Interpretación de los resultados

In this study we restrict our observation to one-month calendar spread options, whose underlying is the difference between two Futures contracts written on the same commodity – in our case light sweet crude oil. However, some of theories and results, we present along the way, can be modified and they might be applied also to more complex basket of underlying assets (Borovkova & Permana, 2010). For example, they might be used for pricing a crack spread option or a spark spread option. For the sake of simplicity we decided to focus on European-style spread options, as we already suggested they are easier to evaluate compared to American-style option.

Before starting our review of the available literature for pricing spread options, we want to spend some words on the first tool investors can use to price options, meaning the famous Black-Scholes formula. This is the building block of option pricing and its one of the most important model in the financial theory. We now give a brief overview of this, which is the starting point of our literature review on spread options.

As we explained earlier, when the investor exercises a European call option he receives the following payoff at maturity (𝑇):

Payoff𝑇 = max(𝑆𝑇− 𝐾; 0)

However, what we are interested in; it is the price of the call option (𝑐_𝑡) with 𝑡 < 𝑇. Black, Scholes and Merton (Black & Scholes, 1973) and (Merton, 1973) proposed a technique to value derivative contracts. Today their studies are still highly appreciated, as much as that in 1997 Merton and Scholes won the Nobel Prize for their researches on pricing derivatives. The basic idea behind their model is that uncertain cash flows, like the one of options, can be replicated by a self-financial strategy. This means that the value of the option today must be equal to the cost today of the replicating portfolio strategy. If this would not be true, rationale investor would take advantage of the mispricing situation and immediately set up an arbitrage strategy. Therefore one of the assumptions of the Black-Scholes model is that there are no riskless arbitrage opportunities, the market is complete instead. Black and

Pricing Crude Oil Calendar Spread Options 79

Scholes heavily relied on the solution of their parabolic PDE (Black & Scholes, 1973) and (Carmona & Durrleman, 2003).

In order to explain the PDE formula we assume that the underlying asset – e.g. a stock – follows a one-factor Geometric Brownian motion as:

𝑑𝑆𝑡= 𝜇𝑆𝑆𝑡𝑑𝑡 + 𝜎𝑆𝑆𝑡𝑑𝑊𝑡

The price of the call option 𝑐_𝑡 must depend on the value of the underlying asset 𝑆_𝑡 and on the current date 𝑡. Therefore, assuming 𝑐_𝑡 to be the value of the call option and applying Itô’s lemma, developed in 3.2 The Mathematical Background section, we can expressed the value of the call in function of its set of inputs: 𝑑𝑐𝑡 = �𝜕𝑐_𝜕𝑆𝑡 𝑡𝜇𝑆𝑆𝑡+ 𝜕𝑐𝑡 𝜕𝑡 + 1 2 𝜕2_𝑐 𝑡 𝜕𝑆𝑡2 𝜎𝑆 2_𝑆 𝑡2� 𝑑𝑡 +𝜕𝑐_𝜕𝑆𝑡 𝑡𝜎𝑆𝑆𝑡𝑑𝑊𝑡

Then the riskless arbitrage portfolio (𝜋) replicating the option will have the following structure: 𝜋 = −𝑐𝑡+𝜕𝑐_𝜕𝑆𝑡

𝑡𝑆𝑡

This mean that the investor goes short one unit of the derivative and at the same times he buys 𝜕𝑐𝑡

𝜕𝑆𝑡

units of the underlying asset. Consequently a change in value of the portfolio is given by:

𝑑𝜋 = −𝑑𝑐𝑡+𝜕𝑐_𝜕𝑆𝑡 𝑡𝑑𝑆𝑡 = �− 𝜕𝑐𝑡 𝜕𝑡 − 1 2 𝜕2_𝑐 𝑡 𝜕𝑆𝑡2𝜎𝑆 2_𝑆 𝑡2� 𝑑𝑡

It is easy to note now that portfolio does not depend anymore on the Wiener processes of the stochastic differential equation (𝑑𝑊_𝑡). This simply means that the portfolio is now riskless and consequently its return has to be equal to the short term risk free rate: 𝑟 (Hull, 2008). In more formal terms we can rewrite the differential equation of the return of the replicating portfolio as:

𝑑𝜋 = 𝑟𝜋𝑑𝑡

If we now substitute the different variables with the results above, we obtain:

�𝜕𝑐_{𝜕𝑡 +}𝑡 1₂𝜕_𝜕𝑆2𝑐𝑡

𝑡2𝜎𝑆 2_𝑆

𝑡2� 𝑑𝑡 = 𝑟 �𝑐𝑡−_𝜕𝑆𝜕𝑐𝑡 𝑡𝑆𝑡� 𝑑𝑡

which it can be rewritten in the following way28

28

This procedure is basically the same we used in

:

3 for defining the closed-form expression of the price of Futures contracts. This PDE is particular for option pricing, the one of the Black-Scholes and Merton model has the same form but for a general function 𝑓(∙).

80 Pricing Crude Oil Calendar Spread Options 𝜕𝑐𝑡 𝜕𝑡 + 𝜕𝑐𝑡 𝜕𝑆𝑡𝑟𝑆𝑡+ 1 2 𝜕2_𝑐 𝑡 𝜕𝑆𝑡2𝜎𝑆 2_𝑆 𝑡2− 𝑟𝑐𝑡 = 0

This is the Black-Sholes PDE with the boundary condition 𝑐_𝑡 = max(𝑆_𝑇− 𝐾; 0).

An alternative method to compute the value of derivative products, which consistently leads to the same result, consists in discounting the uncertain future cash flow for a probability structure, which has the particularity to be risk-neutral (Carmona & Durrleman, 2003). As defined earlier in this study, the structure of risk neutral probabilities consists of the set of probabilities that allows excluding possible source of risk from future cash flows. One example of such source of risk is the market risk. This technique it is less complex and more straightforward than solving the previous PDE (Carmona & Durrleman, 2003), which sometimes require lot of effort or even does not lead to a nice closed-form solution. We do not want to go into details, as we did in the section dedicated to the risk adjusted probabilities, but we just limit ourselves in repeating the results of the previous subchapter, reminding that the drift of the underlying asset under risk-neutral probabilities is equal to the short term risk free interest rate. However, we obviously need to know the evolution of the underlying asset under these risk neutral probabilities, too.

We already explained that the theory, which explains the following passage (or transformation) from historical probabilities to risk-neutral measures, is known as the Girsanov theorem. According to what we have just suggested the value of the call option 𝑐_𝑡 can be expressed in the following way:

𝑐𝑡= 𝐸𝑡𝑄�𝑒−𝑟(𝑇−𝑡)max(𝑆𝑡− 𝐾; 0)�

Both techniques lead to the well-known formula of the Black-Scholes option pricing model, since the stock price is assumed to be log-normal distributed. Therefore we have:

𝑐𝑡 = 𝑆𝑡Φ(𝑑1) − 𝐾𝑒−𝑟(𝑇−𝑡)Φ(𝑑2) where: 𝑑1 = ln �𝑆𝑡𝑒𝑟(𝑇−𝑡) 𝐾 � 𝜎√𝑇 − 𝑡 + 1 2 𝜎√𝑇 − 𝑡 and 𝑑2= 𝑑1− 𝜎√𝑇 − 𝑡 Φ(∙) is the cumulative distribution of the standard normal distribution, 𝒩(0, 1).

We have to point out that the discount rate (𝑟) is assumed to be the risk free interest rate, if we decided to include dividends or even more relevant in energy markets the convenience yield, we might need to adjust this riskless interest rate.

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