Black-Scholes Market

An Explicit Solution for the Problem of Optimal Investment with Random Endowment ArXiv ID: 2506.20506 “View on arXiv” Authors: Michael Donisch, Christoph Knochenhauer Abstract We consider the problem of optimal investment with random endowment in a Black–Scholes market for an agent with constant relative risk aversion. Using duality arguments, we derive an explicit expression for the optimal trading strategy, which can be decomposed into the optimal strategy in the absence of a random endowment and an additive shift term whose magnitude depends linearly on the endowment-to-wealth ratio and exponentially on time to maturity. ...

Machine Learning-powered Pricing of the Multidimensional Passport Option ArXiv ID: 2307.14887 “View on arXiv” Authors: Unknown Abstract Introduced in the late 90s, the passport option gives its holder the right to trade in a market and receive any positive gain in the resulting traded account at maturity. Pricing the option amounts to solving a stochastic control problem that for $d>1$ risky assets remains an open problem. Even in a correlated Black-Scholes (BS) market with $d=2$ risky assets, no optimal trading strategy has been derived in closed form. In this paper, we derive a discrete-time solution for multi-dimensional BS markets with uncorrelated assets. Moreover, inspired by the success of deep reinforcement learning in, e.g., board games, we propose two machine learning-powered approaches to pricing general options on a portfolio value in general markets. These approaches prove to be successful for pricing the passport option in one-dimensional and multi-dimensional uncorrelated BS markets. ...