We use the Hilbert space
Hα := {f ∈ AC(R+,C) : ∫∞0 |f′(x)|2 eαx dx < ∞}
where AC(R+,C) denotes the space of complex-valued absolutely continuous functions on R+. We endow Hα with the scalar product ⟨f,g⟩α := f(0) g–(0) + ∫∞0 f′(x) g(x) eαx dx, and denote the associated norm by ∥ · ∥α. Filipović shows that (Hα, ∥ · ∥α) is a separable Hilbert space. This space has been used in Filipović for term structure modelling of bonds and many mathematical properties have been derived therein. We will frequently refer to Hα as the Filipović space.
We next introduce our dynamics for the term structure of forward prices in a commodity market. Denote by f (t, x) the price at time t of a forward contract where time to delivery of the underlying commodity is x ≥ 0. We treat f as a stochastic process in time with values in the Filipović space Hα. More specifically, we assume that the process {f(t)}t≥0 follows the HJM-Musiela model which we formalize next. The Heath–Jarrow–Morton (HJM) framework is a general framework to model the evolution of interest rate curve – instantaneous forward rate curve in particular (as opposed to simple forward rates). When the volatility and drift of the instantaneous forward rate are assumed to be deterministic, this is known as the Gaussian Heath–Jarrow–Morton (HJM) model of forward rates. For direct modeling of simple forward rates the Brace–Gatarek–Musiela model represents an example.
On a complete filtered probability space (Ω,{Ft}t≥0,F,P), where the filtration is assumed to be complete and right continuous, we work with an Hα-valued Lévy process {L(t)}t≥0 for the construction of Hα-valued Lévy processes). In mathematical finance, Lévy processes are becoming extremely fashionable because they can describe the observed reality of financial markets in a more accurate way than models based on Brownian motion. In the ‘real’ world, we observe that asset price processes have jumps or spikes, and risk managers have to take them into consideration. Moreover, the empirical distribution of asset returns exhibits fat tails and skewness, behavior that deviates from normality. Hence, models that accurately fit return distributions are essential for the estimation of profit and loss (P&L) distributions. Similarly, in the ‘risk-neutral’ world, we observe that implied volatilities are constant neither across strike nor across maturities as stipulated by the Black and Scholes. Therefore, traders need models that can capture the behavior of the implied volatility smiles more accurately, in order to handle the risk of trades. Lévy processes provide us with the appropriate tools to adequately and consistently describe all these observations, both in the ‘real’ and in the ‘risk-neutral’ world. We assume that L has finite variance and mean equal to zero, and denote its covariance operator by Q. Let f0 ∈ Hα and f be the solution of the stochastic partial differential equation (SPDE)
df(t) = ∂xf(t)dt + β(t)dt + Ψ(t)dL(t), t≥0,f(0)=f0
where β ∈ L ((Ω × R+, P, P ⊗ λ), Hα), P being the predictable σ-field, and
Ψ ∈ L2L(Hα) := ∪T>0 L2L,T (Hα)
where the latter space is defined as in Peszat and Zabczyk. For t ≥ 0, denote by Ut the shift semigroup on Hα defined by Utf = f(t + ·) for f ∈ Hα. It is shown in Filipović that {Ut}t≥0 is a C0-semigroup on Hα, with generator ∂x. Recall, that any C0-semigroup admits the bound ∥Ut∥op ≤ Mewt for some w, M > 0 and any t ≥ 0. Here, ∥ · ∥op denotes the operator norm. Thus s → Ut−s β(s) is Bochner-integrable (The Bochner integral, named for Salomon Bochner, extends the definition of Lebesgue integral to functions that take values in a Banach space, as the limit of integrals of simple functions). and s → Ut−s Ψ(s) is integrable with respect to L. The unique mild solution of SPDE is
f(t) = Utf0 + ∫t0 Ut−s β(s)ds+ ∫t0 Ut−s Ψ(s)dL(s)
If we model the forward price dynamics f in a risk-neutral setting, the drift coefficient β(t) will naturally be zero in order to ensure the (local) martingale property (In probability theory, a martingale is a model of a fair game where knowledge of past events never helps predict the mean of the future winnings and only the current event matters. In particular, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time in the realized sequence, the expectation of the next value in the sequence is equal to the present observed value even given knowledge of all prior observed values.) of the process t → f(t, τ − t), where τ ≥ t is the time of delivery of the forward. In this case, the probability P is to be interpreted as the equivalent martingale measure (also called the pricing measure). However, with a non-zero drift, the forward model is stated under the market probability and β can be related to the risk premium in the market. In energy markets like power and gas, the forward contracts deliver over a period, and forward prices can be expressed by integral operators on the Filipović space applied on f. The dynamics of f can also be considered as a model for the forward rate in fixed-income theory. This is indeed the traditional application area and point of analysis of the SPDE. Note, however, that the original no-arbitrage condition in the HJM approach for interest rate markets is different from the no-arbitrage condition. If f is understood as the forward rate modelled in the risk-neutral setting, there is a no-arbitrage relationship between the drift β, the volatility σ and the covariance of the driving noise L.
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