A Formula for Asian Option with Geometric Averaging

Asian options are a type of exotic option that can be tricky to price, but a formula can help simplify the process.

The formula for Asian options with geometric averaging is based on the assumption that the underlying asset's price follows a geometric Brownian motion.

This formula takes into account the average of the asset's price over a certain period, which makes it suitable for options that are sensitive to the underlying asset's volatility.

The formula is as follows: (S0 * (1 - e^(-kT)) + (e^(-kT) - 1) * A) / (k * (e^(-kT) - 1)), where S0 is the initial asset price, A is the average asset price, k is the volatility, and T is the time to maturity.

In practice, this formula can be used to price Asian options with geometric averaging, which can be useful in a variety of financial applications.

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The Option Model is a powerful tool for pricing Asian options. It's based on the idea that the option's payoff depends on the average price of the underlying asset over a certain period.

The key formula for the Option Model is S(0) = (1 - e^(-rT)) * (K * e^(-rT) + σ^2 * T / (2 * (1 - e^(-rT)))). This formula takes into account the underlying asset's price, the risk-free interest rate, the time to expiration, and the volatility of the asset.

In practice, this model has been shown to be more accurate than other models for pricing Asian options, especially when the underlying asset's price is highly volatile.

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The Option Model is a financial framework that helps investors make informed decisions about buying and selling securities.

In the Option Model, the price of an option is determined by its intrinsic value and time value.

The intrinsic value of an option is the difference between the strike price and the current market price of the underlying security.

This value represents the profit an investor can expect if they exercise the option.

Time value, on the other hand, is the amount by which the option price exceeds its intrinsic value.

It reflects the investor's expectation of potential gains or losses from the option.

The Option Model assumes that investors are risk-neutral and that the underlying security's price will follow a geometric Brownian motion.

This assumption allows for the calculation of the option's price using a risk-neutral probability measure.

In the world of options, using a geometric option as a control variate can be a game-changer.

A control variate is a technique used to reduce the variance of an estimate, and in this case, the Asian geometric option can be used to achieve this.

The payoff of an Asian geometric option is given by ΦS=max⁡e1/T∫0Tlog⁡Stdt−K,0.

This means that the payoff is based on the maximum of the geometric average of the stock price over a period of time, minus a strike price K.

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In contrast, the payoff of an Asian arithmetic option is based on the average of the stock price over the same period.

By using the Asian geometric option as a control variate, we can reduce the variance of the estimate of the Asian arithmetic option price under the Black-Scholes model.

The Asian arithmetic option price under the Black-Scholes model is represented by CA(S), and using the geometric option as a control variate can help improve the accuracy of this estimate.

In fact, the geometric option can be used to reduce the variance of the estimate by up to 50% in some cases.

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The payoff of an Asian geometric option is given by ΦS=max⁡e1/T∫0Tlog⁡Stdt−K,0. This type of option uses the geometric mean of the underlying asset's price over a certain period.

The geometric mean is calculated as GT=exp⁡[1T∫0Tlog⁡S(t)dt], where the underlying S(t) follows a standard geometric Brownian motion. This is a key component in deriving the formula for the European Asian call with geometric averaging.

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The value of the European Asian call with geometric averaging CG is given by e−rT2π∫ℓ∞(GT−K)e−x2/2dx, where ℓ is the lower bound of the integral. This formula is useful for deriving fair values for the arithmetic Asian option.

Key parameters for the geometric option include σT∫0T(T−t)dWt∼N(0,σ2T3) and b=12(r−12σG2),σG=σ3. These parameters are used to calculate the value of the European Asian call with geometric averaging.

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The control sample estimator for geometric option Greeks Δ is a crucial concept in pricing Asian geometric options. It's a method used to estimate the Greeks of the option, which are the sensitivities of the option's price to changes in the underlying variables.

The control sample estimator is defined in Step 4 of the process, where it's used to estimate the Greeks Δ of the Asian geometric option. This is done by using a control variate, which is a second option that is used to reduce the variance of the estimate.

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The control variate used in this case is the Asian arithmetic option, which is priced under the Black-Scholes model. The price of the Asian arithmetic option is denoted by CA(S), and it's used as a control variate to estimate the Greeks Δ of the Asian geometric option.

The results of using the control sample estimator with the proposed numerical simulation are compared to the classic Common Random Number (CRN) method in Table 1. The results show that the proposed method has a much smaller standard error than the CRN method, indicating that it's a more accurate and efficient method for estimating the Greeks Δ of the Asian geometric option.

In summary, the control sample estimator for geometric option Greeks Δ is a powerful tool for pricing Asian geometric options. By using a control variate and a numerical simulation, it's possible to estimate the Greeks Δ with high accuracy and efficiency.

Here's a brief summary of the key points:

Creating sample paths is a crucial step in pricing geometric options. We need to generate sample paths for the underlying asset price, denoted as St and S~t.

To do this, we fix some parameters: time to maturity T > 0, time steps m ∈ ℕ, initial price S > 0, and initial underlying price shock ϵ > 0.

We then generate sample paths using the following iterative steps: Sti=Sti−1er−σ2/2T/n+σT/nZi,S0=S,S~ti=S~ti−1er−σ2/2T/n+σT/nZi,S~0=S+ϵ, where Z1,…, Zm are m copies of iid normal random numbers.

Using the same random numbers for both paths St and S~t helps reduce variance. We use these sample paths to approximate the integral 1T∫0TStdt, which is equivalent to 1T∑i=1nStiTn=1n∑i=1nSti.

The Asian arithmetic option payoff is simulated by calculating ΦAjS=max⁡1n∑i=1nStij−K,0j=1,…,m, and Φ~AjS=max⁡1n∑i=1nS~tij−K,0j=1,…,m.

The Asian geometric option payoff is simulated by calculating ΦGjS=max⁡e1/n∑i=1nlog⁡Sti−K,0j=1,…,m, and Φ~GjS=max⁡e1/n∑i=1nlog⁡S~ti−K,0j=1,…,m.

Here's a summary of the sample path generation process:

European Options with Geometric Averaging are a type of option that takes into account the geometric mean of the underlying asset's price over a period of time.

The geometric Asian option is a specific type of European option with geometric averaging, and it can be valued using a closed-form solution in conjunction with control variates in Monte Carlo simulations.

To derive the value of the European Asian call with geometric averaging, we need to calculate the continuous-time geometric mean GT, which is defined as GT = exp[1/T ∫0T log S(t)dt].

The underlying asset S(t) follows a standard geometric Brownian motion, and the value of GT can be calculated as GT = S0e^(1/2(r-1/2σ^2)T)e^(σ/T ∫0T (T-t)dWt).

The stochastic integral in the expression for GT is normally distributed with mean 0 and variance σ^2 T/3, which makes it easier to work with.

We can define a new variable x such that σ/T ∫0T (T-t)dWt = σ√(T/3) x, and since x is normally distributed with mean 0 and variance 1, we can find the value of x such that GT ≥ K.

Using this value of x, we can calculate the value of the European Asian call with geometric averaging using the formula CG = e^(-rT) E[(GT - K)_+].

This formula can be simplified to CG = e^(-rT) / √(2π) ∫ℓ∞ (GT - K) e^(-x^2/2) dx, where ℓ is the value of x such that GT ≥ K.

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The value of ℓ can be found by solving the inequality GT ≥ K, which leads to σ/T ∫0T (T-t)dWt ≥ log(K/S0) - 1/2(r-1/2σ^2)T.

The European Asian put with geometric averaging can be valued using a similar approach, and there exists a version of put-call parity for European Asian options with geometric averaging: CG - PG = S0e^(b-r)T - Ke^(-rT).