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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

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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

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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

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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

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Introduction

Large Deviation Theory

What is Large Deviation Theory about?

Two key concepts:

1

Rare Events

2

Exponential Decay

Significant applications in finance:

Risk management Option pricing Portfolio optimization · · ·

Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 4 / 29

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Introduction

Large Deviation Theory

Asymptotics for Empirical Mean

Let {Xi}∞

i=1 be a seq. of i.i.d. real-valued r.v.s on (Ω, F, P) with

E(X1) = µ and Var(X1) = σ2 = 1. Let Sn = X1 + · · · + Xn and ¯ Sn = X1+···+Xn

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

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Introduction

Large Deviation Theory

WLLN and CLT

Law of Large Numbers: ¯ Sn → µ in probability Central Limit Theorem: √ n (¯ Sn − µ) → Z in distribution Asymptotic Expansion: Sn ≈ µn + Z √ n as n large enough

Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 6 / 29

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Introduction

Large Deviation Theory

Why Large?

CLT: P(|¯ Sn − µ| ≥ b 1 √n) → 2Φ(−b) ⇒ 1/√n is the typical order. A result in large deviation theory: P(|¯ Sn − 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

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Two Branching Processes

GWBP

Definition Galton-Watson Branching Processes A discrete-time Markov chain {Zn}∞

n=0 on the non-negative integers

satisfies Zn+1 = Zn

j=1 Xn,j,

if Zn > 0, 0, if Zn = 0, where Xn,j are i.i.d. over all n and j, following an offspring distribution {pk}∞

k=0.

Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 8 / 29

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Two Branching Processes

GWBP

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Two Branching Processes

GWBP

Definition Probability Generating Function f(s) := E(sZ1|Z0 = 1) = ∞

k=0 pksk

Let fn(s) = f[fn−1(s)], then fn(s) = E(sZn|Z0 = 1) E(Z1) = m and E(Zn) = mn 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

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Two Branching Processes

PRIBP

Definition Poisson randomly indexed branching process Let {Zn}∞

n=0 be a GWBP and {N(t)}t≥0 be a Poisson process with

intensity λ independent of {Zn}∞

n=0. Then {ZN(t)}t≥0 is called the

PRIBP . a continuous-time Markov chain Define FN(s, t) := E(sZN(t)) be the p.g.f. of ZN(t), then E(ZN(t)) = eλ(m−1)t and limt→∞ FN(s, t) = q Define WN(t) := ZN(t)/E(ZN(t)), then limt→∞ WN(t) = W ′ a.s.

Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 11 / 29

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Two Branching Processes

PRIBP

Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 12 / 29

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Main Concern Literature

Athreya(1994) : large deviation behavior of the ratio of successive generation sizes Zn+1

Zn

for a GWBP {Zn}∞

n=0

Epps(1996) : model short-term stock price by a PRIBP {ZN(t)}t≥0

Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 13 / 29

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Main Concern

Goal:

Large deviation behavior of the ratio

ZN(t)+1 ZN(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

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Main Concern Objectives

Asymptotics for the large deviation probabilities:

1

P(|

ZN(t)+1 ZN(t) − m| > ε)

2

P(|

ZN(t)+1 ZN(t) − m| > ε | W ′ ≥ d)

Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 15 / 29

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Main Concern Assumptions

1 < m < ∞ Z0 = 1 and then generalize to Z0 = l, where l ∈ N p0 = 0 or p0 > 0

Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 16 / 29

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Asymptotic Results

Q: What is the asymptotics of FN(s, t) → q? Proposition Assume that m = 1. If p0 = 0, then lim

t→∞ e−λ(f ′(q)−1)t[FN(s, t) − q] = ∞

  • k=0

qksk := Q(s) 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 p0 = 0 and p1 > 0, it becomes limt→∞ e−λ(p1−1)tFN(s, t) = ∞

k=1 ˆ

qksk := ˆ Q(s).

Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 17 / 29

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Asymptotic Results

Theorem 1 Assume that p0 = 0 and p1 > 0. Assume that E(exp(α0Z1)) < ∞ for some α0 > 0. Then for any ε > 0, lim

t→∞ e−λ(p1−1)tP

  • ZN(t)+1

ZN(t) − m

  • > ε
  • =

∞

  • k=1

φ(k, ε)ˆ qk, where φ(k, ε) := P

  • 1

k

k

i=1 Xi − m

  • > ε
  • and {Xi}k

i=1 are i.i.d. copies

  • f Z1.

Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 18 / 29

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Asymptotic Results

Q: What if allowing p0 > 0? Theorem 2 Assume that p0 = 0. Assume that E(exp(α0Z1)) < ∞ for some α0 > 0. Then for any ε > 0, lim

t→∞ e−λ(f ′(q)−1)tP

  • ZN(t)+1

ZN(t) − m

  • > ε
  • ZN(t) > 0
  • =

∞

k=1 ϕ(k, ε)qk

1 − q , where ϕ(k, ε) := P

  • 1

k

k

i=1 Xi − m

  • > ε
  • .

Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 19 / 29

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Asymptotic Results

Q: What if conditioned on ZN(t)+1 > 0 instead of ZN(t) > 0? Theorem 3 Assume that p0 = 0 and that E(exp(α0Z1)) < ∞ for some α0 > 0. Then for any ε > 0, lim

t→∞ e−λ(f ′(q)−1)tP

  • ZN(t)+1

ZN(t) − m

  • > ε
  • ZN(t)+1 > 0
  • =

∞

k=1[ϕ(k,ε)−pk 0]qk

1−q

, if 0 < ε < m,

∞

k=1 ϕ(k,ε)qk

1−q

, if ε ≥ m.

Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 20 / 29

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Asymptotic Results

Theorem 4 Assume that E(exp(α0Z1)) < ∞ for some α0 > 0. Then there exists positive constants, D5 > 0 and τ > 0 such that for any ε > 0 and d > 0, we can find some 0 < I(ε) < ∞ such that P

  • ZN(t)+1

ZN(t) − m

  • > ε
  • W ′ ≥ d
  • ≤

αd

  • D5exp
  • − dγI(ε)eλ(p1−1)t

+D3exp

  • − τ
  • d(1 − γ)

2

3 e 1 3 λ(p1−1)t

for any 0 < γ < 1, where αd =

1 P(W ′≥d).

Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 21 / 29

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PRIBP – A Stock Price Model

Epps 1996: PRIBP ⇒ Stock price ZN(t) represents the stock price St in units of tick size Features:

1

Discrete movement

2

Fat-tailed return distribution

3

Bankruptcy

4

Leverage effect

Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 22 / 29

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PRIBP – A Stock Price Model

To fit the setting of the model, we need:

1

allow Z0 ∈ N

2

consider asymptotics in a finite time horizon [0, T]

Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 23 / 29

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PRIBP – A Stock Price Model

Theorem 5 Assume that p0 = 0. Let Z0 = l. Assume that E(exp(α0Z1)) < ∞ for some α0 > 0. Then for any ε > 0, lim

λ→∞ e−λ(f ′(q)−1)tP

  • ZN(t)+1

ZN(t) − m

  • > ε
  • ZN(t) > 0
  • =

∞

k=1 ϕ(k, ε)lql−1qk

1 − ql .

Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 24 / 29

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PRIBP – A Stock Price Model Mean Reversion

We can rewrite the following probability P

  • ZN(t)+1

ZN(t) − m

  • > ε
  • ZN(t) > 0
  • =

P

  • ZN(t)+1 − ZN(t)

ZN(t) − (m − 1)

  • > ε
  • ZN(t) > 0
  • 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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Conclusion and Future Research Conclusion

The large deviation probabilities concerning the ratio of successive generation sizes decay at least at an exponential rate The large deviation probabilities could be estimated A special mean reversion of high-frequency tick-by-tick return

Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 26 / 29

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Conclusion and Future Research Theoretical Research

Renewal RIBP PRIBP with immigration PRIBP in a random environment Multi-type PRIBP

Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 27 / 29

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Conclusion and Future Research Applied Research in Finance

Lookback Option Pricing by PRIBP payoff function: LCfix(T) = max(Smax − K, 0) and LPfix = max(K − Smin, 0) price at current time: LCfix(0) = e−rTE[max(Smax − K, 0)] and LPfix(0) = e−rTE[max(K − Smin, 0)] key: calculate the probabilities of max and min population size up to each generation

Sheng-Jhih Wu (NCSU) Large Deviations for a RIBP April 5, 2012 28 / 29

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Thanks for your attention!

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