By Jianwei Zhu
The sound modeling of the smile influence is a crucial factor in quantitative finance as, for greater than a decade, the Fourier remodel has validated itself because the best instrument for deriving closed-form alternative pricing formulation in numerous version periods. This booklet describes the functions of the Fourier remodel to the modeling of volatility smile, through a accomplished remedy of alternative valuation in a unified framework, overlaying stochastic volatilities and rates of interest, Poisson and Levy jumps, together with numerous asset periods akin to fairness, FX and rates of interest, in addition to quite a few numberical examples and prototype programming codes. Readers will take advantage of this e-book not just by means of gaining an summary of the complex concept and the giant variety of literature on those subject matters, but additionally through receiving first-hand suggestions at the useful functions and implementations of the speculation. The publication is geared toward monetary engineers, hazard managers, graduate scholars and researchers.
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Additional info for Applications of Fourier Transform to Smile Modeling: Theory and Implementation
Q The characteristic functions of Fj j are defined by f j (φ ) ≡ EQ j [exp(iφ x(T ))] 2 for j = 1, 2. 13) These two processes are also termed as likelihood processes because they imply two probabilities. 14) f2 (φ ) ≡ EQ2 [exp(iφ x(T ))] = EQ [g2 (T ) exp(iφ x(T ))] . 15) Given the CFs f j (φ ) of x(T ) under the two different measures Q j , the density function y j (x(T )) of x(T ) is simply the inverse Fourier transform, y j (x(T )) = 1 2π R f j (φ )e−iφ x(T ) d φ , j = 1, 2. 16) Using the density function y j (x(T )) and denoting a = ln K, we can calculate the exercise probability Fj as follows: ∞ Fj (x(T ) > a) = a y j (x)dx ∞ = a = 1 2π 1 2π R R f j (φ )e−iφ x d φ dx f j (φ ) ∞ a e−iφ x dx d φ , where we have changed the order of two integrations by applying the Fubini theorem.
Different types of jumps may be a good complement to the above setting, and will be discussed later. 5) with r(t) and r∗ (t) as the domestic and foreign interest rate respectively. The foreign interest rate r∗ (t) is governed by a similar process as r(t). In the following, we use a simple functional form of b(v(t),t) for general discussion, namely b(v(t),t) = v(t). A popular choice for b(·) may be b(v(t),t) = V (t) as in the Heston model (1993). Another more complicated and still tractable form for b(·) may be the linear affine structure suggested by Dai and Singleton (2000).
Since the money market account is unique, the risk-neutral measure Q is also unique. 5. Equivalent martingale measure and no arbitrage: Denote G(t) as the process of a self-financing portfolio, P is its historical measure. For any numeraire N(t), there exists an equivalent PN , such that G(t) G0 |F0 , = EPN N0 N(t) ∀ t, and G(t) is arbitrage-free under PN . If N(t) is the money market account, then PN is the risk-neutral measure. Summing up our discussions on no arbitrage (dynamic hedging in Black-Scholes model), risk-neutral valuation and equivalent martingale measure, we could establish a cycle relation as follows: no arbitrage ⇒ risk-neutral valuation ⇒ equivalent martingale measure ⇒ no arbitrage.
Applications of Fourier Transform to Smile Modeling: Theory and Implementation by Jianwei Zhu