In this project I investigated the insight behind the Black-Scholes-Merton 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 wanted to further explore the link between the price and the associated strategy in action.
The main assumptions of the Black-Scholes-Merton model for option pricing are :
- The price of a stock is lognormal or equivalently,
$dS = \mu Sdt+\sigma S dW$ - Riskfree rate
$r$ and volatility$\sigma$ constant - Arbitrarily high frequency trading is possible
- No transaction costs: no fees nor bid/ask spread
Despite these assumptions , I find its conclusions astonishing as it gives a systematic risk management strategy to limit the risks when selling an option and more generally has contributed to the creation of a wide variety of new financial instruments that all participants in the market can use and benefit from. As Robert.C Merton puts it : "It gave us a prescription for how to produce them and this became an efficient production process... it opened the doors to not just dealing with options but to a whole array of financial innovations like the mortgage market... it has become a mainstay of how the whole industry works including central banks and other government agencies. None of these would work today without these complex computer models and the finance technologies and of course the data we have to collect in order to run them". Check the entire video: https://www.youtube.com/watch?v=3guNFc0Hf6M&t=1650s
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. For the details on the derivation of the BSM formula and its link to the replication strategy as well as other interesting information, check the contents of Columbia's Foundations of Financial Engineering course prepared by professor Martin Haugh : (https://martin-haugh.github.io/teaching/foundations-fe/)
In the first table I show graphically the accuracy of the replication strategy for two random stock paths, where the portfolio is rebalanced once a day. The specific parameters used for the simulations are
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| S2 | C2 |
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| S3 | C3 |
On the first path, the stock ends at 119.1 $ paying off 19.1$ to the holder, whereas the portfolio turns out to be worth slightly more (2.29$). However, the opposite happens in the second realisation, where the portfolio ends up being worth slightly less (0.39$). In general, as I observed later, the error is unbiased and its deviation vanishes as the rebalancing frequency increases. To see these facts, I carried out a Monte Carlo simulation where I perform the strategy over 100000 randomly generated stock trajectories. I did this for two different rebalancing frequencies and obtained the following histograms for the replication error : the first one every trading day (252 times) and the second one about every hour (5000 times).
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| H1 | H2 |
The mean of both is almost 0 and their deviations are about 44 cents and 10 cents.
The GoldmanSachs quantitative strategies research team gs-quantstrategies claimed that that the deviation of the error is approximately proportional to
By performing discrete rebalancing, we are essentially approximating the theoretical replication portfolio, whose holdings change continuously (an object that is plausible on the mathematical world but we cannot cope with in reality) with one that is rebalanced finitely often (just like we approximating an arbitrary function using a combination of step functions). The reason for the convergence of these discretely rebalanced portfolios is similar to why the euler method converges when solving a classical DE; the error can be bounded by the second derivative of the price with respect to the stock (gamma). A way to prove this is by showing that the finite accumulation of the errors converges almost surely to 0 as the rebalance frequency becomes large. Section I of convergence_errors does a good job at explaining the steps to prove the convergence of these errors.
The following table shows how the components of the discrete portfolio evolved (stock holdings, cash borrowed/lent at that time) for the second path outlined above. Also, it is accompanied by the
The Binomial model is a simpler model which assumes that prices only evolve through a binary tree corresponding to up and down moves for a discrete set of periods (corresponding to the tree depth). The replication strategy in this model is easy to calculate by solving a linear system of equations and leads to the same conclusions as the BSM model, if the number of periods goes to infinity and with the right choice of up/down factors. I also implemented this pricing model and verified this equivalence numerically, by plotting the interpolated pricing surface obtained from a large tree of prices and comparing it with the BSM one (see figure below). In addition, this scheme is more versatile for replicating path dependent options and can easily be modified to account for a varying volatility and risk free rate. For the details of this model and the mathematical proof of its equivalence to the BSM model I read the following notes: BinomialNotes
I plotted the two price surfaces using the same parameters and for the case of the binomial interpolated in between. The graphical difference is negligible.
- Derive an explicit formula for the hedge error
- Account for transaction costs
- Test with historical data
- Learn higher order hedging strategies like Delta-Gamma hedging
- Explore local volatility and stochastic volatility models
- Learn how to model the risk free rate