Introduction    Mathematical Results          Application to Financial Market Models             Extensions and future wor...
Introduction        Mathematical Results   Application to Financial Market Models   Extensions and future work




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Introduction        Mathematical Results   Application to Financial Market Models   Extensions and future work




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Introduction      Mathematical Results   Application to Financial Market Models   Extensions and future work




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Introduction      Mathematical Results   Application to Financial Market Models   Extensions and future work




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Introduction      Mathematical Results   Application to Financial Market Models   Extensions and future work




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Introduction      Mathematical Results   Application to Financial Market Models   Extensions and future work




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Introduction      Mathematical Results        Application to Financial Market Models      Extensions and future work




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Introduction        Mathematical Results   Application to Financial Market Models   Extensions and future work




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Introduction       Mathematical Results   Application to Financial Market Models   Extensions and future work




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Introduction       Mathematical Results   Application to Financial Market Models   Extensions and future work




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Introduction       Mathematical Results   Application to Financial Market Models   Extensions and future work




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Introduction     Mathematical Results     Application to Financial Market Models   Extensions and future work




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Introduction     Mathematical Results     Application to Financial Market Models   Extensions and future work




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Introduction     Mathematical Results     Application to Financial Market Models   Extensions and future work




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Introduction   Mathematical Results      Application to Financial Market Models   Extensions and future work




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Introduction      Mathematical Results   Application to Financial Market Models   Extensions and future work




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Introduction      Mathematical Results   Application to Financial Market Models   Extensions and future work




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Introduction      Mathematical Results   Application to Financial Market Models   Extensions and future work




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Introduction        Mathematical Results   Application to Financial Market Models   Extensions and future work




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Introduction      Mathematical Results   Application to Financial Market Models   Extensions and future work




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Introduction      Mathematical Results   Application to Financial Market Models   Extensions and future work




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Introduction        Mathematical Results   Application to Financial Market Models   Extensions and future work




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Introduction      Mathematical Results   Application to Financial Market Models   Extensions and future work




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Introduction      Mathematical Results   Application to Financial Market Models   Extensions and future work




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Introduction      Mathematical Results    Application to Financial Market Models   Extensions and future work




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Introduction      Mathematical Results    Application to Financial Market Models   Extensions and future work




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Introduction      Mathematical Results    Application to Financial Market Models   Extensions and future work




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Christmas Talk07

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Christmas Talk07

  1. 1. Introduction Mathematical Results Application to Financial Market Models Extensions and future work The size of the largest fluctuations in a financial market model with Markovian switching Terry Lynch1 (joint work with J. Appleby1 , X. Mao2 and H. Wu1 ) 1 Dublin City University, Ireland. 2 Strathclyde University, Glasgow, U.K. Christmas Talk Dublin City University Dec 14th 2007 Supported by the Irish Research Council for Science, Engineering and Technology
  2. 2. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Outline 1 Introduction 2 Mathematical Results 3 Application to Financial Market Models 4 Extensions and future work
  3. 3. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Outline 1 Introduction 2 Mathematical Results 3 Application to Financial Market Models 4 Extensions and future work
  4. 4. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Introduction We consider the size of the large fluctuations of a stochastic differential equation (S.D.E) with Markovian switching, concentrating on processes which obey the law of the iterated logarithm: |X (t)| lim sup √ = c a.s. t→∞ 2t log log t The results are applied to financial market models which are subject to random regime shifts (confident to nervous) and to changes in market sentiment (investor intuition). Markovian switching: parameters can switch according to a Markov jump process We show that our security model exhibits the same long-run growth and deviation properties as conventional geometric Brownian motion.
  5. 5. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Introduction We consider the size of the large fluctuations of a stochastic differential equation (S.D.E) with Markovian switching, concentrating on processes which obey the law of the iterated logarithm: |X (t)| lim sup √ = c a.s. t→∞ 2t log log t The results are applied to financial market models which are subject to random regime shifts (confident to nervous) and to changes in market sentiment (investor intuition). Markovian switching: parameters can switch according to a Markov jump process We show that our security model exhibits the same long-run growth and deviation properties as conventional geometric Brownian motion.
  6. 6. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Introduction We consider the size of the large fluctuations of a stochastic differential equation (S.D.E) with Markovian switching, concentrating on processes which obey the law of the iterated logarithm: |X (t)| lim sup √ = c a.s. t→∞ 2t log log t The results are applied to financial market models which are subject to random regime shifts (confident to nervous) and to changes in market sentiment (investor intuition). Markovian switching: parameters can switch according to a Markov jump process We show that our security model exhibits the same long-run growth and deviation properties as conventional geometric Brownian motion.
  7. 7. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Introduction We consider the size of the large fluctuations of a stochastic differential equation (S.D.E) with Markovian switching, concentrating on processes which obey the law of the iterated logarithm: |X (t)| lim sup √ = c a.s. t→∞ 2t log log t The results are applied to financial market models which are subject to random regime shifts (confident to nervous) and to changes in market sentiment (investor intuition). Markovian switching: parameters can switch according to a Markov jump process We show that our security model exhibits the same long-run growth and deviation properties as conventional geometric Brownian motion.
  8. 8. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Classical geometric Brownian motion (GBM) Characterised as the unique solution of the linear S.D.E dS∗ (t) = µS∗ (t) dt + σS∗ (t) dB(t) where µ is the instantaneous mean rate of growth of the price, σ its instantaneous volatility and S∗ (0) > 0. S∗ grows exponentially according to log S∗ (t) 1 lim = µ − σ2, a.s. (1) t→∞ t 2 The maximum size of the large deviations from this growth trend obey the law of the iterated logarithm | log S∗ (t) − (µ − 1 σ 2 )t| 2 lim sup √ = σ, a.s. (2) t→∞ 2t log log t
  9. 9. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Classical geometric Brownian motion (GBM) Characterised as the unique solution of the linear S.D.E dS∗ (t) = µS∗ (t) dt + σS∗ (t) dB(t) where µ is the instantaneous mean rate of growth of the price, σ its instantaneous volatility and S∗ (0) > 0. S∗ grows exponentially according to log S∗ (t) 1 lim = µ − σ2, a.s. (1) t→∞ t 2 The maximum size of the large deviations from this growth trend obey the law of the iterated logarithm | log S∗ (t) − (µ − 1 σ 2 )t| 2 lim sup √ = σ, a.s. (2) t→∞ 2t log log t
  10. 10. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Classical geometric Brownian motion (GBM) Characterised as the unique solution of the linear S.D.E dS∗ (t) = µS∗ (t) dt + σS∗ (t) dB(t) where µ is the instantaneous mean rate of growth of the price, σ its instantaneous volatility and S∗ (0) > 0. S∗ grows exponentially according to log S∗ (t) 1 lim = µ − σ2, a.s. (1) t→∞ t 2 The maximum size of the large deviations from this growth trend obey the law of the iterated logarithm | log S∗ (t) − (µ − 1 σ 2 )t| 2 lim sup √ = σ, a.s. (2) t→∞ 2t log log t
  11. 11. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Outline 1 Introduction 2 Mathematical Results 3 Application to Financial Market Models 4 Extensions and future work
  12. 12. Introduction Mathematical Results Application to Financial Market Models Extensions and future work S.D.E with Markovian switching We study the stochastic differential equation with Markovian switching dX (t) = f (X (t), Y (t), t) dt + g (X (t), Y (t), t) dB(t) (3) where X (0) = x0 , f , g : R × S × [0, ∞) → R are continuous functions obeying local Lipschitz continuity and linear growth conditions, Y is an irreducible continuous-time Markov chain with finite state space S and is independent of the Brownian motion B. Under these conditions, existence and uniqueness is guaranteed.
  13. 13. Introduction Mathematical Results Application to Financial Market Models Extensions and future work S.D.E with Markovian switching We study the stochastic differential equation with Markovian switching dX (t) = f (X (t), Y (t), t) dt + g (X (t), Y (t), t) dB(t) (3) where X (0) = x0 , f , g : R × S × [0, ∞) → R are continuous functions obeying local Lipschitz continuity and linear growth conditions, Y is an irreducible continuous-time Markov chain with finite state space S and is independent of the Brownian motion B. Under these conditions, existence and uniqueness is guaranteed.
  14. 14. Introduction Mathematical Results Application to Financial Market Models Extensions and future work S.D.E with Markovian switching We study the stochastic differential equation with Markovian switching dX (t) = f (X (t), Y (t), t) dt + g (X (t), Y (t), t) dB(t) (3) where X (0) = x0 , f , g : R × S × [0, ∞) → R are continuous functions obeying local Lipschitz continuity and linear growth conditions, Y is an irreducible continuous-time Markov chain with finite state space S and is independent of the Brownian motion B. Under these conditions, existence and uniqueness is guaranteed.
  15. 15. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Main results Theorem 1 Let X be the unique continuous solution satisfying dX (t) = f (X (t), Y (t), t) dt + g (X (t), Y (t), t) dB(t). If there exist ρ > 0 and real numbers K1 and K2 such that ∀ (x, y , t) ∈ R × S × [0, ∞) xf (x, y , t) ≤ ρ, 0 < K2 ≤ g 2 (x, y , t) ≤ K1 then X satisfies |X (t)| lim sup √ ≤ K1 , a.s. t→∞ 2t log log t
  16. 16. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Main results Theorem 1 Let X be the unique continuous solution satisfying dX (t) = f (X (t), Y (t), t) dt + g (X (t), Y (t), t) dB(t). If there exist ρ > 0 and real numbers K1 and K2 such that ∀ (x, y , t) ∈ R × S × [0, ∞) xf (x, y , t) ≤ ρ, 0 < K2 ≤ g 2 (x, y , t) ≤ K1 then X satisfies |X (t)| lim sup √ ≤ K1 , a.s. t→∞ 2t log log t
  17. 17. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Main results Theorem 1 Let X be the unique continuous solution satisfying dX (t) = f (X (t), Y (t), t) dt + g (X (t), Y (t), t) dB(t). If there exist ρ > 0 and real numbers K1 and K2 such that ∀ (x, y , t) ∈ R × S × [0, ∞) xf (x, y , t) ≤ ρ, 0 < K2 ≤ g 2 (x, y , t) ≤ K1 then X satisfies |X (t)| lim sup √ ≤ K1 , a.s. t→∞ 2t log log t
  18. 18. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Main results Theorem 1 continued If, moreover, there exists an L ∈ R such that xf (x, y , t) 1 inf 2 (x, y , t) =: L > − (x,y ,t)∈R×S×[0,∞) g 2 Then X satisfies |X (t)| lim sup √ ≥ K2 , a.s. t→∞ 2t log log t
  19. 19. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Outline of proof The proofs rely on time-change and stochastic comparison arguments, constructing upper and lower bounds on |X | which, under an appropriate change of time and scale, are stationary processes whose dynamics are not determined by Y . The large deviations of these processes are determined by means of a classical theorem of Motoo.
  20. 20. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Outline of proof The proofs rely on time-change and stochastic comparison arguments, constructing upper and lower bounds on |X | which, under an appropriate change of time and scale, are stationary processes whose dynamics are not determined by Y . The large deviations of these processes are determined by means of a classical theorem of Motoo.
  21. 21. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Outline of proof The proofs rely on time-change and stochastic comparison arguments, constructing upper and lower bounds on |X | which, under an appropriate change of time and scale, are stationary processes whose dynamics are not determined by Y . The large deviations of these processes are determined by means of a classical theorem of Motoo.
  22. 22. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Outline of proof The proofs rely on time-change and stochastic comparison arguments, constructing upper and lower bounds on |X | which, under an appropriate change of time and scale, are stationary processes whose dynamics are not determined by Y . The large deviations of these processes are determined by means of a classical theorem of Motoo.
  23. 23. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Outline 1 Introduction 2 Mathematical Results 3 Application to Financial Market Models 4 Extensions and future work
  24. 24. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Security Price Model The large deviation results are now applied to a security price model, where the security price S obeys dS(t) = µS(t)dt + S(t)dX (t), t≥0 Here we consider the special case dX (t) = f (X (t), Y (t), t) dt + γ(Y (t)) dB(t), t≥0 with X (0) = 0 and γ : S → R {0}. Under the conditions xf (x, y , t) sup ≤ ρ and (x,y ,t)∈R×S×[0,∞) γ 2 (y ) xf (x, y , t) 1 inf 2 (y ) >− , (x,y ,t)∈R×S×[0,∞) γ 2
  25. 25. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Security Price Model The large deviation results are now applied to a security price model, where the security price S obeys dS(t) = µS(t)dt + S(t)dX (t), t≥0 Here we consider the special case dX (t) = f (X (t), Y (t), t) dt + γ(Y (t)) dB(t), t≥0 with X (0) = 0 and γ : S → R {0}. Under the conditions xf (x, y , t) sup ≤ ρ and (x,y ,t)∈R×S×[0,∞) γ 2 (y ) xf (x, y , t) 1 inf 2 (y ) >− , (x,y ,t)∈R×S×[0,∞) γ 2
  26. 26. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Security Price Model The large deviation results are now applied to a security price model, where the security price S obeys dS(t) = µS(t)dt + S(t)dX (t), t≥0 Here we consider the special case dX (t) = f (X (t), Y (t), t) dt + γ(Y (t)) dB(t), t≥0 with X (0) = 0 and γ : S → R {0}. Under the conditions xf (x, y , t) sup ≤ ρ and (x,y ,t)∈R×S×[0,∞) γ 2 (y ) xf (x, y , t) 1 inf 2 (y ) >− , (x,y ,t)∈R×S×[0,∞) γ 2
  27. 27. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Fluctuations in the Markovian switching model Theorem 2 Let S be the unique continuous process governed by dS(t) = µS(t)dt + S(t)dX (t), t≥0 with S(0) = s0 > 0, where X is defined on the previous slide. Then: 1 log S(t) 1 2 lim = µ − σ∗ , a.s. t→∞ t 2 2 t | log S(t) − (µt − 1 0 γ 2 (Y (s))ds)| 2 lim sup √ = σ∗ t→∞ 2t log log t where σ∗ = j∈S γ 2 (j)πj , and π is the stationary probability 2 distribution of Y .
  28. 28. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Fluctuations in the Markovian switching model Theorem 2 Let S be the unique continuous process governed by dS(t) = µS(t)dt + S(t)dX (t), t≥0 with S(0) = s0 > 0, where X is defined on the previous slide. Then: 1 log S(t) 1 2 lim = µ − σ∗ , a.s. t→∞ t 2 2 t | log S(t) − (µt − 1 0 γ 2 (Y (s))ds)| 2 lim sup √ = σ∗ t→∞ 2t log log t where σ∗ = j∈S γ 2 (j)πj , and π is the stationary probability 2 distribution of Y .
  29. 29. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Fluctuations in the Markovian switching model Theorem 2 Let S be the unique continuous process governed by dS(t) = µS(t)dt + S(t)dX (t), t≥0 with S(0) = s0 > 0, where X is defined on the previous slide. Then: 1 log S(t) 1 2 lim = µ − σ∗ , a.s. t→∞ t 2 2 t | log S(t) − (µt − 1 0 γ 2 (Y (s))ds)| 2 lim sup √ = σ∗ t→∞ 2t log log t where σ∗ = j∈S γ 2 (j)πj , and π is the stationary probability 2 distribution of Y .
  30. 30. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Comments Despite the presence of the Markov process Y (which introduces regime shifts), we have shown that the new market model obeys the same long-term properties of standard geometric Brownian motion models. However, the analysis is now more complicated because the increments are neither stationary nor Gaussian, and it is not possible to obtain an explicit solution. The incorporation of investor sentiment into the model in this manner is one of the important motivations behind the discipline of behavioural finance.
  31. 31. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Comments Despite the presence of the Markov process Y (which introduces regime shifts), we have shown that the new market model obeys the same long-term properties of standard geometric Brownian motion models. However, the analysis is now more complicated because the increments are neither stationary nor Gaussian, and it is not possible to obtain an explicit solution. The incorporation of investor sentiment into the model in this manner is one of the important motivations behind the discipline of behavioural finance.
  32. 32. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Comments Despite the presence of the Markov process Y (which introduces regime shifts), we have shown that the new market model obeys the same long-term properties of standard geometric Brownian motion models. However, the analysis is now more complicated because the increments are neither stationary nor Gaussian, and it is not possible to obtain an explicit solution. The incorporation of investor sentiment into the model in this manner is one of the important motivations behind the discipline of behavioural finance.
  33. 33. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Outline 1 Introduction 2 Mathematical Results 3 Application to Financial Market Models 4 Extensions and future work
  34. 34. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Extensions of the market model We can also get upper and lower bounds on both positive and negative large fluctuations under the assumption that the drift is integrable. A result can be proven about the large fluctuations of the incremental returns when the diffusion coefficient depends on X , Y and t. Can be extended to the case in which the diffusion coefficient in X depends not only on the Markovian switching term but also on a delay term, once that diffusion coefficient remains bounded.
  35. 35. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Extensions of the market model We can also get upper and lower bounds on both positive and negative large fluctuations under the assumption that the drift is integrable. A result can be proven about the large fluctuations of the incremental returns when the diffusion coefficient depends on X , Y and t. Can be extended to the case in which the diffusion coefficient in X depends not only on the Markovian switching term but also on a delay term, once that diffusion coefficient remains bounded.
  36. 36. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Extensions of the market model We can also get upper and lower bounds on both positive and negative large fluctuations under the assumption that the drift is integrable. A result can be proven about the large fluctuations of the incremental returns when the diffusion coefficient depends on X , Y and t. Can be extended to the case in which the diffusion coefficient in X depends not only on the Markovian switching term but also on a delay term, once that diffusion coefficient remains bounded.
  37. 37. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Future work We are currently working on a general d-dimensional S.D.E dX (t) = f (X (t))dt + ΣdB(t) where f is a d × 1 vector, Σ is a d × r matrix and B is an r -dimensional standard Brownian motion. We propose to capture the degree of non-linearity in f by the 1-dimensional scalar function φ. Use techniques from this talk to determine the large fluctuations of X in terms of φ. Can quantify the relation between the degree of mean-reversion and the size of the fluctuations.
  38. 38. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Future work We are currently working on a general d-dimensional S.D.E dX (t) = f (X (t))dt + ΣdB(t) where f is a d × 1 vector, Σ is a d × r matrix and B is an r -dimensional standard Brownian motion. We propose to capture the degree of non-linearity in f by the 1-dimensional scalar function φ. Use techniques from this talk to determine the large fluctuations of X in terms of φ. Can quantify the relation between the degree of mean-reversion and the size of the fluctuations.
  39. 39. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Future work We are currently working on a general d-dimensional S.D.E dX (t) = f (X (t))dt + ΣdB(t) where f is a d × 1 vector, Σ is a d × r matrix and B is an r -dimensional standard Brownian motion. We propose to capture the degree of non-linearity in f by the 1-dimensional scalar function φ. Use techniques from this talk to determine the large fluctuations of X in terms of φ. Can quantify the relation between the degree of mean-reversion and the size of the fluctuations.
  40. 40. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Future work We are currently working on a general d-dimensional S.D.E dX (t) = f (X (t))dt + ΣdB(t) where f is a d × 1 vector, Σ is a d × r matrix and B is an r -dimensional standard Brownian motion. We propose to capture the degree of non-linearity in f by the 1-dimensional scalar function φ. Use techniques from this talk to determine the large fluctuations of X in terms of φ. Can quantify the relation between the degree of mean-reversion and the size of the fluctuations.
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