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Register a new userThe fundamental theorem of asset pricing for self-financing portfolios
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Consider a financial market with nonnegative semimartingales which does not need to have a num{e}raire. We are interested in the absence of arbitrage in the sense that no self-financing portfolio gives rise to arbitrage opportunities, where we are allowed to add a savings account to the market. We will prove that in this sense the market is free of arbitrage if and only if there exists an equivalent local martingale deflator which is a multiplicative special semimartingale. In this case, the additional savings account relates to the finite variation part of the multiplicative decomposition of the deflator.
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A new framework for asset price dynamics is introduced in which the concept of noisy information about future cash flows is used to derive the price processes. In this framework an asset is defined by its cash-flow structure. Each cash flow is modelled by a random variable that can be expressed as a function of a collection of independent random variables called market factors. With each such X-factor we associate a market information process, the values of which are accessible to market agents. Each information process is a sum of two terms; one contains true information about the value of the market factor; the other represents noise. The noise term is modelled by an independent Brownian bridge. The market filtration is assumed to be that generated by the aggregate of the independent information processes. The price of an asset is given by the expectation of the discounted cash flows in the risk-neutral measure, conditional on the information provided by the market filtration. When the cash flows are the dividend payments associated with equities, an explicit model is obtained for the share-price, and the prices of options on dividend-paying assets are derived. Remarkably, the resulting formula for the price of a European call option is of the Black-Scholes-Merton type. The information-based framework also generates a natural explanation for the origin of stochastic volatility.
We propose an extension of the Cox-Ross-Rubinstein (CRR) model based on q-binomial (or Kemp) random walks, with application to default with logistic failure rates. This model allows us to consider time-dependent switching probabilities varying according to a trend parameter, and it includes tilt and stretch parameters that control increment sizes. Option pricing formulas are written using q-binomial coefficients, and we study the convergence of this model to a Black-Scholes type formula in continuous time. A convergence rate of order O(1/N) is obtained when the tilt and stretch parameters are set equal to one.
The main objective of this paper is to present an algorithm of pricing perpetual American put options with asset-dependent discounting. The value function of such an instrument can be described as begin{equation*} V^{omega}_{text{A}^{text{Put}}}(s) = sup_{tauinmathcal{T}} mathbb{E}_{s}[e^{-int_0^tau omega(S_w) dw} (K-S_tau)^{+}], end{equation*} where $mathcal{T}$ is a family of stopping times, $omega$ is a discount function and $mathbb{E}$ is an expectation taken with respect to a martingale measure. Moreover, we assume that the asset price process $S_t$ is a geometric Levy process with negative exponential jumps, i.e. $S_t = s e^{zeta t + sigma B_t - sum_{i=1}^{N_t} Y_i}$. The asset-dependent discounting is reflected in the $omega$ function, so this approach is a generalisation of the classic case when $omega$ is constant. It turns out that under certain conditions on the $omega$ function, the value function $V^{omega}_{text{A}^{text{Put}}}(s)$ is convex and can be represented in a closed form; see Al-Hadad and Palmowski (2021). We provide an option pricing algorithm in this scenario and we present exact calculations for the particular choices of $omega$ such that $V^{omega}_{text{A}^{text{Put}}}(s)$ takes a simplified form.
In a multi-dimensional diffusion framework, the price of a financial derivative can be expressed as an iterated conditional expectation, where the inner conditional expectation conditions on the future of an auxiliary process that enters into the dynamics for the spot. Inspired by results from non-linear filtering theory, we show that this inner conditional expectation solves a backward SPDE (a so-called `conditional Feynman-Kac formula), thereby establishing a connection between SPDE and derivative pricing theory. The benefits of this representation are potentially significant and of both theoretical and practical interest. In particular, this representation leads to an alternative class of so-called mixed Monte-Carlo / PDE numerical methods.
We consider an incomplete multi-asset binomial market model. We prove that for a wide class of contingent claims the extremal multi-step martingale measure is a power of the corresponding single-step extremal martingale measure. This allows for closed form formulas for the bounds of a no-arbitrage contingent claim price interval. We construct a feasible algorithm for computing those boundaries as well as for the corresponding hedging strategies. Our results apply, for example, to European basket call and put options and Asian arithmetic average options.
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