# Option Pricing and Replication under Generic Market Conditions

## Explanation, Consequences, Papers, Resources, and References

# Equity Portfolio Replication

The notion of a replicating portfolio first appears in the argument for the Black-Scholes model (1973). In order to bind an option’s price to their model under a no-arbitrage assumption, they develop an offsetting equity position replicating the opposite value of the option at expiration. Holding such a portfolio is therefore risk-neutral, consequently, the portfolio must earn an appropriate risk-free rate. However, Black-Scholes use geometric Brownian motion to model the underlying asset price. Infinite variation is implied by this stochastic process meaning overtime the initial equity value will stray linearly where the option value strays non-linearly. This creates a hedging error which is corrected by Black-Scholes by continuously revising the offsetting replicating portfolio and maintains the portfolio’s risk-neutrality expiration. In the presence of transaction costs and discrete trading, this is no longer optimal or possible. This article is meant to provide an explanation of prior and modern literature on this subject while including resources and references for the reader.

# Leland, H. (1985)

## Option Pricing and Replication with Transaction Costs

Leland was the first to research the implications of transaction costs on the Black-Scholes model. Relaxing the assumption of no-transaction costs, he developed a modification to variance inclusive of transaction costs and the replicating strategy’s revision interval. His reasoning, transaction costs exaggerate the cost of equity in either direction (long or short) which can be modeled by inflating the Black-Scholes model’s parameterized variance. Leland proves that with his modification to variance a replicating strategy yields the option’s value at expiration when approaching continuous revisions. Once again binding an option’s price to the Black-Scholes model, this time inclusive of transaction costs. Consequently, he shows his modified replicating strategy generally outperforms a standard Black-Scholes replicating strategy. Though this takes a step closer to the conditions practitioners experience, Leland still heavily relies on the ability to continuously trade. Ideally, Leland needs his modified replicating strategy to yield the option’s value at expiration in the presence of both transaction costs and discrete trading. The natural next question arises: when is it optimal to revise a replicating portfolio in the presence of transaction costs and discrete trading?

# Cao, J., Chen, J., Hull, J. C., & Poulos, Z. (2020)

## Deep Hedging of Derivatives Using Reinforcement Learning

This paper focuses on using artificial intelligence to solve the problem of when to revise a hedge (revise an offsetting replicating position). There are three main types of machine learning: supervised learning, unsupervised learning, and reinforcement learning. Reinforcement learning is used to train computational agents to find the optimal solution based on a state/action/reward system. In other words, it learns by trial and error, a delicate balance of exploration and exploration in an environment (very similar to genetic algorithms). The results of this paper conclude that the computational agents found it best to remain over/under hedged with respect to the equity position within a certain tolerance range. This matches with prior literature and strategies implemented by practitioners, generally accepted as a *delta tolerance* strategy. Though in order to train the computational agents to revise a replicating position the researchers used a standard Black-Scholes delta (exclusive of transaction costs). Would Leland’s modified delta in this reinforcement learning model outperform a standard Black-Scholes delta similar to how Leland’s modified replicating strategy outperformed a standard Black-Scholes replicating strategy?

# Paolucci, R. (2020)

## Option Pricing under Generic Market Conditions

I wrote a brief paper this year that discusses option pricing under generic market conditions. This article explores theoretical and practical solutions to replication in the presence of transaction costs and discrete trading.

# Resources

This article discusses many topics in quantitative finance. I have compiled some resources below that are meant to aid in an explanation for any topics discussed herein.

## Geometric Brownian Motion

## Replicating Portfolios

Black-Scholes Algorithmic Delta Hedging

Risk-Neutral Portfolio Management

## Black-Scholes Model

# References

*Black, F., & Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81(3), 637–654.*

*Cao, Jay and Chen, Jacky and Hull, John C. and Poulos, Zissis, Deep Hedging of Derivatives Using Reinforcement Learning (December 20, 2019)*

*LELAND, H.E. (1985), Option Pricing and Replication with Transactions Costs. The Journal of Finance, 40: 1283–1301.*

*Paolucci, Roman, Option Pricing under Generic Market Conditions (December 1, 2020)*