scholarly journals What is the time value of a stream of investments?

2005 ◽  
Vol 42 (03) ◽  
pp. 861-866 ◽  
Author(s):  
Ragnar Norberg ◽  
Mogens Steffensen
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Discrete Time ◽  
Continuous Time ◽  
Stochastic Integral ◽  
Asset Price ◽  
Expected Value ◽  
Integral Expression ◽  
Time Value ◽  
Price Process

The titular question is investigated for fairly general semimartingale investment and asset price processes. A discrete-time consideration suggests a stochastic differential equation and an integral expression for the time value in the continuous-time framework. It is shown that the two are equivalent if the jump part of the price process converges. The integral expression, which is the answer to the titular question, is the sum of all investments accumulated with returns on the asset (a stochastic integral) plus a term that accounts for the possible covariation between the two processes. The arbitrage-free price of the time value is the expected value of the sum (i.e. integral) of all investments discounted with the locally risk-free asset.

scholarly journals What is the time value of a stream of investments?

2005 ◽  
Vol 42 (3) ◽  
pp. 861-866 ◽  
Author(s):  
Ragnar Norberg ◽  
Mogens Steffensen
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Discrete Time ◽  
Continuous Time ◽  
Stochastic Integral ◽  
Asset Price ◽  
Expected Value ◽  
Integral Expression ◽  
Time Value ◽  
Price Process

The titular question is investigated for fairly general semimartingale investment and asset price processes. A discrete-time consideration suggests a stochastic differential equation and an integral expression for the time value in the continuous-time framework. It is shown that the two are equivalent if the jump part of the price process converges. The integral expression, which is the answer to the titular question, is the sum of all investments accumulated with returns on the asset (a stochastic integral) plus a term that accounts for the possible covariation between the two processes. The arbitrage-free price of the time value is the expected value of the sum (i.e. integral) of all investments discounted with the locally risk-free asset.

scholarly journals Homogenization for deterministic maps and multiplicative noise

2013 ◽  
Vol 469 (2156) ◽  
pp. 20130201 ◽  
Author(s):  
Georg A. Gottwald ◽  
Ian Melbourne
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Discrete Time ◽  
Continuous Time ◽  
Multiplicative Noise ◽  
Stochastic Integrals ◽  
Deterministic System ◽  
One Dimensional ◽  
Stable Lévy Process ◽  
Fast Flow

A recent paper of Melbourne & Stuart (2011 A note on diffusion limits of chaotic skew product flows. Nonlinearity 24 , 1361–1367 (doi:10.1088/0951-7715/24/4/018)) gives a rigorous proof of convergence of a fast–slow deterministic system to a stochastic differential equation with additive noise. In contrast to other approaches, the assumptions on the fast flow are very mild. In this paper, we extend this result from continuous time to discrete time. Moreover, we show how to deal with one-dimensional multiplicative noise. This raises the issue of how to interpret certain stochastic integrals; it is proved that the integrals are of Stratonovich type for continuous time and neither Stratonovich nor Itô for discrete time. We also provide a rigorous derivation of super-diffusive limits where the stochastic differential equation is driven by a stable Lévy process. In the case of one-dimensional multiplicative noise, the stochastic integrals are of Marcus type both in the discrete and continuous time contexts.

scholarly journals Portfolio problems based on jump-diffusion models

Filomat ◽  
2012 ◽  
Vol 26 (3) ◽  
pp. 573-583 ◽  
Author(s):  
Tiantian Liu ◽  
Jun Zhao ◽  
Peibiao Zhao
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Asset Price ◽  
Efficient Frontier ◽  
Jump Diffusion ◽  
Natural Generalization ◽  
Optimal Portfolio ◽  
Linear Quadratic ◽  
Price Process ◽  
Portfolio Problem

Zhou and Li [49] by virtue of stochastic linear-quadratic control theory studied the optimal portfolio problems with the asset price process satisfying a diffusion stochastic differential equation, and proposed the celebrated LQ framework and the efficient frontier for the given portfolio problem. In this paper, we consider the optimal portfolio problems based on the asset price process satisfying a jump-diffusion stochastic differential equation. Similarly, we also arrive at the efficient frontier of the optimal portfolio selection problem. The conclusions obtained here can be regarded as a natural generalization of the work by Zhou and Li [49].

scholarly journals Pricing Defaultable Securities under Actual Probability Measure

2014 ◽  
Vol 2 (4) ◽  
pp. 313-334
Author(s):  
Jianfen Feng ◽  
Dianfa Chen ◽  
Mei Yu
Keyword(s):  
Differential Equation ◽  
Probability Measure ◽  
Stochastic Differential Equation ◽  
Credit Risk ◽  
Discrete Time ◽  
Continuous Time ◽  
Backward Stochastic Differential Equation ◽  
New Approach ◽  
Defaultable Securities

AbstractIn this paper, a new approach is developed to estimate the value of defaultable securities under the actual probability measure. This model gives the price framework by means of the method of backward stochastic differential equation. Such a method solves some problems in most of existing literatures with respect to pricing the credit risk and relaxes certain market limitations. We provide the price of defaultable securities in discrete time and in continuous time respectively, which is favorable to practice to manage real credit risk for finance institutes.

The Hamiltonian Generating Quantum Stochastic Evolutions in the Limit from Repeated to Continuous Interactions

2015 ◽  
Vol 22 (04) ◽  
pp. 1550022
Author(s):  
Matteo Gregoratti
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Discrete Time ◽  
Continuous Time ◽  
Schrodinger Equation ◽  
Hamiltonian Operator ◽  
The Other ◽  
Stochastic Evolution ◽  
Quantum Stochastic Differential Equation ◽  
Proper Formulation

We consider a quantum stochastic evolution in continuous time defined by the quantum stochastic differential equation of Hudson and Parthasarathy. On one side, such an evolution can also be defined by a standard Schrödinger equation with a singular and unbounded Hamiltonian operator K. On the other side, such an evolution can also be obtained as a limit from Hamiltonian repeated interactions in discrete time. We study how the structure of the Hamiltonian K emerges in the limit from repeated to continuous interactions. We present results in the case of 1-dimensional multiplicity and system spaces, where calculations can be explicitly performed, and the proper formulation of the problem can be discussed.

scholarly journals Dynkin game under g-expectation in continuous time

2020 ◽  
Vol 9 (2) ◽  
pp. 459-470
Author(s):  
Helin Wu ◽  
Yong Ren ◽  
Feng Hu
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Continuous Time ◽  
Value Function ◽  
Backward Stochastic Differential Equation ◽  
Saddle Points ◽  
Value Functions ◽  
Dynkin Game ◽  
The Value Function

Abstract In this paper, we investigate some kind of Dynkin game under g-expectation induced by backward stochastic differential equation (short for BSDE). The lower and upper value functions $$\underline{V}_t=ess\sup \nolimits _{\tau \in {\mathcal {T}_t}} ess\inf \nolimits _{\sigma \in {\mathcal {T}_t}}\mathcal {E}^g_t[R(\tau ,\sigma )]$$ V ̲ t = e s s sup τ ∈ T t e s s inf σ ∈ T t E t g [ R ( τ , σ ) ] and $$\overline{V}_t=ess\inf \nolimits _{\sigma \in {\mathcal {T}_t}} ess\sup \nolimits _{\tau \in {\mathcal {T}_t}}\mathcal {E}^g_t[R(\tau ,\sigma )]$$ V ¯ t = e s s inf σ ∈ T t e s s sup τ ∈ T t E t g [ R ( τ , σ ) ] are defined, respectively. Under some suitable assumptions, a pair of saddle points is obtained and the value function of Dynkin game $$V(t)=\underline{V}_t=\overline{V}_t$$ V ( t ) = V ̲ t = V ¯ t follows. Furthermore, we also consider the constrained case of Dynkin game.

scholarly journals Hedging error estimate of the american put option problem in jump-diffusion processes

Filomat ◽  
2018 ◽  
Vol 32 (8) ◽  
pp. 2813-2824
Author(s):  
Sultan Hussain ◽  
Salman Zeb ◽  
Muhammad Saleem ◽  
Nasir Rehman
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Diffusion Processes ◽  
Asset Price ◽  
Independent Random Variables ◽  
American Put Option ◽  
Jump Diffusion Processes ◽  
Put Option ◽  
Discrete Time Hedging ◽  
American Put

We consider discrete time hedging error of the American put option in case of brusque fluctuations in the price of assets. Since continuous time hedging is not possible in practice so we consider discrete time hedging process. We show that if the proportions of jump sizes in the asset price are identically distributed independent random variables having finite moments then the value process of the discrete time hedging uniformly approximates the value process of the corresponding continuous-time hedging in the sense of L1 and L2-norms under the real world probability measure.

scholarly journals Superdiffusive limits for deterministic fast–slow dynamical systems

2020 ◽  
Vol 178 (3-4) ◽  
pp. 735-770
Author(s):  
Ilya Chevyrev ◽  
Peter K. Friz ◽  
Alexey Korepanov ◽  
Ian Melbourne
Keyword(s):  
Differential Equation ◽  
Dynamical Systems ◽  
Stochastic Differential Equation ◽  
Lévy Process ◽  
Stochastic Integral ◽  
Levy Process ◽  
Stable Lévy Process

Abstract We consider deterministic fast–slow dynamical systems on $$\mathbb {R}^m\times Y$$ R m × Y of the form $$\begin{aligned} {\left\{ \begin{array}{ll} x_{k+1}^{(n)} = x_k^{(n)} + n^{-1} a\big (x_k^{(n)}\big ) + n^{-1/\alpha } b\big (x_k^{(n)}\big ) v(y_k), \\ y_{k+1} = f(y_k), \end{array}\right. } \end{aligned}$$ x k + 1 ( n ) = x k ( n ) + n - 1 a ( x k ( n ) ) + n - 1 / α b ( x k ( n ) ) v ( y k ) , y k + 1 = f ( y k ) , where $$\alpha \in (1,2)$$ α ∈ ( 1 , 2 ) . Under certain assumptions we prove convergence of the m-dimensional process $$X_n(t)= x_{\lfloor nt \rfloor }^{(n)}$$ X n ( t ) = x ⌊ n t ⌋ ( n ) to the solution of the stochastic differential equation $$\begin{aligned} \mathrm {d} X = a(X)\mathrm {d} t + b(X) \diamond \mathrm {d} L_\alpha , \end{aligned}$$ d X = a ( X ) d t + b ( X ) ⋄ d L α , where $$L_\alpha $$ L α is an $$\alpha $$ α -stable Lévy process and $$\diamond $$ ⋄ indicates that the stochastic integral is in the Marcus sense. In addition, we show that our assumptions are satisfied for intermittent maps f of Pomeau–Manneville type.

Pricing Risky Options Simply

1998 ◽  
Vol 01 (01) ◽  
pp. 1-23 ◽  
Author(s):  
Erik Aurell ◽  
Sergei I. Simdyankin
Keyword(s):  
Markov Process ◽  
Discrete Time ◽  
Expected Value ◽  
Numerical Results ◽  
Simple Description ◽  
Price Process ◽  
Black Scholes ◽  
Time Price

This paper is a follow-up of (Aurell and Życzkowski, 1996) [2] and (Aurell et al. 1996) [1]. We show that the prescription of pricing option by minimizing risk can be solved in a way that is quite similar to the Black–Scholes' approach. For a given discrete-time price process we determine an auxillary process, generally a pseudo-probability taking both negative and positive values, such that the price of the option in our prescription is the expected value upon maturation with respect to the auxillary process. We present a conjecture due to G. Wolczyńska that this auxillary process is in fact a (pseudo)-Markov process which admits a very simple description. Numerical results are presented in favor of the conjecture.

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