# Chapter 9 Asian Options

An Asian option is a derivative with a payoff at maturity that depends on an average of the underlying on a set of predetermined observation dates. Since the payoff of Asian options is based on the average of the underlying asset prices on a set of observation dates, the uncertainty concerning the fluctuations of the underlying price at maturity decreases. Therefore it decreases the risk exposure to spot price and volatility compared to vanilla options.

Also, the higher the number of observations, the lower the volatility and the lower the option price. Because of this lower volatility, asian options are generally cheaper than vanilla options (with same characteristics).

Asian options are commonly used for currencies, interest rates, commodities and energy markets. They are useful in corporate hedging situations, for instance, a company exchanging foreign currency for domestic currency at regular intervals.

## 9.1 Asian-In and Asian-Out

There are typically two types of Asian options: Asian-in and Asian-out options. Be careful when using these terms as they are not used the same way by the entire financial community!

### 9.1.1 First case

I personnaly speak about asian-in options when the averaging observation dates are spread uniformly during the entire life of the option. At the contrary, asian-out options have their averaging observation dates gather during a specific period near the maturity date.

In the general case, asian-in options are less risky than asian-out options. Intuitively, the uncertainty about future spot prices is lower when averaging periodically over the option's life than when averaging periodically over a shorter period near maturity.

### 9.1.2 Second case

Some people speak about asian-out options when an average is computed to determine the settlement price and asian-in options when the an average is computed to determine the strike price.

According to this definition: - Asian-out options pay the difference between the average of the underlying on the predetermined observation dates and the fixed strike price. - Asian-in options pay the difference between the underlying price at expiry and the average of the underlying at these predetermined observation dates (floating strike).

## 9.2 Geometric Average vs Arithmetic Average

There are also several ways to compute an average. More precisely, the average part of the asian options can be either geometric or arithmetic.

Geometric asian options are easy to price since there exists analytical formulae. This is simply due to the fact the geometric average of a lognormally distributed underlying has a lognormal distribution. Therefore, you will be able to use Black-Scholes model with specific volatility and interest rate levels as shown in Exotic Options Trading from Frans de Weert.

This author emphasizes something that appears to be quite confusing for most of the readers. While \(\sigma_i\) and \(r_i\) are always smaller than \(\sigma_{i+1}\) and \(r_{i+1}\) respectively (because any deviation in period i is automatically in period i+1), it does not prevent the annualised volatility of \(\sigma_i\) to be larger than the annualized volatility of \(\sigma_{i+1}\)! It is therefore important to distinguish between annualized IV and the IV associated with the actual term of the option.

This confirms the fact the volatility of an average of observations is always smaller than the volatility of the share price itself.

Arithmetic asian options are far more common and don't have closed formulas to price them since the arithmetic average of log-normal variables is not log-normal.

The pricing of arithmetic asian options would then rely on Monte Carlo simulation with appropriate smile models such as local volatility. You can also use a moment-matching technique. This consists in calculating the exact first two moments of the Asian and then matching them, making the assumption that they are the result of a log normal. Once you have identified this particular log normal, you end up using B&S formula.

If you want to use geometric average as an estimate for the arithmetic average, keep in mind that the geometric average is a lower bound of the arithmetic average as stated by the well-known Jensen inequality.

A more economic way to understand why the IV of an asian option is smaller than the IV of the vanilla option with the same maturity is by recognising than the duration of the asian option is smaller than the duration of the vanilla option with the same maturity.

## 9.3 Numerical Application

In the absence of interest rates and dividends, the price of an ATM Asian call can be approximated by the following formula: \(C_T = \frac{1}{\sqrt{3}} \; 0.4 \; \sigma \sqrt{T} \; S_0\)

This proxy formula shows that the volatility of the Asian Call is lower than the volatility of the European Call: \(\sigma_{Asian} = \frac{1}{\sqrt{3}} \sigma\).

## 9.4 Risk Analysis : The Greeks

### 9.4.1 Delta Hedging

Let us take the practical example from Frans de Weert's book to apprehend the delta hedging of an asian option.

You, as a trader, just sold 90.000 3-month Asian calls on Total stock with a strike price of 40€ and monthly observations (3 observations). Let's assume that each call gives its owner the right to buy one stock at maturity.

#### 9.4.1.1 Inception

At inception, the asian call's delta is quite similar the 3-month european call, so around 50%. You will therefore delta hedge by buying 45.000 shares of Total.

#### 9.4.1.2 First observation date

One day before the first observation date (approximately one month after inception), Total is now trading at 48€.

In this case, the delta of the asian call is larger than the 'equivalent' european call. This is because the first Asian setting will almost certainly be ITM (> 40€).

Let say the asian delta is 5/6 (long 75.000 shares of Total) and the equivalent european delta is 3/4 (long 67.500 shares of Total).

Since the first Asian setting is assured to be ITM (delta = 1), which accounts for one third of the weight in the arithmetic average (3 monthly observations), 30.000 shares of the 75.000 shares serve as a delta hedge for the first Asian setting.

You will sell these 30.000 shares at the close of the first Asian setting and will be left with a long 45.000 shares of Total as your delta hedge.

Once the first Asian setting is taken, the remaining asian call is effectively on 60.000 shares only.

#### 9.4.1.3 Second observation date

After the first observation, Total's stock price goes down sharply. One day before the second observation date, Total is now trading at 32€. As a result, your delta hedge has decreased from a long 45.000 shares position to a long 10.000 shares position.

Since the second Asian setting is assured to be OTM (delta = O), your delta hedge is purely against the third Asian Setting and nothing against the second one. For that reason you don't have to do anything on the close of the second Asian setting. You can therefore deduce that the delta for the third observation is currently 1/3 (10.000/30.000).

#### 9.4.1.4 Third observation date

At maturity, Total stock is trading at 42€ and is therefore ITM (delta = 1). This means that you hold a long position of 30.000 shares as delta hedge. Obviously, you have to sell these shares at the close of the last Asian setting.

#### 9.4.1.5 To sum up

On the day of an Asian setting, the trader needs to unwind part of his delta hedge if the Asian setting is ITM and does not need to do anything if the Asian setting is OTM.

If the Asian setting is ITM, the trader needs to unwind as a share position the number of Asian options multiplied by the weight of the setting.

This practical example also shows that the duration of an Asian option is indeed shorter than the equivalent European option. Effectively, an Asian option is spread out over a set of European options with maturities equal to the Asian observation dates. Therefore the term of an Asian option can be compared to a European option with a shorter term (estimated as the weighted average of the different Asian observation dates). That's normally the moment you should try comparing a 1-year Asian call with quarterly observations and a 6-month European call with the help of our exotic options pricer.

Since the higher the time to maturity, the higher the option price. It is clear (if it was not yet!) that the price of an Asian option should be smaller than the price of its equivalent European option.

### 9.4.2 Vega, Gama and Theta

We have just seen in the previous section that the duration of an Asian option is shorter than the equivalent European option. It is then easy to see that the greeks change accordingly.

It basically means that, compared to an equivalent European option, an Asian option has higher gamma and theta but a smaller vega. If you are not sure why, I invite you to (re)read about the impact of time to maturity on vega, gamma and theta in chapter 5 - The Greeks..