In this project I investigated the insight behind the Black-Scholes formula: if you know the volatility of a stock then you can replicate the payoff of an option by a continuous rebalancing of a portfolio made up of the underlying stock and a risk-free bond. Therefore, to avoid arbitrage, the cost of the option must be that of the replication strategy. This precise idea blew me away when I first heard of it in class, so I first wanted to see by myself the link between the price formula and this powerful theoretical strategy mentioned. To do this, I first read the derivation of the PDE on Shreve II where I gained the right intuition to see how the more realistic discrete-rebalancing strategy naturally follows from. Also, it was very helpful to have previously read about the Binomial model on Shreve I, where the idea of replicating the option is more simply introduced but in a more contrived scenario where stock prices have a binary set of outcomes over a finite amount of time periods. Despite this simplicity, there are surprisingly many parallels, which is why I found this simpler approach so insightful.
The main assumptions of the Black Scholes model for option pricing are :
- The price of a stock is lognormally distributed (
$dS/S = \mu dt+\sigma dW)S$ ) - The interest rate
$r$ and volatility$\sigma$ are constant - Trading can be done infinitely often
- Trading is free (no transaction costs)
Despite these assumptions (all but the third can be dropped) , I find its conclusions remarkable and not at all obvious, giving a systematic and rational framework for a market maker to asses its risk when selling an option and for its overall contribution to the creation of a wide variety of new financial instruments that all participants in the market can use and benefit from. In this project, I analyzed the effects of relaxing the third assumption that is: only a finite amount of rebalancing can be done throughout the life of the option. For this, I performed simulations of the hedging strategy for a European call option and visualized the statistical distribution of the replication error at maturity.
This first table shows how the accuracy of the replication strategy for two realised paths, where the portfolio is rebalanced 200 times across a trading year (about every 30 hours). The specific parameters used for the simulations are
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| S2 | C2 |
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| S3 | C3 |
Due to the fact that the rebalancing is done finitely often the replication is not 100% exact, although remarkably close in both universes. On the first one, the portfolio has turned out to be worth slightly more than the option payoff while the opposite happens in the second realisation. In general, finite rebalancing leads to an unbiased replication error (mean of 0), whose deviation will converge to 0 as the rebalancing frequency increases. To examine the range of possibilities, I carried out a Monte Carlo simulation in which I evaluate the outcome of the discrete delta-hedging strategy over 5000 different, randomly generated scenarios of future stock price evolution. The table below summarizes the histograms for the replication error at expiry with two different rebalancing frequencies: the first one every trading day (252 times) and the second one about every 1 hour (5000 times). The mean of both is almost 0 and their deviations are about 44 cents and 1 cents.
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| H1 | H2 |
After learning what the Black Scholes formula accomplishes and seeing it succeed in the numerical exploration of this project, I have thought of the following follow-up matters on this interesting topic :
- Rigorous analysis of the replication error to determine the optimal spread to charge on top of the theoretical price and guarantee no losses with 99% confidence
- Include transaction costs
- Test this with other underlying models (Heston, OU process) where there is usually no closed formula for their price
- Learn about delta-gamma trading