By Archil Gulisashvili

ISBN-10: 3642312136

ISBN-13: 9783642312137

Asymptotic research of stochastic inventory rate types is the crucial subject of the current quantity. certain examples of such types are stochastic volatility types, which were built as a solution to sure imperfections in a celebrated Black-Scholes version of alternative pricing. In a inventory expense version with stochastic volatility, the random habit of the volatility is defined by way of a stochastic technique. for example, within the Hull-White version the volatility approach is a geometrical Brownian movement, the Stein-Stein version makes use of an Ornstein-Uhlenbeck technique because the stochastic volatility, and within the Heston version a Cox-Ingersoll-Ross technique governs the habit of the volatility. one of many author's major ambitions is to supply sharp asymptotic formulation with errors estimates for distribution densities of inventory costs, choice pricing capabilities, and implied volatilities in a number of stochastic volatility versions. the writer additionally establishes sharp asymptotic formulation for the implied volatility at severe moves in most cases stochastic inventory cost types. the current quantity is addressed to researchers and graduate scholars operating within the quarter of monetary arithmetic, research, or likelihood concept. The reader is predicted to be acquainted with parts of classical research, stochastic research and chance thought.

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**Extra resources for Analytically Tractable Stochastic Stock Price Models**

**Example text**

0 t 0 γs ζ,γ E exp −Et ds. 27) where ζ,γ Et = 1 2 t 0 ζs2 ds + t ζs d W s + 0 t 1 2 0 γs2 ds + t γs dZs . 0 For the sake of simplicity, we will model independent Brownian motions W and Z on two copies (Ω1 , F (1) , P(1) ) and (Ω2 , F (2) , P(2) ) of Wiener spaces. In this interpretation, the stock price process X is defined on the product space of these Wiener spaces, and the physical measure is given by P = P(1) × P(2) . 26) and ζ Et = 1 2 t 0 ζs2 ds + t ζs d W s . s.

36) with n = 1. In order to find χn2 , we have to convolve n copies of χ12 . Let n = 2. 37), we see that χ22 (y) = = ∞ 1 2[ ( 12 )]2 1 2[ −∞ 1 1 y ( 12 )]2 1 z− 2 e− 2 1{z≥0} (y − z)− 2 e− e− 2 z 1 y−z 2 1 u− 2 (1 − u)− 2 du. 38) is a special case of the Beta function. 28 Let Re(x) > 0 and Re(y) > 0. Then the Beta function B is defined by 1 B(x, y) = t x−1 (1 − t)y−1 dt. 0 The Beta function can be represented in terms of the Gamma function (see [AAR99], p. 5, or [SSh03]). The following formula holds for all x ∈ C and y ∈ C with Re(x) > 0 and Re(y) > 0: B(x, y) = (x) (y) .

S. Define, for each 0 ≤ s < ∞, the stopping time U (s) by the following formula: U (s) = inf t : 0 ≤ t ≤ T , M, M t > s , and consider the time-changed process Bs = MU (s) , s ≥ 0. Then Bs is an GU (s) standard Brownian motion for which Mt = B M,M t , t ≥ 0. , and put Yt = XU (t) , t > 0. s. 40. 70), and take Gt = Fφ(t) , Yt = Tt , and Xt = Y M,M t = Tφ(t) , t > 0. 44, we obtain φ(t) 0 t T s d Zs = Tφ(s) φ (s) d Bt . 72) satis- 32 1 Volatility Processes fies d Yt = (a − bYt ) dt + 2e−bt Tφ(t) φ (t) d Bt = (a − bYt ) dt + c Yt d Bt .

### Analytically Tractable Stochastic Stock Price Models by Archil Gulisashvili

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