FBSDE

In this project, we optimally replicate an option portfolio in the presence of transaction costs using a deep learning framework. We use a propagator transaction cost model to account for transaction costs. Finally, we compare the performance of the replicating portfolio with the performance of the option portfolio.

We model this as a stochastic control problem, and we aim to minimize or maximize the functional \begin{equation} \tilde{J}(c)=E\left(\int_0^T \tilde{\mathrm{rc}}\left(s, \tilde{X}_s, c_s\right) d s+\tilde{\mathrm{fc}}\left(\tilde{X}_T\right)\right) \end{equation}

with respect to the control function \(c_t\), where the underlying stochastic evolving factors \(\tilde{X}_t\) satisfy \begin{equation} d \tilde{X}_t=\tilde{\mu}\left(t, \tilde{X}_t ; c_t\right) d t+\tilde{\sigma}\left(t, \tilde{X}_t ; c_t\right) d W_t, \end{equation}

where \(X_t = [S_t, Y_t]\) which follows the following processes,

\begin{equation} d S_t=\mu\left(t, S_t\right) d t+\sigma\left(t, S_t\right) d W_t \end{equation} \begin{equation} d Y_t=-f\left(t, S_t, Y_t, Q_t\right) d t+ Q_t^T \sigma\left(t, S_t\right) d W_t \end{equation}

We model running costs \(\tilde{rc}\) using a propogator model, and final cost is given by \begin{equation} \tilde{fc} = ||{Y_T - g(S_T)}||^2 \end{equation}

Finally, we model transaction costs using (1) linear permanent impact and quadratic temporary impact, and (2) markov impact process as follows: \begin{equation} I_{t + 1}^Q = e^{-\lambda} I_{t}^Q + \gamma v_{t} \end{equation}

\begin{equation} v_{t} = Q_{t + 1} - Q_{t} \end{equation}

Using this, we obtain new underlier processes for \(S_t\) and \(Y_t\) (not shown here). Finally, we optimize the total PnL using a risk averse optimization using a simple neural network and gradient descent on the following objective function:

\begin{equation} \mathcal{L}(w_T) = \frac{\kappa}{2} V\left[w_T\right]-E\left[w_T\right]
\end{equation}

where \(w_T = w_0 + \sum_{t=1}^T \delta w_t,\) and \(\delta w_t= \delta\left(P_t-Y_t\right)\)

• \(S_t\) denotes the prices of the underlying assets at time t
• \(Y_t\) denotes the value of the options replication portfolio at time t
• \(\tilde{X}_t\) is the concatenation of \(S_t\) and \(Y_t\)
• \(Q_t\) is the value of holdings in the underlying stocks at time t
• \(\tilde{rc}\) and \(\tilde{fc}\) denote the running cost and final cost respectively
• \(Q_t\) is the value of holdings in the underlying assets at time t, with unit in dollars
• \(h_t\) denotes the value of holdings in the underlying assets at time t, with unit in number of shares
• \(\delta t\) denotes the length of rebalancing interval in discrete time
• \(\tau\) denotes the length of the time interval for each rebalancing (trade)
• \(v_t\) denotes (1) the average rate of trading at time t in shares for the linear and KO model (2) the turnover in dollar in propagator model
• \(g\) denotes the payoff function
• \(X_t\) denotes the value of the cash account at time t, with unit in dollars
• \(I_{t}^Q\) is the impact process for the propagator model, with unit in basis points
• \(c_t\) denotes the instantaneous cost when rebalancing positon at time t
• \(\lambda\) denotes the exponential decay; \(\frac{\ln 2}{\lambda}\) represents the half-life in the Markov propagator model
• \(\gamma\) denotes trades loading, i.e. the push in propagator model