Pricing with Variance Gamma Information

Abstract

In the information-based pricing framework of Brody, Hughston & Macrina, the market filtration {Ft}t≥0 is generated by an information process {ξt}t≥0 defined in such a way that at some fixed time T an FT-measurable random variable XT is “revealed”. A cash flow HT is taken to depend on the market factor XT, and one considers the valuation of a financial asset that delivers HT at time T. The value of the asset St at any time t∈[0,T) is the discounted conditional expectation of HT with respect to Ft, where the expectation is under the risk neutral measure and the interest rate is constant. Then ST−=HT, and St=0 for t≥T. In the general situation one has a countable number of cash flows, and each cash flow can depend on a vector of market factors, each associated with an information process. In the present work we introduce a new process, which we call the normalized variance-gamma bridge. We show that the normalized variance-gamma bridge and the associated gamma bridge are jointly Markovian. From these processes, together with the specification of a market factor XT, we construct a so-called variance-gamma information process. The filtration is then taken to be generated by the information process together with the gamma bridge. We show that the resulting extended information process has the Markov property and hence can be used to develop pricing models for a variety of different financial assets, several examples of which are discussed in detail.

Information-Based Trading

“…Then if we substitute (20) into (27), a straightforward calculation gives the result (17). The proof for higher values of n is similar.…”

“…Here for simplicity we stick to the case where X represents the cash flow itself, but the reader will be readily able to see how the more general situation can be accommodated. The theory of information processes along with a variety of applications is set out in references [3,5,6,9,10]; see also [20,21,25]. Now, let us write {F t } t≥0 for the smallest right-continuous complete filtration containing the natural filtration of {ξ t }.…”

“…Now write {F X (x)} x∈R + for the distribution of X, where F X (x) = P (X ≤ x), and let Y be a continuous random variable with distribution {F Y (y)} y∈R and density {f Y (y)} y∈R . Then there is a generalized Bayes theorem ( [20], Lemma 3) to the effect that for all y ∈ R at which f Y (y) = 0 it holds that…”

“…It is interesting to note, as we have pointed out, that the tier-2 price ( 27) can be expressed as a function of a single effective parcel of information, given by (20), with effective flow rate (23). This property generalizes to higher n. In particular, we have: Proposition 3.…”

A pricing model for real-estate business in Bangladesh incorporating the uncertainty in buyer’s readiness: considerations during COVID-19 pandemic

“…Some researchers have successfully developed suitable pricing models using gamma distribution (Lam et al 2002 ; Zhang and Watada 2018 ; Febrer and Guerra 2021 ). Moreover, the randomness of the buyers’ behaviors can also be properly explained with the aid of Gamma distribution (Furman and Landsman 2005 ; Hughston and Sánchez-Betancourt 2020 ). Later risk analysis in pricing strategy has been discussed by different researchers (Furman and Landsman 2005 ; Su and Furman 2016 ).…”