Method Documentation
Expected path

16min
This method implements a mechanism for incorporating upper and lower bounds into the forecast of a saving plan, thus effectively capturing the inherent volatility of the respective plan. The calculation is based on the solution bounds derived from stochastic differential equations. Throughout this documentation, we assume that the initial expected return and risk of the portfolio are denoted as µ and σ, respectively, and are provided as inputs.

In this context, υ(t) represents the future price projection of a portfolio, characterized by an expected return following a normal distribution with parameters (µ, σ). The variable t denotes the time of projection, while υ0 denotes the initial value of the portfolio at t = 0.
Here, υ is a stochastic process which satisfies Itô’s lemma [1]:

\begin{align*}
d\upsilon(t) & =\frac{\partial\upsilon}{\partial t}dt+\frac{\partial\upsilon}{\partial w}dw+\frac{1}{2}\frac{\partial^{2}\upsilon}{\partial w^{2}}(dw)^{2}
\end{align*}
\begin{align*}
d\upsilon(t) & =\frac{\partial\upsilon}{\partial t}dt+\frac{\partial\upsilon}{\partial w}dw+\frac{1}{2}\frac{\partial^{2}\upsilon}{\partial w^{2}}(dw)^{2}
\end{align*}
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where w is a Wiener process [2], also known as a Brownian motion, characterized by a normal distribution:
\mathcal{N}(0, \sqrt{dt})

\mathcal{N}(0, \sqrt{dt})
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In this case: 
\left(dw\right)^2 = dt

\left(dw\right)^2 = dt
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Projection with cashflow: 
For a portfolio with initial investment υ0, the solution by
\nu(t) = \nu_{0} e^{\left(\mu - \frac{1}{2}\sigma^{2}\right)t + \sigma W(t)}

\nu(t) = \nu_{0} e^{\left(\mu - \frac{1}{2}\sigma^{2}\right)t + \sigma W(t)}
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using Itô’s lemma follows geometric Brownian motion
d\nu = \mu \nu dt + \sigma \nu dw

d\nu = \mu \nu dt + \sigma \nu dw
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The differential form dυ describes the change of investment value. For a given confidence interval Z, using the solution, the bounds to υ are given by
\begin{align*}
\upsilon^{U}(t) & =\upsilon_{0}e^{\left(\mu-\frac{1}{2}\sigma^{2}\right)t+\sigma Z\sqrt{t}}\\
\upsilon^{L}(t) & =\upsilon_{0}e^{\left(\mu-\frac{1}{2}\sigma^{2}\right)t-\sigma Z\sqrt{t}}
\end{align*}
\begin{align*}
\upsilon^{U}(t) & =\upsilon_{0}e^{\left(\mu-\frac{1}{2}\sigma^{2}\right)t+\sigma Z\sqrt{t}}\\
\upsilon^{L}(t) & =\upsilon_{0}e^{\left(\mu-\frac{1}{2}\sigma^{2}\right)t-\sigma Z\sqrt{t}}
\end{align*}
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where:
\begin{align*}
\upsilon^{U}(t) & : \text{the upper bound of the confidence interval for } \upsilon \text{ at time } t, \\
\upsilon^{L}(t) & : \text{the lower bound of the confidence interval for } \upsilon
\text{ at time } t. 
\end{align*}

\begin{align*}
\upsilon^{U}(t) & : \text{the upper bound of the confidence interval for } \upsilon \text{ at time } t, \\
\upsilon^{L}(t) & : \text{the lower bound of the confidence interval for } \upsilon
\text{ at time } t. 
\end{align*}
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Projection without cashflow
In this section, we consider a portfolio with initial investment υ0 and future cashflows given by
\left\{ (t_{c,1}, s_{1}), (t_{c,2}, s_{2}), \ldots, (t_{c,n}, s_{n}) \right\}

\left\{ (t_{c,1}, s_{1}), (t_{c,2}, s_{2}), \ldots, (t_{c,n}, s_{n}) \right\}
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where s_i >=0 is the amount of cashflow at each instant time t_c,i>=0. To represent these discrete values in continuous time, we use the continuous function of cashflows given by
c_i(t) = s_i \delta_{t_i}(t)

c_i(t) = s_i \delta_{t_i}(t)
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where δ is the Dirac delta distribution [3].
For this kind of portfolio with initial investment and cashflows, to calculate the evolution of portfolio value, we adapt the method in [4] where the solution given by
\upsilon(t) = \upsilon_{0}e^{(\mu-\frac{1}{2}\sigma^{2})t+\sigma w(t)} + \sum_{i}\int_{t_{c,i}}^{t}c_{i}(\tau)e^{(\mu-\frac{1}{2}\sigma^{2})(t-\tau)+\sigma(w(t)-w(\tau))}d\tau

\upsilon(t) = \upsilon_{0}e^{(\mu-\frac{1}{2}\sigma^{2})t+\sigma w(t)} + \sum_{i}\int_{t_{c,i}}^{t}c_{i}(\tau)e^{(\mu-\frac{1}{2}\sigma^{2})(t-\tau)+\sigma(w(t)-w(\tau))}d\tau
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follows
d\upsilon = (\mu\upsilon + \sum_{i}c_{i})dt + \sigma\upsilon dw.

d\upsilon = (\mu\upsilon + \sum_{i}c_{i})dt + \sigma\upsilon dw.
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To calculate the lower and upper bound of the solution for a given confidence interval Z, we use numerical approximation as follow. The numerical time step is denoted by ∆t and the index of step is given by
c_{i}(t) = s_{i}\delta_{t_{i}}(t)

c_{i}(t) = s_{i}\delta_{t_{i}}(t)
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The Dirac delta is approximated by
\delta_{t_{c,i}}(k\Delta t) \approx \begin{cases}
\frac{1}{\Delta t}, & \text{for } k_{c,i} \leq k \leq k_{c,i}+1 \\
0, & \text{otherwise}
\end{cases}

\delta_{t_{c,i}}(k\Delta t) \approx \begin{cases}
\frac{1}{\Delta t}, & \text{for } k_{c,i} \leq k \leq k_{c,i}+1 \\
0, & \text{otherwise}
\end{cases}
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where
k_{c,i} = \frac{t_{c,i}}{\Delta t}.

k_{c,i} = \frac{t_{c,i}}{\Delta t}.
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Here, we can denote the set which contains the step index where cashflow occurs
\Omega_{c} = \left\{ \frac{t_{c,1}}{\Delta t}, \frac{t_{c,2}}{\Delta t}, \ldots, \frac{t_{c,n}}{\Delta t} \right\}.

\Omega_{c} = \left\{ \frac{t_{c,1}}{\Delta t}, \frac{t_{c,2}}{\Delta t}, \ldots, \frac{t_{c,n}}{\Delta t} \right\}.
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The approximation to the lower and upper bound of the solution can be described by
\begin{align*}
\upsilon^{U}(k\Delta t) \approx & \upsilon_{0}e^{\left(\mu-\frac{1}{2}\sigma^{2}\right)k\Delta t+\sigma Z\sqrt{k\Delta t}} + \sum_{k_{c,i} \in \Omega_{k}} s_{i}e^{\left(\mu-\frac{1}{2}\sigma^{2}\right)(k-k_{c,i})\Delta t+\sigma Z\left(\sqrt{k\Delta t}-\sqrt{k_{c,i}\Delta t}\right)} \\
\upsilon^{L}(k\Delta t) \approx & \upsilon_{0}e^{\left(\mu-\frac{1}{2}\sigma^{2}\right)k\Delta t-\sigma Z\sqrt{k\Delta t}} + \sum_{k_{c,i} \in \Omega_{k}} s_{i}e^{\left(\mu-\frac{1}{2}\sigma^{2}\right)(k-k_{c,i})\Delta t-\sigma Z\left(\sqrt{k\Delta t}-\sqrt{k_{c,i}\Delta t}\right)}
\end{align*}

\begin{align*}
\upsilon^{U}(k\Delta t) \approx & \upsilon_{0}e^{\left(\mu-\frac{1}{2}\sigma^{2}\right)k\Delta t+\sigma Z\sqrt{k\Delta t}} + \sum_{k_{c,i} \in \Omega_{k}} s_{i}e^{\left(\mu-\frac{1}{2}\sigma^{2}\right)(k-k_{c,i})\Delta t+\sigma Z\left(\sqrt{k\Delta t}-\sqrt{k_{c,i}\Delta t}\right)} \\
\upsilon^{L}(k\Delta t) \approx & \upsilon_{0}e^{\left(\mu-\frac{1}{2}\sigma^{2}\right)k\Delta t-\sigma Z\sqrt{k\Delta t}} + \sum_{k_{c,i} \in \Omega_{k}} s_{i}e^{\left(\mu-\frac{1}{2}\sigma^{2}\right)(k-k_{c,i})\Delta t-\sigma Z\left(\sqrt{k\Delta t}-\sqrt{k_{c,i}\Delta t}\right)}
\end{align*}
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where υL and υU are the lower and upper bound of future projection, respec- tively, given by the model. The set Ωk contains the step index where the effect of its corresponding cashflow is added to the solution, i.e.
\Omega_{k} = \left\{ k_{c} \in \Omega_{c} \,|\, k_{c} \leq k \right\}
\Omega_{k} = \left\{ k_{c} \in \Omega_{c} \,|\, k_{c} \leq k \right\}
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Note: This section generalize the method in the previous section (without cashflow). The method in previous section is identical to the method in this section when all the cashflows are set to zero.
References
[1]	Bagchi, A. (1993). Optimal Control of Stochastic Systems. International Series in Systems and Control Engineering. Prentice Hall International.
[2]	Wiener, N. (1981). Collected Works, Edited by: Masani, P. Vol. 3, MIT Press.
 
[3]	Dirac, Paul (1930). The Principles of Quantum Mechanics (1st ed.), Oxford University Press.
[4]	Shreve, S. E. and Soner, H. M. (1994). Optimal investment and consumption with transaction costs. The Annals of Applied Probability.
Updated 05 Jun 2023
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Expected value
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TABLE OF CONTENTS
Projection with cashflow:
Projection without cashflow
References