\left( {t, T}\right) \) follow a predictable process. The combined spot-futures market is arbitrage-free if and only if the futures and forward prices agree; that is, \( {f}_{S}\left( {t, T}\right) = {F}_{S}\left( {t, T}\right) \) for every \( t \leq T \) . | Proof. We first aim show that the asserted equality is necessary for the absence of arbitrage in the combined spot-futures market. As mentioned above, the \( T \) -maturity bond is considered as the basic spot asset; in particular, all proceeds from futures contracts are immediately reinvested in this bond. Let us cons... | No |
Lemma 2.8.2 The process \( Y \) given by formula (2.80) is the Snell envelope of the payoff process \( Z \) . | Proof. First, we note that \( Y \) is an \( \mathbb{F} \) -adapted process and that (2.80) implies immediately that \( {Y}_{t - 1} \geq {\mathbb{E}}_{\mathbb{P}}\left( {{Y}_{t} \mid {\mathcal{F}}_{t - 1}}\right) \) for \( t = 1,2,\ldots, n \) . This shows that \( Y \) is an \( \mathbb{F} \) -supermartingale. Formula (2... | Yes |
Lemma 2.8.3 The stopped process \( {Y}^{{\tau }_{0}^{ * }} \) is an \( \mathbb{F} \) -martingale. | Proof. Let us denote \( M = {Y}^{{\tau }_{0}^{ * }} \) . Then we have, for any \( t = 0,1,\ldots, n - 1 \) ,\n\n\[ \n{\mathbb{E}}_{\mathbb{P}}\left( {{M}_{t + 1} \mid {\mathcal{F}}_{t}}\right) = {\mathbb{E}}_{\mathbb{P}}\left( {{\mathbb{1}}_{\left\{ {\tau }_{0}^{ * } > t\right\} }{M}_{t + 1} + {\mathbb{1}}_{\left\{ {\t... | Yes |
Lemma 2.8.4 The stopping time \( {\widehat{\tau }}_{0} \) is a solution to the optimal stopping problem (A). | Proof. In view of (2.83), the inequality \( {A}_{t + 1} - {A}_{t} > 0 \) holds if and only if \( {Y}_{t} > \) \( {\mathbb{E}}_{\mathbb{P}}\left( {{Y}_{t + 1} \mid {\mathcal{F}}_{t}}\right) \), which in turn implies that \( {Y}_{t} = {Z}_{t} \) . Hence the equality \( {Y}_{{\widehat{\tau }}_{0}} = {Z}_{{\widehat{\tau }}... | Yes |
Lemma 2.8.5 (i) The stopping time \( {\tau }_{0}^{ * } \) is the minimal solution to problem (A), specifically, if \( \tau \) is any stopping time such that \( \mathbb{P}\left\{ {\tau < {\tau }_{0}^{ * }}\right\} > 0 \) then \( {Y}_{0} > {\mathbb{E}}_{\mathbb{P}}\left( {Z}_{\tau }\right) \) . | Proof. For part (i), we note that on the event \( \left\{ {\tau < {\tau }_{0}^{ * }}\right\} \) we have \( {Y}_{\tau } > {Z}_{\tau } \) (by the definition of \( {\tau }_{0}^{ * } \) ). It is thus easy to check that the equalities\n\n\[ \n{Y}_{0} = {\mathbb{E}}_{\mathbb{P}}\left( {Z}_{{\tau }_{0}^{ * }}\right) = {\mathb... | Yes |
Corollary 2.8.1 (i) A stopping time \( \tau \in {\mathcal{T}}_{\left\lbrack 0, n\right\rbrack } \) belongs to \( {\mathcal{T}}_{\left\lbrack 0, n\right\rbrack }^{ * } \) whenever \( {\tau }_{0}^{ * } \leq \tau \leq \) \( {\widehat{\tau }}_{0} \) and \( {Y}_{\tau } = {Z}_{\tau } \) . | Proof. For part (i), it suffices to note that the stopped process \( {Y}^{\tau } \) is a martingale and thus\n\n\[ \n{Y}_{0} = {\mathbb{E}}_{\mathbb{P}}\left( {Y}_{\tau }\right) = {\mathbb{E}}_{\mathbb{P}}\left( {Z}_{\tau }\right) \n\] \n\nwhere the last equality holds if and only if \( {Y}_{\tau } = {Z}_{\tau } \) (re... | No |
Corollary 2.8.2 For a fixed \( t = 0,1,\ldots, n \), a stopping time \( \tau \) belongs to \( {\mathcal{T}}_{\left\lbrack t, n\right\rbrack }^{ * } \) whenever \( {\tau }_{t}^{ * } \leq \tau \leq {\widehat{\tau }}_{t} \) and \( {Y}_{\tau } = {Z}_{\tau } \) . | For any stopping time \( \tau \in {\mathcal{T}}_{\left\lbrack t, n\right\rbrack } \) that does not belong to \( {\mathcal{T}}_{\left\lbrack t, n\right\rbrack }^{ * } \) the optimality of \( {\tau }_{t}^{ * } \) and \( {\widehat{\tau }}_{t} \) implies immediately that\n\n\[ \mathbb{P}\left\{ {{\mathbb{E}}_{\mathbb{P}}\l... | "Yes" |
Proposition 2.8.4 Assume that an American claim \( {X}^{a} \) was sold at time 0 at the initial price \( {\pi }_{0}^{s}\left( {X}^{a}\right) \) . Then there exists an arbitrage opportunity for the seller of the claim if and only if the holder does not exercise the claim at some rational exercise time. | Proof. Suppose first that the claim is exercised by its holder at some rational exercise time \( \tau \) . Then the wealth process \( V\left( {\phi }^{ * }\right) = U \) of the optimal superhedging strategy \( {\phi }^{ * } \) for the seller matches exactly the payoff process \( X \) at time \( \tau \) .\n\nIf, on the ... | Yes |
Lemma 2.8.6 Suppose that a trading strategy \( \phi \in \Phi \) is such that \( {V}_{0}\left( \phi \right) = - {\pi }_{0}^{s}\left( {X}^{a}\right) \) and for some stopping time \( \tau \in {\mathcal{T}}_{\left\lbrack 0, T\right\rbrack } \) we have \( {V}_{\tau }\left( \phi \right) + {X}_{\tau } \geq 0 \) . Then \( {V}_... | Proof. Assume, on the contrary, that the inequality \( \mathbb{P}\left\{ {{V}_{\tau }\left( \phi \right) + {X}_{\tau } > 0}\right\} > 0 \) is valid or, equivalently, \( {\mathbb{P}}^{ * }\left\{ {{V}_{\tau }^{ * }\left( \phi \right) + {X}_{\tau }^{ * } > 0}\right\} > 0 \) . Then the martingale property of the relative ... | Yes |
Lemma 2.9.1 Let \( {L}^{0}\left( {\Omega ,\mathcal{F},\mathbb{P}}\right) \) stand for the class of all random variables defined on a probability space \( \left( {\Omega ,\mathcal{F},\mathbb{P}}\right) \) . Let \( A \) and \( B \) be two finite sets and let \( g : A \times B \rightarrow \) \( {L}^{0}\left( {\Omega ,\mat... | Proof. It is clear that, for any \( {a}_{0} \in A \) and \( {b}_{0} \in B \) ,\n\n\[ G\left( {a}_{0}\right) \mathrel{\text{:=}} \mathop{\max }\limits_{{b \in B}}g\left( {{a}_{0}, b}\right) \geq g\left( {{a}_{0},{b}_{0}}\right) \geq \mathop{\min }\limits_{{a \in A}}g\left( {a,{b}_{0}}\right) = : H\left( {b}_{0}\right) ,... | Yes |
Lemma 2.9.2 Assume that the Nash equilibrium holds. Then the Stackelberg equilibrium holds and\n\n\[ \n{\bar{Y}}_{t} = {\mathbb{E}}_{\mathbb{P}}\left( {Z\left( {{\sigma }_{t}^{ * },{\tau }_{t}^{ * }}\right) \mid {\mathcal{F}}_{t}}\right) \n\]\n\nso that \( {\sigma }_{t}^{ * } \) and \( {\tau }_{t}^{ * } \) are optimal ... | Proof. From (2.90), we obtain\n\n\[ \n\mathop{\max }\limits_{{\tau \in {\mathcal{T}}_{\left\lbrack t, n\right\rbrack }}}{\mathbb{E}}_{\mathbb{P}}\left( {Z\left( {{\sigma }_{t}^{ * },\tau }\right) \mid {\mathcal{F}}_{t}}\right) \leq {\mathbb{E}}_{\mathbb{P}}\left( {Z\left( {{\sigma }_{t}^{ * },{\tau }_{t}^{ * }}\right) ... | Yes |
Lemma 2.9.3 Let us set, for any fixed \( t = 0,1,\ldots, n \) , \[ {\sigma }_{t}^{ * } = \min \left\{ {u \in \{ t, t + 1,\ldots, n\} \mid {Y}_{u} = {H}_{u}}\right\} \land n \] and \[ {\tau }_{t}^{ * } = \min \left\{ {u \in \{ t, t + 1,\ldots, n\} \mid {Y}_{u} = {L}_{u}}\right\} . \] Then the following representations a... | Proof. We have \[ {\sigma }_{t - 1}^{ * } = \min \left\{ {u \in \{ t - 1, t,\ldots, n\} \mid {Y}_{u} = {H}_{u}}\right\} \land n \] \[ = \left( {t - 1}\right) {\mathbb{1}}_{\left\{ {Y}_{t - 1} = {H}_{t - 1}\right\} } + \left( {\min \left\{ {u \geq t \mid {Y}_{u} = {H}_{u}}\right\} \land n}\right) {\mathbb{1}}_{\left\{ {... | Yes |
Lemma 2.9.4 For every \( t = 0,1,\ldots, n - 1 \), we have that \( {Y}_{t} \leq {\mathbb{E}}_{\mathbb{P}}\left( {{Y}_{t + 1} \mid {\mathcal{F}}_{t}}\right) \) on the event \( \left\{ {{\tau }_{t}^{ * } > t}\right\} \) and \( {Y}_{t} \geq {\mathbb{E}}_{\mathbb{P}}\left( {{Y}_{t + 1} \mid {\mathcal{F}}_{t}}\right) \) on ... | Proof. Let us fix \( t \) and let us write \( a = {L}_{t}, b = {H}_{t} \) and \( c = {\mathbb{E}}_{\mathbb{P}}\left( {{Y}_{t + 1} \mid {\mathcal{F}}_{t}}\right) \) . We wish to examine the relationship between the value of \( c \) and \( h\left( c\right) = \min \left( {b,\max \left( {a, c}\right) }\right) \) for arbitr... | Yes |
Theorem 2.9.1 (i) Let the stopping times \( {\sigma }_{t}^{ * },{\tau }_{t}^{ * } \) be given by (2.92)-(2.93). Then we have, for arbitrary stopping times \( \tau ,\sigma \in {\mathcal{T}}_{\left\lbrack t, n\right\rbrack } \), \[ {\mathbb{E}}_{\mathbb{P}}\left( {Z\left( {{\sigma }_{t}^{ * },\tau }\right) \mid {\mathcal... | Proof. We will first show that the inequality holds for \( t = n \). Of course, if \( \tau ,\sigma \in \) \( {\mathcal{T}}_{\left\lbrack t, n\right\rbrack } \) then \( \tau = \sigma = n \). As \[ {\sigma }_{n}^{ * } = \min \left\{ {u \geq n \mid {Y}_{n} = {H}_{n}}\right\} \land n = n = \min \left\{ {u \geq n \mid {Y}_{... | Yes |
Proposition 2.9.1 For any stopping time \( \sigma \in {\mathcal{T}}_{\left\lbrack t, T\right\rbrack } \), there exists a minimal seller’s super-hedging strategy for the American contingent claim \( {X}^{a,\sigma } \) with the payoff \( {X}_{t}^{\sigma } = \) \( Z\left( {\sigma, t}\right) \) for \( t = 0,1,\ldots, T \) ... | \[ {\pi }_{t}^{s}\left( {X}^{a,\sigma }\right) = \mathop{\max }\limits_{{\tau \in {\mathcal{T}}_{\left\lbrack t, T\right\rbrack }}}{B}_{t}{\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {{Z}^{ * }\left( {\sigma ,\tau }\right) \mid {\mathcal{F}}_{t}}\right) . \] | Yes |
Theorem 2.9.2 The seller’s price \( {\pi }_{t}^{s}\left( {X}^{g}\right) \) at time \( t \) of a game contingent claim is equal to the seller’s price \( {\pi }^{s}\left( {X}^{a,{\sigma }_{t}^{ * }}\right) \) of an American claim \( {X}^{a,{\sigma }_{t}^{ * }} \) with the payoff process \( {X}^{{\sigma }_{t}^{ * }} \) gi... | Proof. Let us fix some date \( t = 0,1,\ldots, T \) . The seller of a game contingent claim is set to choose her cancelation time \( \sigma \in {\mathcal{T}}_{\left\lbrack t, T\right\rbrack } \) as well as a trading strategy \( \phi \) . We will first show that for any seller’s super-hedging strategy \( \left( {\sigma ... | Yes |
Proposition 2.9.2 The buyer’s price \( {\pi }^{b}\left( {X}^{g}\right) \) of a game contingent claim \( {X}^{g} \) is equal to the seller’s price \( {\pi }^{s}\left( {X}^{g}\right) \), that is, \( {\pi }_{t}^{b}\left( {X}^{g}\right) = {U}_{t} \) for every \( t = 0,1,\ldots, T \) . Hence the arbitrage price \( \pi \left... | Proof. Since the super-hedging problems for the seller and the buyer of a game contingent claim are essentially symmetric, the equality \( {\pi }^{b}\left( {X}^{g}\right) = U \) can be established using the same arguments as those employed in the proof of Theorem 2.9.2. We thus obtain the asserted equality \( {\pi }_{t... | No |
Lemma 3.1.1 The process \( M \) is a martingale under \( \mathbb{P} \) with respect to the filtration \( \mathbb{F} \) . | Proof. In order to show that \( M \) is a martingale under \( \mathbb{P} \) with respect to \( \mathbb{F} \), it suffices to note that\n\n\[ \n{\mathbb{E}}_{\mathbb{P}}\left( {{e}^{\sigma \left( {{W}_{t} - {W}_{u}}\right) } \mid {\mathcal{F}}_{u}}\right) = {\mathbb{E}}_{\mathbb{P}}\left( {e}^{\sigma \left( {{W}_{t} - {... | Yes |
Example 3.1.1 The following example of a suicide strategy is borrowed from Harrison and Pliska (1981). It can be modified easily to provide an example of an arbitrage opportunity in an unconstrained Black-Scholes setting. For simplicity, we take \( r = 0, T = 1 \), and \( {S}_{0} = 1 \) . For a strictly positive consta... | To show this, we shall modify the strategy as follows. On the interval \( \left\lbrack {0,1/2}\right\rbrack \) , we follow the strategy above with \( b = 1 \) . The probability of bankruptcy during \( \left\lbrack {0,1/2}\right\rbrack \) thus equals \( p = \mathbb{P}\{ \tau \left( 1\right) \leq 1/2\} \) . If \( \tau \l... | Yes |
Lemma 3.1.2 A probability measure is a spot martingale measure if and only if it is a martingale measure for the discounted stock price \( {S}^{ * } \) . | Proof. Let \( \phi \) be a self-financing strategy and let \( {V}^{ * } = {V}^{ * }\left( \phi \right) \) . Using Itô’s integration by parts formula, we get (note that \( d{B}_{t}^{-1} = - r{B}_{t}^{-1}{dt} \) )\n\n\[ d{V}_{t}^{ * } = d\left( {{V}_{t}{B}_{t}^{-1}}\right) = {V}_{t}d{B}_{t}^{-1} + {B}_{t}^{-1}d{V}_{t} \]... | Yes |
Lemma 3.1.3 (i) The unique martingale measure \( \mathbb{Q} \) for the discounted stock price process \( {S}^{ * } \) is given by the Radon-Nikodým derivative\n\n\[ \frac{d\mathbb{Q}}{d\mathbb{P}} = \exp \left( {\frac{r - \mu }{\sigma }{W}_{{T}^{ * }} - \frac{1}{2}\frac{{\left( r - \mu \right) }^{2}}{{\sigma }^{2}}{T}^... | Proof. Let us first prove part (i). Recall that we assumed that the filtration \( \mathbb{F} \) is the \( \mathbb{P} \) -augmentation of the natural filtration of \( W \), that is, \( \mathbb{F} = {\mathbb{F}}^{W} \) (see Sect. A.2). We thus know from Proposition A.15.3 that for any probability measure \( \mathbb{Q} \)... | Yes |
Corollary 3.1.3 Let \( X \) be an attainable European contingent claim that settles at time \( T \) . Then the arbitrage price \( {\pi }_{t}\left( X\right) \) at time \( t \in \left\lbrack {0, T}\right\rbrack \) in \( {\mathcal{M}}_{BS} \) is given by the risk-neutral valuation formula\n\n\[ \n{\pi }_{t}\left( X\right)... | Proof. Let \( \phi \) be an admissible strategy replicating \( X \) . We have\n\n\[ \n\frac{{\pi }_{t}\left( X\right) }{{B}_{t}} = {V}_{t}^{ * }\left( \phi \right) = {\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {{V}_{T}^{ * }\left( \phi \right) \mid {\mathcal{F}}_{t}}\right) = {\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {{B}_{T... | Yes |
Theorem 3.1.1 The arbitrage price at time \( t < T \) of the European call option with expiry date \( T \) and strike price \( K \) in the Black-Scholes model is given by the formula\n\n\[ \n{C}_{t} = c\left( {{S}_{t}, T - t}\right) ,\;\forall t \in \lbrack 0, T), \n\]\n\nwhere the function \( c : {\mathbb{R}}_{ + } \t... | Proof. We provide two alternative proofs of the Black-Scholes result. The first relies on the direct determination of the replicating strategy. It thus gives not only the valuation formula (this, however, requires solving the Black-Scholes PDE (3.26)), but also explicit formulas for the replicating strategy.\n\nThe sec... | Yes |
Proposition 3.1.2 The arbitrage prices of European call and put options with the same expiry date \( T \) and strike price \( K \) satisfy the put-call parity relationship\n\n\[ \n{C}_{t} - {P}_{t} = {S}_{t} - K{e}^{-r\left( {T - t}\right) }\n\]\n\n(3.43)\n\nfor every \( t \in \left\lbrack {0, T}\right\rbrack \) . | Proof. It is sufficient to observe that the payoffs of the call and put options at expiry satisfy the equality\n\n\[ \n{\left( {S}_{T} - K\right) }^{ + } - {\left( K - {S}_{T}\right) }^{ + } = {S}_{T} - K.\n\]\n\nRelationship (3.43) now follows from the risk-neutral valuation formula. Alternatively, one may derive (3.4... | Yes |
Corollary 3.1.4 The Black-Scholes price at time \( t < T \) of a European put option with strike price \( K \) equals \( {P}_{t} = p\left( {{S}_{t}, T - t}\right) \), where the function \( p : {\mathbb{R}}_{ + } \times \left\lbrack {0, T}\right\rbrack \rightarrow \) \( \mathbb{R} \) is given by (3.44). More explicitly,... | \[ {P}_{t} = {KB}\left( {t, T}\right) N\left( {-{d}_{2}\left( {{S}_{t}, T - t}\right) }\right) - {S}_{t}N\left( {-{d}_{1}\left( {{S}_{t}, T - t}\right) }\right) . \] | Yes |
Lemma 3.1.4 Let \( W \) be a one dimensional Brownian motion defined on a probability space \( \left( {\Omega ,\mathcal{F},\mathbb{P}}\right) \). For a Borel-measurable function \( h : \mathbb{R} \rightarrow \mathbb{R} \), we define the function \( u : \mathbb{R} \times \left\lbrack {0, T}\right\rbrack \rightarrow \mat... | Proof. From the fundamental properties of the Brownian motion, it is clear that \( u \) is given by the expression\n\n\[ u\left( {x, t}\right) = \frac{1}{\sqrt{{2\pi }\left( {T - t}\right) }}{\int }_{-\infty }^{+\infty }{e}^{-r\left( {T - t}\right) }h\left( y\right) {e}^{-\frac{{\left( y - x\right) }^{2}}{2\left( {T - ... | No |
Consider a call option on a stock \( S \), with strike price \( \$ {30} \) and with 3 months to expiry. Suppose, in addition, that the current stock price equals \( \$ {31} \) , the stock price volatility is \( \sigma = {10}\% \) per annum, and the risk-free interest rate is \( r = 5\% \) per annum with continuous comp... | This means that to hedge a short position in the call option sold at the arbitrage price \( {C}_{0} = \$ {1.52} \), an investor needs to purchase at time 0 the number \( \delta = {0.82} \) shares of stock (this transaction requires an additional borrowing of 23.9 units of cash). The elasticity at time 0 of the call opt... | Yes |
Lemma 3.2.1 The unique spot martingale measure \( {\mathbb{P}}^{ * } \) for the Black-Scholes model in which the stock \( S \) continuously pays dividends at some fixed rate \( \kappa \) is given by formula (3.11) with \( \mu \) replaced by \( {\mu }_{\kappa } \) . For any self-financing trading strategy \( \phi \), th... | \[ d{V}_{t}^{ * }\left( \phi \right) = \sigma {\phi }_{t}^{1}{e}^{-{\kappa t}}{\widetilde{S}}_{t}^{ * }d{\widetilde{W}}_{t} = {\phi }_{t}^{1}{e}^{-{\kappa t}}d{\widetilde{S}}_{t}^{ * }, \] where in turn the dynamics of the process \( {\widetilde{S}}_{t}^{ * } = {e}^{\left( {\kappa - r}\right) t}{S}_{t} \) are \[ d{\wid... | Yes |
Proposition 3.2.1 The arbitrage price at time \( t < T \) of a call option on a stock which pays dividends at a constant rate \( \kappa \) during the option’s lifetime is given by the risk-neutral formula\n\n\[ \n{C}_{t}^{\kappa } = {B}_{t}{\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {{B}_{T}^{-1}{\left( {S}_{T} - K\right) ... | Proof. The first equality is obvious. For the second, note first that we may rewrite (3.61) as follows\n\n\[ \n{C}_{t}^{\kappa } = {e}^{-r\left( {T - t}\right) }{\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {{\left( {S}_{T} - K\right) }^{ + } \mid {\mathcal{F}}_{t}}\right) = {e}^{-{\kappa T}}{e}^{-r\left( {T - t}\right) }{\m... | Yes |
Consider a European call option with strike price \( K \) and expiry date \( T \), written on a stock \( S \) that pays deterministic dividends \( {\kappa }_{1},\ldots ,{\kappa }_{m} \) at times \( {T}_{1},\ldots ,{T}_{m} \). Assume that the stock price \( S \) satisfies (3.70)-(3.71). Then the arbitrage price at time ... | Proof. It is sufficient to consider the case of \( t = 0 \). An application of the risk-neutral valuation formula yields \[ {C}_{0}^{\kappa } = {e}^{-{rT}}{\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {\left( {S}_{T} - K\right) }^{ + }\right) = {\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {\left( {G}_{T}^{ * } - {e}^{-{rT}}\left(... | Yes |
Consider a European call option with strike price \( K \) and expiry date \( T \), written on a stock \( S \) paying deterministic dividends \( {\kappa }_{1},\ldots ,{\kappa }_{m} \) at times \( {T}_{1},\ldots ,{T}_{m} \) . Assume that the stock price \( S \) satisfies (3.70) and (3.72). Then the arbitrage price at tim... | Proof. Once again we consider the case of \( t = 0 \) . We need to find\n\n\[ \n{\widetilde{C}}_{0}^{\kappa } = {e}^{-{rT}}{\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {\left( {S}_{T} - K\right) }^{ + }\right) = {e}^{-{rT}}{\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {\left( {G}_{T} - K\right) }^{ + }\right) .\n\]\n\nBut now und... | Yes |
Theorem 3.3.1 Assume that \( r = 0 \) . Then the arbitrage price at time \( t < T \) of the European call option with expiry date \( T \) and strike price \( K \) in the Bachelier market is given by the formula\n\n\[ \n{C}_{t} = c\left( {{S}_{t}, T - t}\right) ,\;\forall t \in \lbrack 0, T), \n\]\n\nwhere the function ... | Proof. We shall only derive the arbitrage price, using the risk-neutral valuation formula. For \( t = 0 \), we obtain\n\n\[ \n{C}_{0} = {\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {\left( {S}_{T} - K\right) }^{ + }\right) = {\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {{S}_{T}{\mathbb{1}}_{D}}\right) - K{\mathbb{P}}^{ * }\{ D\}... | Yes |
Lemma 3.4.1 Let \( \widetilde{\mathbb{P}} \) be a probability measure on \( \left( {\Omega ,{\mathcal{F}}_{T}}\right) \) equivalent to \( \mathbb{P} \) . Then \( \widetilde{\mathbb{P}} \) is a futures martingale measure if and only if the futures price \( f \) is a local martingale under \( \widetilde{\mathbb{P}} \) . | Proof. The discounted wealth \( {\widetilde{V}}^{f} \) for any trading strategy \( \phi \in {\Phi }^{f} \) satisfies\n\n\[ d{\widetilde{V}}_{t}^{f}\left( \phi \right) = {B}_{t}^{-1}\left( {{\phi }_{t}^{1}d{f}_{t} + {\phi }_{t}^{2}d{B}_{t}}\right) - r{B}_{t}^{-1}{V}_{t}^{f}\left( \phi \right) {dt} = {\phi }_{t}^{1}{B}_{... | Yes |
The unique martingale measure \( \widetilde{\mathbb{P}} \) on \( \left( {\Omega ,{\mathcal{F}}_{T}}\right) \) for the process \( f \) is given by the Radon-Nikodým derivative | \[ \frac{d\widetilde{\mathbb{P}}}{d\mathbb{P}} = \exp \left( {-\frac{\mu }{\sigma }{W}_{T} - \frac{1}{2}\frac{{\mu }^{2}}{{\sigma }^{2}}T}\right) ,\;\mathbb{P}\text{-a.s. } \] The dynamics of the futures price \( f \) under \( \widetilde{\mathbb{P}} \) are \[ d{f}_{t} = \sigma {f}_{t}d{\widetilde{W}}_{t} \] (3.84) and ... | Yes |
Corollary 3.4.1 The arbitrage price in \( {\mathcal{M}}^{f} \) of any attainable contingent claim \( X = \) \( g\left( {f}_{T}\right) \) settling at time \( T \) is given by \( {\pi }_{t}^{f}\left( X\right) = v\left( {{f}_{t}, t}\right) \), where the function \( v : {\mathbb{R}}_{ + } \times \) \( \left\lbrack {0, T}\r... | \[ \frac{\partial v}{\partial t} + \frac{1}{2}{\sigma }^{2}{f}^{2}\frac{{\partial }^{2}v}{\partial {f}^{2}} - {rv} = 0,\;\forall \left( {f, t}\right) \in \left( {0,\infty }\right) \times \left( {0, T}\right) , \] subject to the terminal condition \( v\left( {f, T}\right) = g\left( f\right) \) | Yes |
Corollary 3.4.2 The following relationship, known as the put-call parity for futures options, holds for every \( t \in \left\lbrack {0, T}\right\rbrack \)\n\n\[ \n{C}_{t}^{f} - {P}_{t}^{f} = {c}^{f}\left( {{f}_{t}, T - t}\right) - {p}^{f}\left( {{f}_{t}, T - t}\right) = {e}^{-r\left( {T - t}\right) }\left( {{f}_{t} - K... | Consequently, the price of a futures put option equals\n\n\[ \n{P}_{t}^{f} = {e}^{-r\left( {T - t}\right) }\left( {{KN}\left( {-{\widetilde{d}}_{2}\left( {{f}_{t}, T - t}\right) }\right) - {f}_{t}N\left( {-{\widetilde{d}}_{1}\left( {{f}_{t}, T - t}\right) }\right) }\right) ,\n\]\n\nwhere \( {\widetilde{d}}_{1}\left( {f... | No |
To find the arbitrage price of the corresponding futures put option, we make use of the put-call parity relationship. | We find that \( {P}_{0}^{f} = {0.23} \) ; moreover, for the replicating portfolio of the put option we have \( {\phi }_{0}^{1} = - {0.25} \) and \( {\phi }_{0}^{2} = {0.23} \) . Since now \( {\phi }_{0}^{1} < 0 \), we deal here with the short hedge - a strategy typical for an investor who expects to sell a given asset ... | Yes |
Corollary 3.5.1 Assume that the short-term interest rate \( r \) is constant and a constant \( \sigma > 0 \) is such that the inequality \( {\bar{\sigma }}_{t} \geq \sigma \) holds for every \( t \in \left\lbrack {0, T}\right\rbrack \) . If the initial endowment satisfies \( {V}_{0}\left( \phi \right) \geq - c\left( {{... | \[ {\phi }_{t}^{1} = N\left( {{d}_{1}\left( {{S}_{t}, t}\right) }\right) = - {\bar{c}}_{s}\left( {{S}_{t}, t}\right) ,\;\forall t \in \left\lbrack {0, T}\right\rbrack ,\] we have \( {V}_{T}\left( \phi \right) \geq - {\left( {S}_{T} - K\right) }^{ + } \). | Yes |
Proposition 3.5.2 Let the short-term interest rate \( r\left( t\right) \) be deterministic. Assume that \( \widetilde{\sigma }\left( {{S}_{t}, t}\right) \geq {\bar{\sigma }}_{t} \) for every \( t \in \left\lbrack {0, T}\right\rbrack \), where the process \( S \) is given by (3.97). Consider a European claim \( X = h\le... | Proof. The proof is based on arguments similar to those of the proof of Proposition 3.5.1. We shall focus on the case of a convex payoff function \( h \) . First, Itô’s formula yields\n\n\[ d\widetilde{v}\left( {{S}_{t}, t}\right) = {\widetilde{v}}_{t}\left( {{S}_{t}, t}\right) {dt} + {\widetilde{v}}_{s}\left( {{S}_{t}... | Yes |
Corollary 3.5.2 Suppose that instead of dynamics (3.97), we postulate that \( S \) is governed under some probability \( \overline{\mathbb{P}} \) by the following SDE\n\n\[ d{S}_{t} = {S}_{t}\left( {r\left( t\right) {dt} + {\bar{\sigma }}_{t}d{\bar{W}}_{t}}\right) \]\n\n(3.101)\n\nwhere \( \bar{W} \) is a Brownian moti... | Proof of Corollary 3.5.2. It suffices to observe that Proposition 3.5.2 yields, for \( {V}_{0}\left( \phi \right) = {\widetilde{\pi }}_{0}\left( X\right) \)\n\n\[ {V}_{T}\left( \phi \right) {B}_{T}^{-1} = {\widetilde{\pi }}_{0}\left( X\right) + {\int }_{0}^{T}{\phi }_{u}^{1}d{S}_{u}^{ * } \geq {B}_{T}^{-1}h\left( {S}_{... | Yes |
Corollary 3.5.3 Assume that \( {\widetilde{S}}_{0} = {\bar{S}}_{0} \) and the inequality \( \widetilde{\sigma }\left( {s, t}\right) \geq \bar{\sigma }\left( {s, t}\right) \) holds for every \( \left( {s, t}\right) \) . Then for any path-independent European contingent claim \( X \) represented by a convex function \( h... | Proof. It suffices to make use of Corollary 3.5.2. | No |
Proposition 3.5.3 Assume that the short-term interest rate \( r \) is constant. Suppose that the inequality \( {\sigma }_{\min } \leq {\bar{\sigma }}_{u} \leq {\sigma }_{\min } \) holds for every \( t \in \left\lbrack {0, T}\right\rbrack \), and the initial endowment satisfies \( {V}_{0}\left( \phi \right) \geq \bar{v}... | Proof. The proof goes along similar lines to the proof of Proposition 3.5.1. In particular, we obtain the following equality\n\n\[ \n{V}_{T}^{ * }\left( \phi \right) - {V}_{0}\left( \phi \right) = {B}_{T}^{-1}h\left( {S}_{T}\right) - \bar{v}\left( {{S}_{0},0}\right) \n\]\n\n\[ \n+ \frac{1}{2}{\int }_{0}^{T}\left( {\mat... | Yes |
Proposition 4.1.1 The following formula is valid for any \( {\mathcal{F}}_{T} \) -measurable random variable \( X \) (provided that the conditional expectation is well-defined)\n\n\[ \n{\mathbb{E}}_{\widetilde{\mathbb{P}}}\left( {X \mid {\mathcal{F}}_{t}}\right) = {\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {\left. {X\exp ... | Proof. Applying the abstract Bayes formula, we obtain\n\n\[ \n{\mathbb{E}}_{\widetilde{\mathbb{P}}}\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 4.2.1 The forward exchange rate \( {F}_{Q}\left( {t, T}\right) \) at time \( t \) for the settlement date \( T \) is given by the following formula\n\n\[ \n{F}_{Q}\left( {t, T}\right) = {e}^{\left( {{r}_{d} - {r}_{f}}\right) \left( {T - t}\right) }{Q}_{t},\;\forall t \in \left\lbrack {0, T}\right\rbrack .\n... | Proof. It is easily seen that if (4.16) does not hold, risk-free profitable opportunities arise between the domestic and the foreign market. | No |
Our aim is now to show that it is possible to express \( {f}_{Sf}^{d}\left( {t, T}\right) \) in terms of the futures exchange rate \( {f}_{Q}\left( {t, T}\right) \) and the foreign market futures price \( {f}_{Sf}\left( {t, T}\right) \) . | Indeed, we have the following simple result. | No |
Lemma 4.4.1 The domestic futures price \( {f}_{Sf}^{d}\left( {t, T}\right) \) of the foreign market asset \( {S}^{f} \) for the settlement date \( T \) satisfies\n\n\[ \n{f}_{{S}^{f}}^{d}\left( {t, T}\right) = {f}_{Q}\left( {t, T}\right) {f}_{{S}^{f}}\left( {t, T}\right) ,\;\forall t \in \left\lbrack {0, T}\right\rbrac... | Proof. It is clear that (4.27) implies the terminal equality (4.26). Furthermore, using (4.24)-(4.25) and Itô's formula, we get\n\n\[ \nd{Z}_{t} = {Z}_{t}\left( {{\sigma }_{f} + {\sigma }_{Q}}\right) \cdot d{W}_{t}^{ * },\n\]\n\nwhere we write \( {Z}_{t} = {f}_{Q}\left( {t, T}\right) {f}_{{S}^{f}}\left( {t, T}\right) \... | No |
The terminal payoff from a foreign equity call struck in foreign currency equals\n\n\[ \n{C}_{T}^{1}\overset{\text{ def }}{ = }{Q}_{T}{\left( {S}_{T}^{f} - {K}^{f}\right) }^{ + }.\n\] | This means, in particular, that the terminal payoff is assumed to be converted into domestic currency at the spot exchange rate that prevails at the expiry date. By reasoning in much the same way as in the previous section, one can check that the arbitrage price of a European call option at time \( t \) equals\n\n\[ \n... | Yes |
Proposition 4.5.1 The arbitrage price at time \( t \) of a European quanto call option with expiry date \( T \) and strike price \( {K}^{f} \) equals (in units of domestic currency) | Proof. Using (4.31), we obtain\n\n\[ \n{C}_{t}^{3} = \bar{Q}{e}^{\left( {\delta - {r}_{d}}\right) \left( {T - t}\right) }{\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {{e}^{-\delta \left( {T - t}\right) }{\left( {S}_{T}^{f} - {K}^{f}\right) }^{ + } \mid {\mathcal{F}}_{t}}\right) .\n\]\n\nSince the dynamics of \( {S}^{f} \) u... | Yes |
Proposition 4.5.2 The arbitrage price, expressed in domestic currency, of a European equity-linked foreign exchange call option, with strike exchange rate \( K \) and expiry date \( T \), is given by the following formula\n\n\[ \n{C}_{t}^{4} = {S}_{t}^{f}\left( {{Q}_{t}N\left( {{w}_{1}\left( {{Q}_{t}, T - t}\right) }\r... | Proof. As usual, it is sufficient to consider the case of \( t = 0 \) . In view of (4.13), we have\n\n\[ \n{S}_{T}^{f} = {S}_{0}^{f}\exp \left( {{\sigma }_{{S}^{f}} \cdot {W}_{T}^{ * } - \frac{1}{2}{\left| {\sigma }_{{S}^{f}}\right| }^{2}T + \left( {{r}_{f} - {\sigma }_{Q} \cdot {\sigma }_{{S}^{f}}}\right) T}\right) . ... | Yes |
Proposition 5.1.1 Let \( V \) be an adapted process defined by formula (5.4) for some reward function \( g \) . Then there exists an admissible trading and consumption strategy \( \left( {\phi, A}\right) \) such that \( {V}_{t} = {V}_{t}\left( {\phi, A}\right) \) for every \( t \in \left\lbrack {0, T}\right\rbrack \) . | Proof. We shall give the outline of the proof (for technical details, we refer to Karatzas (1988) and Myneni (1992)). Let us introduce the Snell envelope \( J \) of the discounted reward process \( {Z}_{t}^{ * } = {e}^{-{rt}}g\left( {{S}_{t}, t}\right) \) . By definition, the process \( J \) is the smallest supermartin... | Yes |
Theorem 5.1.1 There is absence of arbitrage (in the sense of Definition 5.1.5) in the market model with trading in an American claim if and only if the price \( {\pi }_{0}\left( {X}^{a}\right) \) is given by the formula\n\n\[ \n{\pi }_{0}\left( {X}^{a}\right) = \mathop{\sup }\limits_{{\tau \in {\mathcal{T}}_{\left\lbra... | Proof. We shall follow Myneni (1992). Let us assume that the \ | No |
Corollary 5.2.1 The discounted reward \( {X}^{ * } \) (respectively, \( {Y}^{ * } \) ) of the American call option (respectively, put option) with constant strike price follows a submartingale under \( {\mathbb{P}}^{ * } \) if \( r \geq 0 \) (respectively, if \( r \leq 0 \) ). | 8 This property follows directly from Jensen's conditional inequality. | No |
Proposition 5.3.1 The Snell envelope \( {J}^{p} \) admits the following decomposition | \[ {J}_{t}^{p} = {\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {{e}^{-{rT}}{\left( {K}_{T} - {S}_{T}\right) }^{ + } \mid {\mathcal{F}}_{t}}\right) + {\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {{\int }_{t}^{T}{e}^{-{ru}}{\mathbb{1}}_{\left\{ {\tau }_{u} = u\right\} }\left( {r{K}_{u} - {k}_{u}}\right) {du} \mid {\mathcal{F}}_{t}}... | Yes |
Proposition 5.4.1 The American put value function \( {P}^{a}\left( {s, t}\right) \) is smooth on the continuation region \( \mathcal{C} \) with \( - 1 \leq {P}_{s}^{a}\left( {s, t}\right) \leq 0 \) for all \( \left( {s, t}\right) \in \mathcal{C} \). The optimal stopping boundary \( {b}^{ * } \) is a continuous and non-... | \[ \mathop{\lim }\limits_{{T - t \downarrow 0}}{b}^{ * }\left( {T - t}\right) = \mathop{\lim }\limits_{{t \uparrow T}}{c}^{ * }\left( t\right) = K. \] On \( \mathcal{C} \), the function \( {P}^{a} \) satisfies \( {P}_{t}^{a}\left( {s, t}\right) = \mathcal{L}{P}^{a}\left( {s, t}\right) \) ; that is, \[ {P}_{t}^{a}\left(... | Yes |
Theorem 5.4.2 Assume that \( v : {\mathbb{R}}_{ + } \times \left\lbrack {0, T}\right\rbrack \) is a continuous function such that the function \( g\left( {s, t}\right) = v\left( {{e}^{s}, t}\right) \) satisfies certain growth conditions. Suppose that for every \( \left( {s, t}\right) \in {\mathbb{R}}_{ + } \times \left... | As one might expect, a solution to the problem above is not known explicitly (for the proof of existence and uniqueness we refer to Bensoussan and Lions (1978)). Numerical methods of solving variational inequalities associated with American options were developed by Jaillet et al. (1990). | No |
Proposition 5.6.1 The arbitrage price \( {\widetilde{C}}_{t}^{a}\left( {T, K}\right) \) of an unprotected American call option with expiry date \( T > {T}_{D} \) and strike price \( K \), written on a stock paying a known dividend \( D \) at time \( {T}_{D} \), equals\n\n\[ \n{\widetilde{C}}_{t}^{a}\left( {T, K}\right)... | Proof. Note that the first term in (5.13) represents the price of an option written on a dividend-paying stock, hence it is not given by the standard Black-Scholes formula. On the other hand, on the ex-dividend date \( {T}_{D} \) we have\n\n\[ \n{\widetilde{C}}_{{T}_{D}}\left( {T, K}\right) = C\left( {{S}_{{T}_{D}} - {... | No |
Theorem 5.7.1 The value process \( \bar{Y} \) the Dynkin game associated with a game contingent claim satisfies \( {\bar{Y}}_{t} = v\left( {{S}_{t}, t}\right) \), where the function \( v\left( {s, t}\right) \) is a solution to the linear complementarity problem | \[ \left( \begin{matrix} {\mathcal{L}}_{t}v\left( {s, t}\right) = 0 \\ L\left( {s, t}\right) \leq v\left( {s, t}\right) \leq H\left( {s, t}\right) \end{matrix}\right) \vee \left( \begin{matrix} {\mathcal{L}}_{t}v\left( {s, t}\right) \geq 0 \\ v\left( {s, t}\right) = L\left( {s, t}\right) \end{matrix}\right) \vee \left(... | Yes |
Proposition 6.7.1 Assume that \( r > 0 \) . Then the price at time \( t < T \) of a European lookback call option equals\n\n\[ \n{\mathbf{{LC}}}_{t} = {sN}\left( \frac{\ln \left( {s/m}\right) + {r}_{1}\tau }{\sigma \sqrt{\tau }}\right) - m{e}^{-{r\tau }}N\left( \frac{\ln \left( {s/m}\right) + {r}_{2}\tau }{\sigma \sqrt... | Proof. Since the discounted stock price \( {S}^{ * } \) is a martingale under \( {\mathbb{P}}^{ * } \), we have\n\n\[ \n{I}_{1} = {e}^{-r\left( {T - t}\right) }{\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {{S}_{T} \mid {\mathcal{F}}_{t}}\right) = {e}^{rt}{\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {{S}_{T}^{ * } \mid {\mathcal{... | Yes |
Proposition 6.8.1 The price of an Asian call option admits the representation\n\n\[ \n{C}_{t}^{A} = \frac{4{e}^{-r\left( {T - t}\right) }{S}_{t}}{{\sigma }^{2}\left( {T - {T}_{0}}\right) }{C}^{v}\left( {h, q}\right) ,\n\]\n\nwhere\n\n\[ \nv = \frac{2r}{{\sigma }^{2}} - 1,\;h = \frac{{\sigma }^{2}}{4}\left( {T - t}\righ... | In order to apply the last result to price the option, one needs to find (at least numerically) the inverse Laplace transform of the function \( g\left( \lambda \right) \) . For this purpose, we write \( f\left( h\right) = {C}^{v}\left( {h, q}\right) \) and we introduce an auxiliary function \( \widetilde{f}\left( h\ri... | Yes |
Proposition 6.9.1 The approximate value \( {\widehat{C}}_{t}^{B} \) of the price \( {C}_{t}^{B} \) of a basket call option with strike price \( K \) and expiry date \( T \) equals\n\n\[ \n{\widehat{C}}_{t}^{B} = \left( {\mathop{\sum }\limits_{{j = 1}}^{k}{w}_{j}{S}_{t}^{j}}\right) \left( {{cN}\left( {{l}_{1}\left( {T -... | Suppose that \( k = 1 \) . In this case, \( {w}_{1} = 1 \), and the arithmetic average agrees with the geometric one. Consequently, \( c = 1 \) and (6.20) reduces to the standard Black-Scholes formula. | Yes |
Proposition 7.1.1 For any \( p > 0 \) and \( K > 0 \), for a call option we have\n\n\[ \n{C}_{0}\left( {T, K}\right) \leq \frac{{\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {S}_{T}^{p + 1}\right) }{\left( {p + 1}\right) {K}^{p}}{\left( \frac{p}{p + 1}\right) }^{p} \n\]\n\n(7.6)\n\nand the price of a put option satisfies\n\n... | Proof. It is not difficult to check that\n\n\[ \n{\left( s - K\right) }^{ + } \leq \frac{{s}^{p + 1}}{\left( {p + 1}\right) {K}^{p}}{\left( \frac{p}{p + 1}\right) }^{p}.\n\]\n\nIndeed, both sides have the same value for \( s = \left( {p + 1}\right) K/p \), and the right-hand side has positive second order derivative. S... | Yes |
Lemma 7.1.1 There exists a real number \( {x}^{ * } > 0 \) such that for every \( x > {x}^{ * } \) we have \( {\widehat{\sigma }}_{0}\left( x\right) < \sqrt{{2x}/T} \) . | Proof. Since the Black-Scholes price \( c\left( {{S}_{0}, T, K,\sigma }\right) \) is increasing in \( \sigma \), it suffices to check that \( c\left( {{S}_{0}, T,{S}_{0}{e}^{x},{\widehat{\sigma }}_{0}\left( x\right) }\right) < c\left( {{S}_{0}, T,{S}_{0}{e}^{x},\sqrt{{2x}/T}}\right) \) for \( x \) large enough. This in... | Yes |
Lemma 7.1.3 We have\n\n\[ \n{\mathbf{{FS}}}_{t} = {b}_{t}{\mathbb{E}}_{\widehat{\mathbb{P}}}\left( {{\left( Y - K\right) }^{ + } \mid {\mathcal{F}}_{t}}\right) \n\] | Proof. We have, for every \( t \in \left\lbrack {0,{T}_{0}}\right\rbrack \)\n\n\[ \n{\mathbf{{FS}}}_{t} = {B}_{t}{\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {{B}_{T}^{-1}{\left( {S}_{T} - K{S}_{{T}_{0}}\right) }^{ + } \mid {\mathcal{F}}_{t}}\right) = a{S}_{0}{B}_{t}{\mathbb{E}}_{{\mathbb{P}}^{ * }}\left( {{\eta }_{T}{\left... | Yes |
Proposition 7.1.4 The arbitrage price at time \( t \in \left\lbrack {0,{T}_{0}}\right\rbrack \) of a forward-start option equals\n\n\[ \n{\mathbf{{FS}}}_{t} = {S}_{t}{\mathbb{E}}_{\widehat{\mathbb{P}}}{\left( {e}^{{\int }_{{T}_{0}}^{T}{\sigma }_{t}d{\widehat{W}}_{t} - \frac{1}{2}{\int }_{{T}_{0}}^{T}{\sigma }_{t}^{2}{d... | Proof. The fact that \( \widehat{W} \) is a standard Brownian motion is a consequence of equality (7.11) and Girsanov's theorem. Indeed, (7.11) yields\n\n\[ \n{\eta }_{T} = \exp \left( {{\int }_{0}^{T}{\sigma }_{t}{\mathbb{1}}_{\left\lbrack 0,{T}_{0}\right\rbrack }\left( t\right) d{W}_{t}^{ * } - \frac{1}{2}{\int }_{0}... | Yes |
Lemma 7.3.1 For any bounded Borel measurable function \( g : \mathbb{R} \rightarrow \mathbb{R} \), we have\n\n\[ \n{\int }_{\mathbb{R}}g\left( s\right) {dF}\left( {s, T}\right) = {e}^{rT}{\int }_{\mathbb{R}}g\left( s\right) {c}_{KK}\left( {{ds}, T}\right) ,\n\]\n\nand thus for every Borel set \( A \) in \( \mathbb{R} \... | Proof of Lemma 7.3.1. Let us take an arbitrary twice continuously differentiable function \( h \) with compact support. As explained above we have, by virtue of the definition of the distributional derivative,\n\n\[ \n{\mu }^{c}\left( {h}^{\prime \prime }\right) = - {\mu }_{K}^{c}\left( {h}^{\prime }\right) = {\mu }_{K... | Yes |
Corollary 7.3.1 Suppose that the random variable \( {S}_{T} \) admits a continuous probability density function \( f\left( {\cdot, T}\right) \) . Then the function \( c\left( {\cdot, T}\right) \) is twice continuously differentiable with respect to \( K \) and\n\n\[ \n{c}_{KK}\left( {s, T}\right) = {e}^{-{rT}}f\left( {... | By virtue of Lemma 7.3.1, we conclude that, under mild technical conditions, all one-dimensional marginal distributions of the price process \( S \) are uniquely determined by the prices \( {C}_{0}\left( {T, K}\right) \) of European call options (or equivalently, by the knowledge of the Black-Scholes implied volatility... | No |
Corollary 7.3.2 Let the call prices \( {C}_{0}\left( {T, K}\right) = c\left( {K, T}\right) \) be given by (7.25) for some martingale \( M \) . We assume that the function \( c\left( {K, T}\right) \) is once differentiable with respect to \( T \) and twice differentiable with respect to \( K \) . Suppose, in addition, t... | \[ {\sigma }^{2}\left( {s, t}\right) = \frac{2\left( {{c}_{T}\left( {s, t}\right) + {rs}{c}_{K}\left( {s, t}\right) }\right) }{{s}^{2}{c}_{KK}\left( {s, t}\right) } \] (7.30) provided that the function \( \sigma \left( {s, t}\right) \), implicitly defined by the last equation, is sufficiently regular to guarantee the e... | Yes |
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