Dai2011HMM (NMDEV 1.2.9 API)

java.lang.Object
- tech.nmfin.trend.dai2011.Dai2011HMM

```
public class Dai2011HMM
extends Object
```
Creates a two-state Geometric Brownian Motion with a constant volatility. There are five parameters: two drifts, one volatility, and two transition intensities.

See Also:

Min Dai, Qing Zhang and Qiji Jim Zhu, "Optimal Trend Following Trading Rules," 2011.

Nested Class Summary

Nested Classes
Modifier and Type Class and Description

static class Dai2011HMM.CalibrationParam

static class Dai2011HMM.ModelParam

Nested Classes
Modifier and Type	Class and Description
`static class`	`Dai2011HMM.CalibrationParam`
`static class`	`Dai2011HMM.ModelParam`

Constructor Summary

Constructors
Constructor and Description
`Dai2011HMM(Dai2011HMM.ModelParam model)`
`Dai2011HMM(Dai2011HMM that)` Copy constructor.
`Dai2011HMM(double mu1, double mu2, double lambda1, double lambda2, double sigma)` Constructs a two-state Markov switching Geometric Brownian Motion.

Method Summary

All Methods Instance Methods Concrete Methods
Modifier and Type	Method and Description
`HMMRNG`	`getHMM()`
`Dai2011HMM`	`getReciprocal()` Gets the reciprocal state switching GBM.
`double`	`lambda1()` Gets the transition intensity from bull market to bear market.
`double`	`lambda2()` Gets the transition intensity from bear market to bull market.
`double`	`mu1()` Gets the drift in the bull market.
`double`	`mu2()` Gets the drift in the bear market.
`double`	`nextP(double pt, double St, double St1)` Gets the evolution of \(p_{t}\), the conditional probability of being in an uptrend given all information, i.e., \(p_t = P(\alpha_t = 1 \| \mathcal{F}_t)\).
`double`	`nextX(double xt, double St, double newSt)` Gets the evolution of \(x_{t}\), logit of the conditional probability from (0, 1) onto \((-\infty, +\infty)\), i.e., \(x_t = \log{\frac{p_t}{1-p_t}}\).
`double`	`sigma()` Gets the diffusion volatility.
`String`	`toString()`

Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, wait, wait, wait

- Constructor Detail
  - Dai2011HMM
```
public Dai2011HMM(double mu1,
                  double mu2,
                  double lambda1,
                  double lambda2,
                  double sigma)
```
    Constructs a two-state Markov switching Geometric Brownian Motion. \(\frac{dS_t}{S_t} = \mu_i dt + \sigma dW_t,~i = 0, 1.\)
    
    Parameters:
    
    mu1 - bull market return
    
    mu2 - bear market return
    
    lambda1 - bull-to-bear intensity
    
    lambda2 - bear-to-bull intensity
    
    sigma - the volatility, assume to be constant
  - Dai2011HMM
```
public Dai2011HMM(Dai2011HMM.ModelParam model)
```
  - Dai2011HMM
```
public Dai2011HMM(Dai2011HMM that)
```
    Copy constructor.
    
    Parameters:
    
    that - a two-state Markov switching GBM instance
- Method Detail
  - lambda1
```
public double lambda1()
```
    Gets the transition intensity from bull market to bear market.
    
    Returns:
    
    the bull to bear transition intensity
  - lambda2
```
public double lambda2()
```
    Gets the transition intensity from bear market to bull market.
    
    Returns:
    
    the bear to bull transition intensity
  - mu1
```
public double mu1()
```
    Gets the drift in the bull market.
    
    Returns:
    
    the drift in the bull market
  - mu2
```
public double mu2()
```
    Gets the drift in the bear market.
    
    Returns:
    
    the drift in the bear market
  - sigma
```
public double sigma()
```
    Gets the diffusion volatility.
    
    Returns:
    
    the volatility
  - getReciprocal
```
public Dai2011HMM getReciprocal()
```
    Gets the reciprocal state switching GBM. \(S_t d\frac{1}{S_t} = (\sigma^2 -\mu_i) dt - \sigma dW_t,~i = 1, 2.\) Note that the indices 0 and 1 are switched.
    
    Returns:
    
    the reciprocal GBM
    
    See Also:
    
    "eq. 15"
  - nextP
```
public double nextP(double pt,
                    double St,
                    double St1)
```
    Gets the evolution of \(p_{t}\), the conditional probability of being in an uptrend given all information, i.e., \(p_t = P(\alpha_t = 1 | \mathcal{F}_t)\).
    See (eq.13) in "Optimal Trend Following Trading Rules", M. Dai, 2011. The evolution of \(p_t\) is \[ dp_t = g(p_t) dt + \frac{(\mu_1-\mu_2)p_t(1-p_t)}{\sigma^2} d\log{S_t}, \] where \[ g(p_t) = -\lambda_1 p_t + \lambda_2 (1-p_t) - \frac{(\mu_1-\mu_2)p_t(1-p_t)((\mu_1-\mu_2)p_t+\mu_2-\sigma^2/2)}{\sigma^2}. \]
    In discrete form, it becomes \[ p_{t+1} = \min{\left(1,\max{\left(0, p_t + g(p_t)dt + \frac{(\mu_1-\mu_2)p_t(1-p_t)}{\sigma^2}\log{S_{t+1}/S_t}\right)}\right)}. \]
    
    Parameters:
    
    pt - the previous conditional probability
    
    St - the previous stock price
    
    St1 - the latest stock price
    
    Returns:
    
    the evolved conditional probability
    
    See Also:
    
    "eqs. 13, 14"
  - nextX
```
public double nextX(double xt,
                    double St,
                    double newSt)
```
    Gets the evolution of \(x_{t}\), logit of the conditional probability from (0, 1) onto \((-\infty, +\infty)\), i.e., \(x_t = \log{\frac{p_t}{1-p_t}}\).
    The evolution of \(x_t\) can be derived as \[ dx_t = -(\frac{\mu_1-\mu_2}{\sigma^2} (\mu_1-\frac{\sigma^2}{2}) + \lambda_1 - \lambda_2) dt -(\lambda_1 e^{x_t} - \lambda_2 e^{-x_t} + \frac{(\mu_1-\mu_2)^2}{\sigma^2} \frac{1}{1+e^{x_t}}) dt + \frac{\mu_1-\mu_2}{\sigma^2} d\log{S_t}. \]
    
    Parameters:
    
    xt - the previous value
    
    St - the previous stock price
    
    newSt - the latest stock price
    
    Returns:
    
    the evolved value
  - getHMM
```
public HMMRNG getHMM()
```
  - toString
```
public String toString()
```
    Overrides:
    
    toString in class Object

Class Dai2011HMM

Nested Class Summary

Constructor Summary

Method Summary

Methods inherited from class java.lang.Object

Constructor Detail

Dai2011HMM

Dai2011HMM

Dai2011HMM

Method Detail

lambda1

lambda2

mu1

mu2

sigma

getReciprocal

nextP

nextX

getHMM

toString