Geometrical Approximation and Perturbative Methods for Pdes in Finance

The most popular market model in continuous time is the Black-Scholes model. It assumes for the underlying process, a geometric Brownian motion with constant volatility, that is dS_t = rS_t dt σS_t dW_t, dB_t = rB_t dt, where r is the constant risk-free rate, St is the stock and σ is the constant volatility of the stock.Under these assumptions, closed form solutions for the values of European call and put options, are derived by use of the PDE method. We want to discuss by present work, the PDE approach in most complicated cases of market models. Our objective is to use three different techniques that are respectively Spectral Methods, Geometrical Approximation and Perturbative method (these last introduced by us), in order to compute the price of the derivatives for the following kinds of contracts: Vanilla Options, Barrier Options and Double Barrier Options.We have structured the work in five chapters: Chapters 1 and 2, respectively show the theoretical foundations of the parabolic PDE, and the Black Scholes market model. In chapter 3, we are going to consider Double Barrier Options, in Black Scholes model, that belongs to the kind of exotic options, in which case we have a deterministic volatility function σ(t).For example we consider the value of a Knock-out, down-and-out Call option, that is given by the solution of the Black-Scholes equation with appropriate boundary conditions, but we are able to discuss also the cases in which we have Knock-in options, or we have a Put option and do not a Call. To grant the existence and uniqueness of the solution, it is necessary to define the boundary condition and the initial condition. Also we require that when the value of the underlying asset hits the two barriers, lower (L) and upper (H), the option is cancelled in our case, but it could be activated for knock-in options . The best method to solve the above problem, is the using of the Spectral Theory, which allows to write the price of the Knock-out or Knock-in options, as series expansion. We are going to compare the spectral method with others, studying also the computational complexity.In chapter 4, we propose a new technique, that we have called the Geometrical Approximation method, applying this one to the stochastic volatility market models as Heston and SABR. The assumption of constant volatility isn’t reasonable in a real market, since we require different values for the volatility parameter for different strikes and different expiries to match market prices. The volatility parameter that is required in the Black-Scholes formula to reproduce market prices is called the implied volatility. To obtain market prices of options maturing at a certain date, volatility needs to be a function of the strike. This function is the so called volatility skew or smile.Furthermore for a fixed strike we also need different volatility parameters to match the market prices of options maturing on different dates written on the same underlying, hence volatility is a function of both the strike and the expiry date of the derivative security. This bivariate function is called the volatility surface. There are two prominent ways of working around this problem, namely, local volatility models and stochastic volatility models. For local volatility models the assumption of constant volatility made in Black and Scholes (1973) is relaxed. The underlying risk-neutral stochastic process becomes dS_t = r(t) S_t dt σ(t, S_t) S_t dW_t, where r(t) is the instantaneous forward rate of maturity t implied by the yield curve and the function σ(St, t) is chosen (calibrated) such that the model is consistent with market data, see Dupire (1994), Derman and Kani (1994) and (Wilmott, 2000). It is claimed in Hagan et al. (2002) that local volatility models predict that the smile shifts to higher prices (resp. lower prices) when the price of the underlying decreases (resp. increases). This is in contrast with the market behaviour where the smile shifts to higher prices (resp. lower prices) when the price of the underlying increases (resp. decreases). Another way of working around the inconsistency introduced by constant volatility is by introducing a stochastic process for the volatility itself; such models are called stochastic volatility models. The major advances in stochastic volatility models are Hull and White (1987), Heston (1993) and Hagan et al. (2002). Such models have a general form and varying its parameters we can obtain them: for δ = 1, j = 1, α = 0, a2(S) = Sβ, β ∈ (0, 1] and b1 = 0, we get the SABR model, by Hagan; for δ = 1, j = 2, α = 0, a2(S) = S and b1 = k(θ − σjt ), we get Heston model, by Heston; for δ = 1, α = 0 and b1 = 0 we get Black-Scholes model with constant volatility, by Black Scholes Merton; where the tradable security S_t and its volatility σ_t are correlated. Using the above indicated general market model, from Itˆo’s lemma, it is possible to derive, under mild additional assumptions, the partial differential equation satisfied by the value function of a European contingent claim. For this purpose, one needs first to specify the market price of volatility risk λ(σ, t). The market price for the risk is associated with the Girsanov transformation of the underlying probability measure leading to a particular martingale measure. Let us observe that pricing of contingent claims using the market price of volatility risk is not preferences-free.The price function f = f(t, S, σ) of a European contingent claim has to satisfy a specific PDE of parabolic kind, with the terminal condition Φ(ST) = f(T,S, σ) for every S ∈ R and σ ∈ R . We are going to use some geometrical transformations in order to simplify the above pricing PDE. Our idea is to determine, by Ito’s lemma, the exact PDE for derivative pricing in Heston and SABR market model and instead to use the exact pay-off function, for example, for a Vanilla Call option (S_T −E) , we consider this (S_T*e^(e_T) − E) , where (e_T) is a stochastic process linked to volatility (or variance). What mean eT will be clear later. Hence, we are able to solve the exact PDE, but with different Cauchy’s condition (with respect to original problem).In other words it is possible to approximate our closed form solution obtained by considerations on property of continuity of Feynman-Kac formula, with the solutions computed using the numerical techniques known in literature. Finally, in Chapter 5 we present another approximation technique, again for the Heston model, based on different idea, respect to Geometrical Approximation method. In fact in this case we are going to choose a particular volatility risk price, so that, the drift term of the variance processes is equal to zero. Also by the latter procedure, that we name Perturbative Method, we are able to evaluate the Vanilla Options, and not only, through an approximate solution in closed form, that can be used also for pricing several kinds of derivatives contracts, and we have used here also for computing the price of the knock-out Barrier Options.