Fx options pricer
Spread options are based on the spread between two commodity prices. They are commonly used to model physical investments as "real options" or to mark-to-market contracts that hedge physical assets. For example, a natural gas fueled electrical generation unit can be used to convert fuel natural gas into electricity.
Option Pricing Theory
Whenever this conversion is profitable, it would be rational to operate the unit. This type of conversion is readily modeled by a spread option. When the spread of electricity prices - fuel costs is greater than the conversion cost, then the unit would operate. In this example, the conversion cost, which might be called the Variable Operations and Maintenance VOM for a generation unit, would represent the strike price.
Analytic formulas similar to the Black Scholes equation are commonly used to value commodity spread options. While an exact closed form solution does not exist to value spread options, approximate solutions can give reasonably accurate results. In a Black Scholes equation, the distribution of price returns is assumed to be normally distributed on the expiration date.
This ratio can then be converted into a formula very similar to the Generalized Black Scholes formulas. The ratio implies that the option is profitable to exercise in the money whenever the ratio of prices F in the formula above is greater than 1. The key complexity is determining the appropriate volatility that needs to be used in the equation.
That assumption makes it possible to estimate the volatility needed for the modified Black Scholes style equation see Figure 7. A second complexity is that the prices of two assets F1 and F2 have to be in the same units. For example, in a heat rate option, the option represents the ability to convert fuel natural gas into electricity.
Black-Scholes-Merton (BSM) Option Valuation Model
To use the approximation, it is necessary to convert the price of the second asset into the units of the first asset See Example 1 - a Heat Rate Option. This conversion rate will typically be specified as part of the contract. Before being placed into a spread option model, the price of natural gas would need to be converted into the correct units. Another important consideration not discussed in this write-up is that volatility and correlation need to be matched to the tenor of the underlying assets.
This means that it is necessary to measure the volatility of forward prices rather than spot prices. It may also be necessary to match the volatility and correlation to the correct month. For example, power prices in August may behave very differently than power prices in October or May in certain regions. However, the relative complexity of modeling commodity prices the typical underlying for spread options makes calibrating inputs a key part of the model. American options differ from European options because they can be exercised at any time.
If there is a possibility that it will be more profitable to exercise the option than sell it, an American option will have more value than a corresponding European option.
Early exercise typically occurs only when an option is in the money. If an option is out of the money, there is usually no reason to exercise early - it would be better to sell the option in the case of a put option, to sell the option and the underlying asset. The decision of whether to exercise early is primarily a question of interest rates and carrying costs.
Carrying costs, or cost of carry , is a term that means an intermediate cash flows that is the result of holding an asset. For example, dividends on stocks are a positive cost of carry owning the asset gives the owner a cash flow. A commodity might have a negative cost of carry.
FX Options & Pricing Sheet
For example, a commodity that requires its owner to pay for storage would cause the owner of the physical commodity to pay cash to hold the asset. Note: Commodity options are typically written on forwards or futures which have zero cost of carry instead of the actual underlying commodity.
Cost of carry is cash flow that affects the owner of the underlying commodity and not the owner of the option.
For example, when a stock pays a dividend, the owner of a call option does not receive the dividend - just the owner of the stock. For the perspective for the owner of a call option on a stock, the cost of carry will be the interest received from holding cash r less any dividends paid to owners of the stock q. Since an option has some value the extrinsic value that would be given up by exercising the option, exercising an option prior to maturity is a trade off between the option's extrinsic value the remaining optionality and the relative benefit of holding cash time value of money versus the benefit of holding the asset carrying costs.
With commodities, things are a slightly different. There is typically no cost of carry since the underlying is a forward or a futures contract. It does not cost any money to enter an at-the-money commodity forward, swap, or futures contract. Also, these contracts don't have any intermediate cash flows.
FX Options Analytics: Vols, Risk Reversals & Pin Risk | Saxo Group
As a result, the primary benefit of early exercise is to get cash immediately exercising an in-the-money option rather than cash in the future. In high interest rate environments, the money received immediately from immediate execution may exceed the extrinsic value of the contract. This is due to strike price not being present valued in immediate execution it is specified in the contract and a fixed number but the payoff of a European option is discounted forward price - strike price. The overall result is that early exercise is fairly uncommon for most commodity options.
Typically, it only occurs when interest rates are high.
Deriscope Resources
There is no exact closed-form solution for American options. However, there are many approximations that are reasonably close to prices produced by open-form solutions like binomial tree models. Two models are shown below, both created by Bjerksund and Stensland.
The first was produced in and the second in The second model is a refinement of the first model, adding more complexity, in exchange for better accuracy. There is no closed form solution for American options, and there are multiple people who have developed closed-form approximations to value American options. This is one such approximation. However, this approximation is no longer in active use by the public interface. It is primarily included as a secondary test on the BS calculation.
This function uses a numerical approximation to estimate the value of an American option. It does this by estimating a early exercise boundary and analytically estimating the probability of hitting that boundary. This uses the same inputs as a Generalized Black Scholes model:.
Intermediate Calculations. To be consistent with the Bjerksund Stensland paper, this write-up uses similar notation. Calculate the Early Exercise Boundary i. The lower case i represents the early exercise boundary. Alpha is an intermediate calculation. Compare to European Value. Due to the approximation, it is sometime possible to get a value slightly smaller than the European value. If so, set the value equal to the European value estimated using Generalized Black Scholes. Intermediate calculations. The Psi function has a large number of intermediate calculations.
For clarity, these are loosely organized into groups with each group used to simplify the next set of intermediate calculations. The calculation of Psi. This is the actual calculation of the Psi function. In the function below, M represents the cumulative bivariate normal distribution described a couple of paragraphs below this section. Phi is an intermediate calculation used by both the Bjerksun Stensland and approximations. Many of the parameters are the same as the GBS model. This is the normal cumulative density function.