In the top down description of theoretical finance, a security S(t) follows a random walk described by a Ito-Weiner process (or Langevin equation) as
dS(t)/S(t) = φdt + σ R(t) dt —– (1)
where R(t) is a Gaussian white noise with zero mean and uncorrelated values at time t and t′ ⟨R(t)R(t′)⟩ = δ(t−t′). φ is the drift term or expected return, while σ is a constant factor multiplying the random source R(t), termed volatility.
As a consequence of Ito calculus, differentials of functions of random variables, say f(S,t), do not satisfy Leibnitz’s rule, and for a Ito-Weiner process with drift (2) one easily obtains for the time derivative of f(S,t)
df/dt = ∂f/∂t + 1/2 σ2 S2 ∂2f/∂S2 + φS∂f/∂S + σS∂f/∂S R —– (2)
The Black-Scholes model is obtained by removing the randomness of the stochastic process shown above by introducing a random process correlated to equation 2. This operation, termed hedging, allows to remove the dependence on the white noise function R(t), by constructing a portfolio Π, whose evolution is given by the short-term risk free interest rate r
dΠ/dt = rΠ —– (3)
A possibility is to choose Π = f – S∂f/∂S. This is a portfolio in which an investor holds an option f and short sells an amount of the underlying security S proportional to ∂f/∂S. A combination of equations 2 and 3 yields the Black-Scholes equation
∂f/∂t + 1/2 σ2 S2 ∂2f/∂S2 + rS∂f/∂S = rf —– (4)
There are some assumptions underlying this result. We have assumed absence of arbitrage, constant spot rate r, continuous balance of the portfolio, no transaction costs and infinite divisibility of the stock. The quantum mechanical version of this equation is obtained by a change of variable S = ex, with x a real variable. This yields
∂f/∂t = HBSf —– (5)
with an Hamiltonian HBS given by
HBS = – σ2/2 ∂2/∂x2 + (1/2 σ2 – r) ∂/∂x + r —– (6)
Notice that one can introduce a quantum mechanical formalism and interpret the option price as a ket |f⟩ in the basis of |x⟩, the underlying security price. Using Dirac notation, we can formally reinterpret f (x, t) = ⟨x|f (t)⟩, as a projection of an abstract quantum state |f(t)⟩ on the chosen basis.
In this notation, the evolution of the option price can be formally written as |f, t⟩ = etH |f, 0⟩, for an appropriate Hamiltonian H.
In general, the description of these processes is driven by two correlated white noise functions R1 and R2
dV/dt = λ + μV + ζVαR1
dV/dt = φS + σ√V + μV + ζVαR2 —– (7)
with V = √σ and ⟨R1(t)R2(t′)⟩ = 1/ρ δ(t − t′)
ρ being the correlation parameter. However, since volatility is not traded in the market (the market is said to be incomplete), perfect hedging is not possible, and an additional term, the market price of volatility risk β(S,V,t,r), is in this case introduced. β can be modeled appropriately. In some models, a redefinition of the drift term μ in (7) in the evolution of the volatility is sufficient to hedge such more complex portfolios, which amounts to an implicit choice of β(S, V, t, r). We just quote the result for the evolution of an option price in the presence of stochastic volatility, which, in the Hamiltonian formulation are given by
∂f/∂t = HMGf —– (8),
HMG = -(r – ey/2) ∂/∂x – (λe-y + μ – ζ2/2 e2y(α – 1)) ∂/∂y – ey/2 ∂2/∂x2 -ρζ ey(α – 1)/2 ∂2/∂x∂y – ζ2 e2y(α – 1) /2 ∂2/∂y2 + r —– (9)
which is nonlinear in the variables x = log(S) and y = log(V ). For general values of the parameters, the best way to obtain the pricing of the options in this model is by a simulation of the path integral.