The Black-Scholes (BS) analysis of derivative pricing is one of the most beautiful results in financial economics. There are several assumptions in the basis of BS analysis such as the quasi-Brownian character of the underlying price process, constant volatility and, the absence of arbitrage.
let us denote V (t, S) as the price of a derivative at time t condition to the underlying asset price equal to S. We assume that the underlying asset price follows the geometrical Brownian motion,
dS/S = μdt + σdW —– (1)
with some average return μ and the volatility σ. They can be kept constant or be arbitrary functions of S and t. The symbol dW stands for the standard Wiener process. To price the derivative one forms a portfolio which consists of the derivative and ∆ units of the underlying asset so that the price of the portfolio is equal to Π:
Π = V − ∆S —– (2)
The change in the portfolio price during a time step dt can be written as
dΠ = dV − ∆dS = (∂V/∂t + σ2S2∂2V/2∂S2) dt + (∂V/∂S – ∆) dS —– (3)
from of Ito’s lemma. We can now chose the number of the underlying asset units ∆ to be equal to ∂V/∂S to cancel the second term on the right hand side of the last equation. Since, after cancellation, there are no risky contributions (i.e. there is no term proportional to dS) the portfolio is risk-free and hence, in the absence of the arbitrage, its price will grow with the risk-free interest rate r:
dΠ = rΠdt —– (4)
or, in other words, the price of the derivative V(t,S) shall obey the Black-Scholes equation:
(∂V/∂t + σ2S2∂2V/2∂S2) dt + rS∂V/∂S – rV = 0 —– (5)
In what follows we use this equation in the following operator form:
LBSV = 0, LBS = ∂/∂t + σ2S2∂2V/2∂S2 + rS∂/∂S – r —– (6)
To formulate the model we return back to Eqn(1). Let us imagine that at some moment of time τ < t a fluctuation of the return (an arbitrage opportunity) appeared in the market. It happened when the price of the underlying stock was S′ ≡ S(τ). We then denote this instantaneous arbitrage return as ν(τ, S′). Arbitragers would react to this circumstance and act in such a way that the arbitrage gradually disappears and the market returns to its equilibrium state, i.e. the absence of the arbitrage. For small enough fluctuations it is natural to assume that the arbitrage return R (in absence of other fluctuations) evolves according to the following equation:
dR/dt = −λR, R(τ) = ν(τ,S′) —– (7)
with some parameter λ which is characteristic for the market. This parameter can be either estimated from a microscopic theory or can be found from the market using an analogue of the fluctuation-dissipation theorem. The fluctuation-dissipation theorem states that the linear response of a given system to an external perturbation is expressed in terms of fluctuation properties of the system in thermal equilibrium. This theorem may be represented by a stochastic equation describing the fluctuation, which is a generalization of the familiar Langevin equation in the classical theory of Brownian motion. In the last case the parameter λ can be estimated from the market data as
λ = -1/(t -t’) log [〈LBSV/(V – S∂V/∂S) (t) LBSV/(V – S∂V/∂S) (t’)〉market / 〈(LBSV/(V – S∂V/∂S)2 (t)〉market] —– (8)
and may well be a function of time and the price of the underlying asset. We consider λ as a constant to get simple analytical formulas for derivative prices. The generalization to the case of time-dependent parameters is straightforward.
The solution of Equation 7 gives us R(t,S) = ν(τ,S)e−λ(t−τ) which, after summing over all possible fluctuations with the corresponding frequencies, leads us to the following expression for the arbitrage return at time t:
R (t, S) = ∫0t dτ ∫0∞ dS’ P(t, S|τ, S’) e−λ(t−τ) ν (τ, S’), t < T —– (9)
where T is the expiration date for the derivative contract started at time t = 0 and the function P (t, S|τ, S′) is the conditional probability for the underlying price. To specify the stochastic process ν(t,S) we assume that the fluctuations at different times and underlying prices are independent and form the white noise with a variance Σ2 · f (t):
⟨ν(t, S)⟩ = 0 , ⟨ν(t, S) ν (t′, S′)⟩ = Σ2 · θ(T − t) f(t) δ(t − t′) δ(S − S′) —– (10)
The function f(t) is introduced here to smooth out the transition to the zero virtual arbitrage at the expiration date. The quantity Σ2 · f (t) can be estimated from the market data as:
∑2/2λ· f (t) = 〈(LBSV/(V – S∂V/∂S)) 2 (t)⟩ market —– (11)
and has to vanish as time tends to the expiration date. Since we introduced the stochastic arbitrage return R(t, S), equation 4 has to be substituted with the following equation:
dΠ = [r + R(t, S)]Πdt, which can be rewritten as
LBSV = R (t, S) V – (S∂V/∂S) —– (12)
using the operator LBS.
It is worth noting that the model reduces to the pure BS analysis in the case of infinitely fast market reaction, i.e. λ → ∞. It also returns to the BS model when there are no arbitrage opportunities at all, i.e. when Σ = 0. In the presence of the random arbitrage fluctuations R(t, S), the only objects which can be calculated are the average value and other higher moments of the derivative price.