Brownian Motion and Ito s Lemma

Transcription

1 Brownian Motion and Ito s Lemma 1 Introduction 2 Geometric Brownian Motion 3 Ito s Product Rule 4 Some Properties of the Stochastic Integral 5 Correlated Stock Prices 6 The Ornstein-Uhlenbeck Process

2 Brownian Motion and Ito s Lemma 1 Introduction 2 Geometric Brownian Motion 3 Ito s Product Rule 4 Some Properties of the Stochastic Integral 5 Correlated Stock Prices 6 The Ornstein-Uhlenbeck Process

3 Samuelson s Model The Black-Scholes Assumption About Stock Prices The original paper by Black and Scholes assumes that the price of the underlying asset is a stochastic process {S t } which is solves the following stochastic differential equation (in the differential form): where ds t = S t [α dt + σ dz t ] α... denotes the continuously compounded expected return on the stock; σ... denotes the volatility; {Z t }... is a standard Brownian motion In other words, {S t } is a geometric Brownian motion

4 Samuelson s Model The Black-Scholes Assumption About Stock Prices The original paper by Black and Scholes assumes that the price of the underlying asset is a stochastic process {S t } which is solves the following stochastic differential equation (in the differential form): where ds t = S t [α dt + σ dz t ] α... denotes the continuously compounded expected return on the stock; σ... denotes the volatility; {Z t }... is a standard Brownian motion In other words, {S t } is a geometric Brownian motion

5 Samuelson s Model The Black-Scholes Assumption About Stock Prices The original paper by Black and Scholes assumes that the price of the underlying asset is a stochastic process {S t } which is solves the following stochastic differential equation (in the differential form): where ds t = S t [α dt + σ dz t ] α... denotes the continuously compounded expected return on the stock; σ... denotes the volatility; {Z t }... is a standard Brownian motion In other words, {S t } is a geometric Brownian motion

6 Samuelson s Model The Black-Scholes Assumption About Stock Prices The original paper by Black and Scholes assumes that the price of the underlying asset is a stochastic process {S t } which is solves the following stochastic differential equation (in the differential form): where ds t = S t [α dt + σ dz t ] α... denotes the continuously compounded expected return on the stock; σ... denotes the volatility; {Z t }... is a standard Brownian motion In other words, {S t } is a geometric Brownian motion

7 Samuelson s Model The Black-Scholes Assumption About Stock Prices The original paper by Black and Scholes assumes that the price of the underlying asset is a stochastic process {S t } which is solves the following stochastic differential equation (in the differential form): where ds t = S t [α dt + σ dz t ] α... denotes the continuously compounded expected return on the stock; σ... denotes the volatility; {Z t }... is a standard Brownian motion In other words, {S t } is a geometric Brownian motion

8 On the distribution of the stock price at a given time Recall the example from class to conclude that ( ln(s t ) N ln(s ) + (α 1 ) 2 )σ2 )t, σ 2 t, for every t In other words, at any time t the stock-price random variable S t is log-normal The above means that we assume that the continuously compounded returns are modeled by a normally distributed random variable.

9 On the distribution of the stock price at a given time Recall the example from class to conclude that ( ln(s t ) N ln(s ) + (α 1 ) 2 )σ2 )t, σ 2 t, for every t In other words, at any time t the stock-price random variable S t is log-normal The above means that we assume that the continuously compounded returns are modeled by a normally distributed random variable.

10 On the distribution of the stock price at a given time Recall the example from class to conclude that ( ln(s t ) N ln(s ) + (α 1 ) 2 )σ2 )t, σ 2 t, for every t In other words, at any time t the stock-price random variable S t is log-normal The above means that we assume that the continuously compounded returns are modeled by a normally distributed random variable.

11 Brownian Motion and Ito s Lemma 1 Introduction 2 Geometric Brownian Motion 3 Ito s Product Rule 4 Some Properties of the Stochastic Integral 5 Correlated Stock Prices 6 The Ornstein-Uhlenbeck Process

12 More Heuristics: Relative Importance of the Drift and Noise Recall the SDE which defines the geometric B.M. ds t = S t [α dt + σ dz t ] Consider a time period of length h and the ratio of the per-period standard deviation to the per-period drift, i.e., σs t h αs t h = σ α h For h infinitesimaly small the above ration diverges. We may interpret this by saying that for short time-periods the random component of the process {S t } is dominant. As the observed period grows longer, the drift (mean) of the stochastic process {S t } has a greater effect

13 More Heuristics: Relative Importance of the Drift and Noise Recall the SDE which defines the geometric B.M. ds t = S t [α dt + σ dz t ] Consider a time period of length h and the ratio of the per-period standard deviation to the per-period drift, i.e., σs t h αs t h = σ α h For h infinitesimaly small the above ration diverges. We may interpret this by saying that for short time-periods the random component of the process {S t } is dominant. As the observed period grows longer, the drift (mean) of the stochastic process {S t } has a greater effect

14 More Heuristics: Relative Importance of the Drift and Noise Recall the SDE which defines the geometric B.M. ds t = S t [α dt + σ dz t ] Consider a time period of length h and the ratio of the per-period standard deviation to the per-period drift, i.e., σs t h αs t h = σ α h For h infinitesimaly small the above ration diverges. We may interpret this by saying that for short time-periods the random component of the process {S t } is dominant. As the observed period grows longer, the drift (mean) of the stochastic process {S t } has a greater effect

15 More Heuristics: Relative Importance of the Drift and Noise Recall the SDE which defines the geometric B.M. ds t = S t [α dt + σ dz t ] Consider a time period of length h and the ratio of the per-period standard deviation to the per-period drift, i.e., σs t h αs t h = σ α h For h infinitesimaly small the above ration diverges. We may interpret this by saying that for short time-periods the random component of the process {S t } is dominant. As the observed period grows longer, the drift (mean) of the stochastic process {S t } has a greater effect

16 More Heuristics: Relative Importance of the Drift and Noise Recall the SDE which defines the geometric B.M. ds t = S t [α dt + σ dz t ] Consider a time period of length h and the ratio of the per-period standard deviation to the per-period drift, i.e., σs t h αs t h = σ α h For h infinitesimaly small the above ration diverges. We may interpret this by saying that for short time-periods the random component of the process {S t } is dominant. As the observed period grows longer, the drift (mean) of the stochastic process {S t } has a greater effect

17 Brownian Motion and Ito s Lemma 1 Introduction 2 Geometric Brownian Motion 3 Ito s Product Rule 4 Some Properties of the Stochastic Integral 5 Correlated Stock Prices 6 The Ornstein-Uhlenbeck Process

18 Theorem [Ito s Product Rule] Consider two Ito proocesses {X t } and {Y t }. Then d(x t Y t ) = X t dy t + Y t dx t + dx t dy t. Note: We calculate the last term using the multiplication table with dt s and db t s

19 Theorem [Ito s Product Rule] Consider two Ito proocesses {X t } and {Y t }. Then d(x t Y t ) = X t dy t + Y t dx t + dx t dy t. Note: We calculate the last term using the multiplication table with dt s and db t s

20 Brownian Motion and Ito s Lemma 1 Introduction 2 Geometric Brownian Motion 3 Ito s Product Rule 4 Some Properties of the Stochastic Integral 5 Correlated Stock Prices 6 The Ornstein-Uhlenbeck Process

21 Martingality Under some integrability and regularity conditions on the integrand, the process {Y t } defined by Y t = ν s db S, where {B s } is a standard B.M. is a martingale. In particular [ ] E[Y t ] = E ν s db S =, for every t

22 Martingality Under some integrability and regularity conditions on the integrand, the process {Y t } defined by Y t = ν s db S, where {B s } is a standard B.M. is a martingale. In particular [ ] E[Y t ] = E ν s db S =, for every t

23 Ito Isometry Under some integrability and regularity conditions on the integrand ν, let us define the process {Y t } as Y t = where {B s } is a standard B.M. Then ν s db S, E[Yt 2 ] = E[ νs 2 ds]

24 Continuity Under some integrability and regularity conditions on the integrand ν, let us define the process {Y t } as Y t = ν s db S, where {B s } is a standard B.M. Then the paths of {Y t } are (almost surely) continuous.

25 Linearity Moreover, for a constant c, we have that cy t = (cν s ) db S, Additionally, if {A t } is a stochastic process given as A t = ξ s db s, for an integrand {ξ t } conforming to the integrability and regularity conditions necessary for the sotchastic integral to be well-defined, then Y t ± A t = (ν s ± ξ S ) db S

26 Linearity Moreover, for a constant c, we have that cy t = (cν s ) db S, Additionally, if {A t } is a stochastic process given as A t = ξ s db s, for an integrand {ξ t } conforming to the integrability and regularity conditions necessary for the sotchastic integral to be well-defined, then Y t ± A t = (ν s ± ξ S ) db S

27 Brownian Motion and Ito s Lemma 1 Introduction 2 Geometric Brownian Motion 3 Ito s Product Rule 4 Some Properties of the Stochastic Integral 5 Correlated Stock Prices 6 The Ornstein-Uhlenbeck Process

28 The Set-Up Consider two stock prices {S t } and {Q t }. Suppose that they satisfy the following system of SDEs ds t = α S dt + σ s dw t S t dq [ t = α Q dt + σ Q ρdw t + ] 1 ρ Q 2 dw t t where ρ [ 1, 1], α S, α Q, σ S > and σ Q > are given constants and {W t } and {W t } are independent standard Brownian motions. Theorem: If W and W are independent, then dw t dw t =. We can now add the above to our multiplication table.

29 The Set-Up Consider two stock prices {S t } and {Q t }. Suppose that they satisfy the following system of SDEs ds t = α S dt + σ s dw t S t dq [ t = α Q dt + σ Q ρdw t + ] 1 ρ Q 2 dw t t where ρ [ 1, 1], α S, α Q, σ S > and σ Q > are given constants and {W t } and {W t } are independent standard Brownian motions. Theorem: If W and W are independent, then dw t dw t =. We can now add the above to our multiplication table.

30 The Set-Up Consider two stock prices {S t } and {Q t }. Suppose that they satisfy the following system of SDEs ds t = α S dt + σ s dw t S t dq [ t = α Q dt + σ Q ρdw t + ] 1 ρ Q 2 dw t t where ρ [ 1, 1], α S, α Q, σ S > and σ Q > are given constants and {W t } and {W t } are independent standard Brownian motions. Theorem: If W and W are independent, then dw t dw t =. We can now add the above to our multiplication table.

31 Define A New Standard Brownian Motion W t = ρw t + 1 ρ 2 W t. { W t } is an almost everywhere continuous process with W = One can prove that W is a standard Brownian motion. Now, we can write dq t Q t = α Q dt + σ Q d W t According to Ito s Product Rule and the fact that W and W are independent d(w t Wt ) = W t d W t + W t dw t + ρ dt In the integral form the above reads as W t W t = W s d W s + W s dw s + ρt

32 Define A New Standard Brownian Motion W t = ρw t + 1 ρ 2 W t. { W t } is an almost everywhere continuous process with W = One can prove that W is a standard Brownian motion. Now, we can write dq t Q t = α Q dt + σ Q d W t According to Ito s Product Rule and the fact that W and W are independent d(w t Wt ) = W t d W t + W t dw t + ρ dt In the integral form the above reads as W t W t = W s d W s + W s dw s + ρt

33 Define A New Standard Brownian Motion W t = ρw t + 1 ρ 2 W t. { W t } is an almost everywhere continuous process with W = One can prove that W is a standard Brownian motion. Now, we can write dq t Q t = α Q dt + σ Q d W t According to Ito s Product Rule and the fact that W and W are independent d(w t Wt ) = W t d W t + W t dw t + ρ dt In the integral form the above reads as W t W t = W s d W s + W s dw s + ρt

34 Define A New Standard Brownian Motion W t = ρw t + 1 ρ 2 W t. { W t } is an almost everywhere continuous process with W = One can prove that W is a standard Brownian motion. Now, we can write dq t Q t = α Q dt + σ Q d W t According to Ito s Product Rule and the fact that W and W are independent d(w t Wt ) = W t d W t + W t dw t + ρ dt In the integral form the above reads as W t W t = W s d W s + W s dw s + ρt

35 Define A New Standard Brownian Motion W t = ρw t + 1 ρ 2 W t. { W t } is an almost everywhere continuous process with W = One can prove that W is a standard Brownian motion. Now, we can write dq t Q t = α Q dt + σ Q d W t According to Ito s Product Rule and the fact that W and W are independent d(w t Wt ) = W t d W t + W t dw t + ρ dt In the integral form the above reads as W t W t = W s d W s + W s dw s + ρt

36 On the Correlation of the two Brownian Motions For covenience, let us repeat that W t Wt = W s d W s + W s dw s + ρt Using the fact that the stochastic integral is a martingale, for every t, we have E[W t Wt ] = ρt. Recalling that the quadratic variaton of any standard B.M. is t, we see that ρ is the correlation between the Brownian motions W and W

37 On the Correlation of the two Brownian Motions For covenience, let us repeat that W t Wt = W s d W s + W s dw s + ρt Using the fact that the stochastic integral is a martingale, for every t, we have E[W t Wt ] = ρt. Recalling that the quadratic variaton of any standard B.M. is t, we see that ρ is the correlation between the Brownian motions W and W

38 On the Correlation of the two Brownian Motions For covenience, let us repeat that W t Wt = W s d W s + W s dw s + ρt Using the fact that the stochastic integral is a martingale, for every t, we have E[W t Wt ] = ρt. Recalling that the quadratic variaton of any standard B.M. is t, we see that ρ is the correlation between the Brownian motions W and W

39 Brownian Motion and Ito s Lemma 1 Introduction 2 Geometric Brownian Motion 3 Ito s Product Rule 4 Some Properties of the Stochastic Integral 5 Correlated Stock Prices 6 The Ornstein-Uhlenbeck Process

40 The Ornstein-Uhlenbeck Process Along with the processes we discussed so far, consider a stochastic process {X t } which satisfies dx t = [α x t ] dt + σ dz t, where α and σ are given constants and {Z t } is a standard Brownian motion. The process above is called the mean reverting process (Why??) In particular, if we set α =, the resulting process is called the Ornstein-Uhlenbeck process

41 The Ornstein-Uhlenbeck Process Along with the processes we discussed so far, consider a stochastic process {X t } which satisfies dx t = [α x t ] dt + σ dz t, where α and σ are given constants and {Z t } is a standard Brownian motion. The process above is called the mean reverting process (Why??) In particular, if we set α =, the resulting process is called the Ornstein-Uhlenbeck process

42 The Ornstein-Uhlenbeck Process Along with the processes we discussed so far, consider a stochastic process {X t } which satisfies dx t = [α x t ] dt + σ dz t, where α and σ are given constants and {Z t } is a standard Brownian motion. The process above is called the mean reverting process (Why??) In particular, if we set α =, the resulting process is called the Ornstein-Uhlenbeck process