By Anatoliy Swishchuk
This booklet is dedicated to the historical past of switch of Time equipment (CTM), the connections of CTM to stochastic volatilities and finance, primary facets of the speculation of CTM, simple thoughts, and its houses. An emphasis is given on many functions of CTM in monetary and effort markets, and the provided numerical examples are in keeping with actual information. The switch of time approach is utilized to derive the well known Black-Scholes formulation for eu name techniques, and to derive an particular alternative pricing formulation for a ecu name choice for a mean-reverting version for commodity costs. particular formulation also are derived for variance and volatility swaps for monetary markets with a stochastic volatility following a classical and not on time Heston version. The CTM is utilized to cost monetary and effort derivatives for one-factor and multi-factor alpha-stable Levy-based models.
Readers must have a uncomplicated wisdom of chance and facts, and a few familiarity with stochastic procedures, resembling Brownian movement, Levy method and martingale.
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Extra info for Change of Time Methods in Quantitative Finance
6. The Ho and Lee process: dS(t) = θ (t)dt + σ dW (t). Solution S(t) = S(0) + Wˆ (σ 2t) + 0t θ (s)ds. 7. The Hull and White process: dS(t) = (a(t) − b(t)S(t))dt + σ (t)dW (t). t 2 ˆ ˆ ˆ Solution S(t) = exp[− 0t b(s)ds][S(0)− a(s) b(s) + W (Tt )], where Tt = 0 σ (s)[S(0)− a(s) b(s) + Wˆ (Tˆs ) + exp[ a(s) 2 s 0 b(u)du] b(s) ] ds. 8. The Heath, Jarrow, and Morton process: f (t, u) = f (0, u) + 0t a(v, u)dv + t t ˆ ˆ ˆ 0 b( f (v, u))dW (v). Solution f (t, u) = f (0, u) + W (Tt ) + 0 a(v, u)dv, where Tt = t 2 s ˆ ˆ b ( f (0, u) + W ( T ) + a(v, u)dv)ds.
Proof. 18). 1) C(T ) = e−rT EQ [max(S(T ) − K, 0)] = e−rT EQ [max(ert [S(0) + W˜ ∗ (Tˆt )] − K, 0)] ∗ (T )− σ 2 T 2 2 σ ∗ σ W (T )+(r− 2 )T = e−rT EQ [max(ert S(0)eσ W = = − K, 0)] e−rT EQ [max(S(0)e − K, 0)] √ σ 2 )T u2 +∞ 1 σ u T +(r− −rT √ 2 e max[S(0)e − K, 0]e− 2 du. 21) 40 4 Change of Time Method (CTM) and Black-Scholes Formula namely, y0 = K ) − (r − σ 2 /2)T ln( S(0) √ . 21) may be presented in the following form: 1 C(T ) = e−rT √ 2π +∞ y0 (S(0)eσ u √ 2 T +(r− σ2 )T u2 − K)e− 2 du. 5). D.
We present a variance drift-adjusted version of the Heston model which leads to a significant improvement of the market volatility surface fitting by 44% (compared to Heston). The numerical example we performed with recent market data shows a significant reduction of the average absolute calibration error1 (calibration on 12 dates ranging from 19 Sep. to 17 Oct. 2011 for the FOREX underlying EURUSD). Our model has two additional parameters compared to the Heston model, can be implemented very easily and was initially introduced for volatility derivatives pricing.