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This example also demonstrates how variations in spot prices, volatility, and strike prices affect option prices on European Vanilla and Asian options. Asian options are securities with payoffs that depend on the average value of an underlying asset over a specific period of time.

Underlying assets can be stocks, commodities, or financial indices. Two types of Asian options are found in the market: Average price options have a fixed strike value and the average used is the asset price.

Average strike options have a strike equal to the average value of the underlying asset. The Levy model is based on the arithmetic mean of the price of the underlying during the life of the option [1]. The Kemna-Vorst method provides a closed form pricing solution to geometric averaging options [2].

The pricing functions asianbylevy and asianbykv take an interest rate term structure and stock structure as inputs. The lattice pricing function asianbycrr takes an interest rate tree CRRTree and stock structure as inputs. We can price the previous options by building a CRRTree using the interest rate term structure and stock specification from the example above.

The results above compare the findings from calculating both geometric and arithmetic Asian options, using CRR trees with 20 and 40 levels. It can be seen that as the number of levels increases, the results approach the closed form solutions. The pricing function asianbyls takes an interest rate term structure and stock structure as inputs. The output and execution time of the Monte Carlo simulation depends on the number of paths NumTrials and the number of time periods per path NumPeriods.

We can create a plot to display the difference between the geometric Asian price using the Kemna-Vorst model, standard Monte Carlo and antithetic Monte Carlo. The graph above shows how oscillation in simulated price is reduced through the use of variate antithetic.

Prices calculated by the Monte Carlo method will vary depending on the outcome of the simulations. Increase NumTrials and analyze the results. The table above contrasts the results from closed approximation models against price simulations implemented via CRR trees and Monte Carlo. Asian options are popular instruments since they tend to be less expensive than comparable Vanilla calls and puts. This is because the volatility in the average value of an underlier tends to be lower than the volatility of the value of the underlier itself.

Let's compare the price of Asian options against their Vanilla counterpart. We can analyze options prices at different levels of the underlying asset. Consider for example, the effect of variations in the price of the underlying asset. It can be observed that the price of the Asian option is cheaper than the price of the Vanilla option. Additionally, it is possible to observe the effect of changes in the volatility of the underlying asset.

The table below shows what happens to Asian and Vanilla option prices when the constant volatility changes. A comparison of the calculated prices show that Asian options are less sensitive to volatility changes, since averaging reduces the volatility of the value of the underlying asset.

Also, Asian options that use arithmetic average are more expensive than those that use geometric average. The figure above displays the option price with respect to strike price.

Since call option value decreases as strike price increases, the Asian call curve is under the Vanilla call curve.

It can be observed that the Asian call option is less expensive than the Vanilla call. Hedging is an insurance to minimize exposure to market movements on the value of a position or portfolio. As the underlying changes, the proportions of the instruments forming the portfolio may need to be adjusted to keep the sensitivities within the desired range.

Delta measures the option price sensitivity to changes in the price of the underlying. Assume that we have a portfolio of two options with the same strike and maturity. The following graph demonstrates the behavior of Delta for the Vanilla and Asian options as a function of the underlying price.

If the asset price is deep in the money, then it is more likely to be exercised. The opposite occurs for an out of the money option. Asian delta is lower for out of the money options and is higher for in the money options than its Vanilla European counterpart. The geometric Asian delta is lower than the arithmetic Asian delta. Choose your country to get translated content where available and see local events and offers.

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Overview of Asian Options Asian options are securities with payoffs that depend on the average value of an underlying asset over a specific period of time. The payoff at maturity of an average price European Asian option is: The average can be arithmetic or geometric.

Consider the following example: Comparison of Asian Arithmetic and Geometric Prices: Asian Prices using the CRR lattice model: Asian Prices using Monte Carlo Method: Arithmetic Asian Standard Monte Carlo: Comparison of Asian call prices: Comparison of Vanilla and Asian Prices: Comparison of Vanilla and Asian Delta: Was this topic helpful? Select Your Country Choose your country to get translated content where available and see local events and offers.

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