Large Deviations for a Randomly Indexed Branching Process with Applications in Finance Sheng-Jhih Wu NCSU April 5, 2012 Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 1 / 29
Outline Introduction Branching Process Large Deviation Theory Two Branching Processes Galton-Watson Branching Process (GWBP) Poisson Randomly Indexed Branching Process (PRIBP) Main concern Asymptotic Results PRIBP – A Stock Price Model Conclusion and Future Research Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 2 / 29
Introduction What is a branching process? Main Concern: Evolution of population size Applications: Biology, Physics, Chemistry, Finance · · · Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 3 / 29
Introduction Large Deviation Theory What is Large Deviation Theory about? Two key concepts: Rare Events 1 Exponential Decay 2 Significant applications in finance: Risk management Option pricing Portfolio optimization · · · Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 4 / 29
Introduction Large Deviation Theory Asymptotics for Empirical Mean Let { X i } ∞ i = 1 be a seq. of i.i.d. real-valued r.v.s on (Ω , F , P ) with E ( X 1 ) = µ and Var ( X 1 ) = σ 2 = 1. Let S n = X 1 + · · · + X n and ¯ S n = X 1 + ··· + X n . n Two standard theorems: Law of Large Numbers (LLN) and Central Limit Theorem (CLT) Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 5 / 29
Introduction Large Deviation Theory WLLN and CLT Law of Large Numbers: ¯ S n → µ in probability Central Limit Theorem: √ n (¯ S n − µ ) → Z in distribution Asymptotic Expansion: √ S n ≈ µ n + Z n as n large enough Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 6 / 29
Introduction Large Deviation Theory Why Large? CLT: S n − µ | ≥ b 1 P ( | ¯ √ n ) → 2 Φ( − b ) ⇒ 1 / √ n is the typical order. A result in large deviation theory: P ( | ¯ S n − µ | ≥ b ) ≈ e − 2 n I ( b ) ⇒ 1 is the order. Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 7 / 29
Two Branching Processes GWBP Definition Galton-Watson Branching Processes A discrete-time Markov chain { Z n } ∞ n = 0 on the non-negative integers satisfies � � Z n j = 1 X n , j , if Z n > 0 , Z n + 1 = 0 , if Z n = 0 , where X n , j are i.i.d. over all n and j, following an offspring distribution { p k } ∞ k = 0 . Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 8 / 29
Two Branching Processes GWBP Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 9 / 29
Two Branching Processes GWBP Definition Probability Generating Function f ( s ) := E ( s Z 1 | Z 0 = 1 ) = � ∞ k = 0 p k s k Let f n ( s ) = f [ f n − 1 ( s )] , then f n ( s ) = E ( s Z n | Z 0 = 1 ) E ( Z 1 ) = m and E ( Z n ) = m n m > 1- supercritical, m = 1- critical, m < 1- subcritical Let q be the smallest root of f ( s ) = s in [ 0 , 1 ] , then q is the extinction probability Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 10 / 29
Two Branching Processes PRIBP Definition Poisson randomly indexed branching process Let { Z n } ∞ n = 0 be a GWBP and { N ( t ) } t ≥ 0 be a Poisson process with intensity λ independent of { Z n } ∞ n = 0 . Then { Z N ( t ) } t ≥ 0 is called the PRIBP . a continuous-time Markov chain Define F N ( s , t ) := E ( s Z N ( t ) ) be the p.g.f. of Z N ( t ) , then E ( Z N ( t ) ) = e λ ( m − 1 ) t and lim t →∞ F N ( s , t ) = q Define W N ( t ) := Z N ( t ) / E ( Z N ( t ) ) , then lim t →∞ W N ( t ) = W ′ a.s. Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 11 / 29
Two Branching Processes PRIBP Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 12 / 29
Main Concern Literature Athreya(1994) : large deviation behavior of the ratio of successive generation sizes Z n + 1 for a GWBP { Z n } ∞ Z n n = 0 Epps(1996) : model short-term stock price by a PRIBP { Z N ( t ) } t ≥ 0 Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 13 / 29
Main Concern Goal: Z N ( t )+ 1 Large deviation behavior of the ratio Z N ( t ) for a PRIBP Motivations: Large deviations of the ratio in more general settings Applications to finance and other areas Contributions: First large deviation results for a PRIBP Continuous-time, possibility of extinction, arbitrary number of ancestors Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 14 / 29
Main Concern Objectives Asymptotics for the large deviation probabilities: Z N ( t )+ 1 P ( | Z N ( t ) − m | > ε ) 1 Z N ( t )+ 1 Z N ( t ) − m | > ε | W ′ ≥ d ) P ( | 2 Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 15 / 29
Main Concern Assumptions 1 < m < ∞ Z 0 = 1 and then generalize to Z 0 = l , where l ∈ N p 0 = 0 or p 0 > 0 Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 16 / 29
Asymptotic Results Q: What is the asymptotics of F N ( s , t ) → q ? Proposition Assume that m � = 1 . If p 0 � = 0 , then ∞ q k s k := Q ( s ) t →∞ e − λ ( f ′ ( q ) − 1 ) t [ F N ( s , t ) − q ] = � lim k = 0 for all 0 ≤ s < 1 . Moreover, Q ( s ) is the unique solution of the functional equation, Q ( f ( s )) = f ′ ( q ) Q ( s ) for all 0 ≤ s < 1 . Remark When p 0 = 0 and p 1 > 0 , it becomes q k s k := ˆ lim t →∞ e − λ ( p 1 − 1 ) t F N ( s , t ) = � ∞ k = 1 ˆ Q ( s ) . Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 17 / 29
Asymptotic Results Theorem 1 Assume that p 0 = 0 and p 1 > 0 . Assume that E ( exp ( α 0 Z 1 )) < ∞ for some α 0 > 0 . Then for any ε > 0 , ∞ � Z N ( t )+ 1 � > ε � t →∞ e − λ ( p 1 − 1 ) t P �� � � φ ( k , ε )ˆ lim − m = q k , Z N ( t ) k = 1 � k � > ε � 1 �� � � and { X i } k where φ ( k , ε ) := P i = 1 X i − m i = 1 are i.i.d. copies k of Z 1 . Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 18 / 29
Asymptotic Results Q: What if allowing p 0 > 0? Theorem 2 Assume that p 0 � = 0 . Assume that E ( exp ( α 0 Z 1 )) < ∞ for some α 0 > 0 . Then for any ε > 0 , � ∞ � Z N ( t )+ 1 k = 1 ϕ ( k , ε ) q k t →∞ e − λ ( f ′ ( q ) − 1 ) t P � > ε � Z N ( t ) > 0 �� � � � lim − m = , Z N ( t ) 1 − q � k � > ε �� � 1 � � where ϕ ( k , ε ) := P i = 1 X i − m . k Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 19 / 29
Asymptotic Results Q: What if conditioned on Z N ( t )+ 1 > 0 instead of Z N ( t ) > 0? Theorem 3 Assume that p 0 � = 0 and that E ( exp ( α 0 Z 1 )) < ∞ for some α 0 > 0 . Then for any ε > 0 , � Z N ( t )+ 1 t →∞ e − λ ( f ′ ( q ) − 1 ) t P � > ε � Z N ( t )+ 1 > 0 �� � � � lim − m Z N ( t ) � � ∞ k = 1 [ ϕ ( k ,ε ) − p k 0 ] q k , if 0 < ε < m , 1 − q = � ∞ k = 1 ϕ ( k ,ε ) q k , ε ≥ m . if 1 − q Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 20 / 29
Asymptotic Results Theorem 4 Assume that E ( exp ( α 0 Z 1 )) < ∞ for some α 0 > 0 . Then there exists positive constants, D 5 > 0 and τ > 0 such that for any ε > 0 and d > 0 , we can find some 0 < I ( ε ) < ∞ such that � Z N ( t )+ 1 � W ′ ≥ d � > ε �� � � � P − m Z N ( t ) − d γ I ( ε ) e λ ( p 1 − 1 ) t � � � ≤ α d D 5 exp � 2 1 3 e 3 λ ( p 1 − 1 ) t �� � � + D 3 exp − τ d ( 1 − γ ) 1 for any 0 < γ < 1 , where α d = P ( W ′ ≥ d ) . Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 21 / 29
PRIBP – A Stock Price Model Epps 1996: PRIBP ⇒ Stock price Z N ( t ) represents the stock price S t in units of tick size Features: Discrete movement 1 Fat-tailed return distribution 2 Bankruptcy 3 Leverage effect 4 Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 22 / 29
PRIBP – A Stock Price Model To fit the setting of the model, we need: allow Z 0 ∈ N 1 consider asymptotics in a finite time horizon [ 0 , T ] 2 Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 23 / 29
PRIBP – A Stock Price Model Theorem 5 Assume that p 0 � = 0 . Let Z 0 = l. Assume that E ( exp ( α 0 Z 1 )) < ∞ for some α 0 > 0 . Then for any ε > 0 , � Z N ( t )+ 1 λ →∞ e − λ ( f ′ ( q ) − 1 ) t P � > ε � Z N ( t ) > 0 �� � � � lim − m Z N ( t ) � ∞ k = 1 ϕ ( k , ε ) lq l − 1 q k = . 1 − q l Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 24 / 29
PRIBP – A Stock Price Model Mean Reversion We can rewrite the following probability � Z N ( t )+ 1 � > ε � Z N ( t ) > 0 �� � � � P − m Z N ( t ) � Z N ( t )+ 1 − Z N ( t ) � > ε � Z N ( t ) > 0 �� � � � = P − ( m − 1 ) Z N ( t ) Mean reversion of high-frequency tick-by-tick return Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 25 / 29
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