Options Pricing: Black-Scholes Model

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The price of a contingent claim can be expressed as the expectation, under the so-called risk-neutral measure, of the payoff discounted to present value. Such representation of the price is important for theoretical and practical purposes. It suggests a straightforward Monte Carlo based method for its calculation: This approach is ubiquitous in the financial practice.

For an introduction to risk-neutral pricing of financial derivatives, see [1]. We have not been able to locate references to a procedure like the one we present here but we wouldn't be surprised if one day we come across them, as it relies on little else than basic probability. Admittedly, Chebfun makes it look particularly simple.

To focus our ideas, we consider the quintessential example of a derivative contract in finance texts: A model for the dynamics of the stock price, with a history stretching back to Sprenkle in [5], is the geometric Brownian motion. We wrap the function handle in a chebfun constructor, specifying a right end point large enough for our calculations to be very accurate. Our choice for the right end-point should make the area under the curve very close to 1.

We check this on the latest lognPDFwhich has the heaviest tail to the right:. So far we have only presented the model that the asset price will follow.

The new measure is known as the "risk-neutral measure" and is equivalent in some sense not discussed here to the original one. The possibility of making this change of measure guarantees the lack of arbitrage opportunities in the market i.

For all its conceptual complexity, implementing the change of measure in the case of a GBM process could not be simpler: The moneyness of the option refers to the position of the stock price at any point in time before maturity with respect to the strike: In our particular setting, if the stock price is, for example, at 50, we would say that the option is OOM, and if it is at we would say that it is deep ITM, highlighting in this way that its is to the right, far from the strike.

Notice that vertical scales of both lines are different, and the Y-axes are omitted to avoid confusion. What is the probability that the option expires OOM? To calculate this value, first we use the cumsum command to obtain the cumulative distribution function CDF and then we evaluate it in the strike:. How can we calculate the distribution of a random variable which is itself the function of another random variable?

The answer of this question appears in every book of basic probability: The usual way of dealing with this requirement is to split the domain of the function in regions with this behaviour and then putting them together.

As before, we have used the value RHS to specify the right end-point. In the case of the call option, this corresponds to a Dirac delta at zero with weight equal to the probability of ending OOM. As before, we can check the accuracy by calculating the area under the curve of the undiscounted PDF:. The final step for the pricing of the call option is the calculation of the expected value of the distribution we just obtained:. Notice that since the location of the Dirac delta in this example is at zero, the inclusion of the OOM component makes no difference when computing the expected value and we could have skip its construction.

The whole calculation we just did can be done analytically and the result is the celebrated Black-Scholes formula very similar expressions had been produced before by Sprenke and Samuelson, but without the key insight of changing the measure [5]:. We leave the interpretation of the different elements in this formula to another example, but we can use it now to check the accuracy of our calculation:. In this example we have shown a Chebfun-based method for the pricing of a European call option.

The method is conceived in the framework of risk- neutral pricing theory, and by "computing with density functions instead of numbers", we have managed to avoid the Monte Carlo simulation approach.

The easy implementation in Chebfun and the high accuracy obtained are promising features but further experiments need to be done, with different contracts and distributions, to have a better understanding of its pros and cons. We end this example by listing the steps of the method when applied to a general European derivative that is, one that is not path-dependent:. Split the payoff profile in regions where it is monotonically increasing, monotonically decreasing, or constant.

Construct chebfuns for each piece. Calculate the total probability of ending in constant regions and use them as weight of Dirac deltas located at points equal to the constant values.

Use the sum command to obtain the expected value of the distribution and discount it by the risk-free rate. Scholes, "The pricing of options and corporate liabilities", Journal of Political Economy 81 Kreps, "Martingales and arbitrage in multiperiod securities markets", Journal of Economic Theory 20 Pliska, "Martingales and stochastic integrals in the theory of continuous trading", Stochastic Processes and their Applications 11 MacKenzie, An Engine, not a Camera: Introduction The price of a contingent claim can be expressed as the expectation, under the so-called risk-neutral measure, of the payoff discounted to present value.

A call option on an asset following a GBM To focus our ideas, we consider the quintessential example of a derivative contract in finance texts: We check this on the latest lognPDFwhich has the heaviest tail to the right: To calculate this value, first we use the cumsum command to obtain the cumulative distribution function CDF and then we evaluate it in the strike: The whole calculation we just did can be done analytically and the result is the celebrated Black-Scholes formula very similar expressions had been produced before by Sprenke and Samuelson, but without the key insight of changing the measure [5]: We leave the interpretation of the different elements in this formula to another example, but we can use it now to check the accuracy of our calculation: We end this example by listing the steps of the method when applied to a general European derivative that is, one that is not path-dependent: Add all contributions to obtain the payoff's distribution at maturity.

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