FAST FOURIER TRANSFORM AND FORMULATION OF ECONOMIC RECESSION INDUCED STOCHASTIC VOLATILITY MODELS FOR AMERICAN OPTIONS COMPUTATION

Abstract

Economic recession has become a global and reccurring phenomenon which poses worrisome uncertainties on assets’ returns in financial markets. Various stochastic models have been formulated in response to price instability in financial markets. However, the existing stochastic volatility models did not incorporate the concept of economic recession and induced volatility-uncertainty for options price valuation in a recessed economic setting. Therefore, this study was geared towards the formulation of economic recession-induced stochastic models for price computation. Stochastic modelling methods with probabilistic uncertainty measure were used to formulate two new volatility models incorporating economy recession volatility uncertainties. The Feynman-Kac formula was applied to derive the characteristic functions for the two novel models. The derived characteristic functions were used to obtain an inverse-Fourier analytic formula for European and American-style options. A modified Carr and Madan Fast-Fourier Transform (FFT)-algorithm was used to obtain an approximate solution for the American-call option, and a class of Multi-Assets option in multi-dimensions. Itˆo Calculus was used to obtain an explicit formula for a Factorial function Black-Scholes Partial Differential Equations (BS-PDE) for American options subject to moving boundary conditions. The FFT call-prices accuracy test at varied fineness grid points N was investigated using an FFT-algorithm via Maple, taking BS-prices as benchmark. Sample paths and numerical simulations were generated via software codes for the control regimeswitching Triple Stochastic Volatility Heston-like (TSVH) model. The derived Uncertain Affine Exponential-Jump Model (UAEM) with recession, induced stochastic-volatility and stochastic-intensity, and a control regime-switching Triple Stochastic Volatility Heston-like (TSVH) formulated with respect to economy recession volatility uncertainties are:  d ln S(t) = r − q − λ(t)m dt + pσ(t)dWs(t) + (eν − 1)dN(t), S(0) = S0 > 0 dσ(t) = κσ β∗ + βrec − σ(t) dt + ξσpσ(t)dWσ(t), σ(0) = σ0 > 0 dλ(t) = κλθ − λ(t) dt + ξλpλ(t)dWλ(t), λ(0) = λ0 > 0. and  dyt = r − q dt + pv1(t)dW1(t) + pv2(t)dW2(t) + αpv3(t)dW3(t) , S(0) = S0 > 0 dv1(t) = κ1θ1 − v1(t) dt + σ1pv1(t)dWc1(t), v1(0) = v10 > 0. dv2(t) = κ2θ2 − v2(t) dt + σ2pv2(t)dWc2(t), v2(0) = v20 > 0. dv3(t) = α κ3θ3 − v3(t) dt + σ3pv3(t)dWc3(t) recession, v3(0) = v30 > 0 respectively, where α was a binary control parameter defined as: α := 0, if the economy is not in recession; 1, if the economy is in recession. The inverse-Fourier analytic formulae for European-style and American-style calloptions obtained for the UAEM-process are: Ecall T (k) = exp(−αk) π Z ∞ 0 e−(rT +iuk)φτ u − (α + 1)i × α2 + α − u2 − i(2α + 1)u α4 + 2α3 + α2 + 2(α2 + α) + 1 u2 + u4 du ivand At(k) = exp(−αk) π Z ∞ 0 e−(rT +iuk) × φτu − (α + 1)i α2 + α − u2 − i(2α + 1)u α4 + 2α3 + α2 + 2(α2 + α) + 1 u2 + u4 du + Pt, respectively where Pt is premium price. The approximate solution obtained for American-call option via FFT-algorithm for the UAEM-process was: A τ (ku) ≈ exp(−αk) π NX j =1 e−iuj ζη(j−1)(u−1) eiϖujψT (uj)η + Pt, where 1 ≤ u ≤ N and ζη = 2π N . The derived multi-Assets options prices formula in n-dimension was: VT (k1,p1, k2,p2 · · · , kn,pn) ≈ e−(α1k1,p1 +α2k2,p2 +...+αnkn,pn ) (2π)n Ω(k1,p1, k2,p2, · · · , kn,pn) nY j =1 hj, where 0 ≤ p1, p2, · · · , pn ≤ N − 1 and Ω(k1,p1, k2,p2, · · · , kn,pn) = N1−1 X m1 =1 N1−1 X m2 =1 · · · N1−1 X mn=1 e − 2π N (m1− N2 )(p1− N2 )+(m2− N2 )(p2− N2 )+···+(mn− N2 )(pn− N2 ) × ψT (u1, u2, · · · , un). The derived explicit formula for the Factorial function BS-PDE was: S(T) = S(t0) exp hn! r + 1 2(n − 1)σ2 T − t0 + n!σ W(t) − W(t0) i, S(t0) ̸= 0), and the TSVH call pricing formula derived was: C(K) = Ste−qτP1 − Ke−rτP2 such that P1 = 1 2 + 1 π Z ∞ 0 ℜ exp(−iω ln K) fω − i; yt, v1(t), v2(t), v3(t) iωSte(r−q)τ dω P2 = 1 2 + 1 π Z ∞ 0 ℜ exp(−iω ln K) fω; yt, v1(t), v2(t), v3(t) iω dω, and fω − i; yt, v1(t), v2(t), v3(t) = exp A(τ, ω) + B0(τ, ω)yt + B1(τ, ω)v1(t) + B2(τ, ω)v2(t) + B3(τ, ω)v3(t) where A, B0, B1, B2, B3 are coefficient terms of the stochastic processes yt, v1(t), v2(t), v3(t). The options prices obtained from An Uncertain Affine Exponential-Jump Model with Recession, induced stochastic-volatility and stochastic-intensity and a control regimeswitching Triple Stochastic Volatility Heston-like model, were efficient in terms of probable future payoffs and applicable in financial markets, for options valuation in recessed and recession-free economy states.