Following the first article of the series in which we analyzed the greeks of a Binary option, in this second part of the series we would like to introduce Asian options and their greeks.

Asian options are part of a broader macro-class of exotic derivatives named Path Dependent Derivatives; in fact *exos *can be divided into three main categories:

1. Contracts with discontinuous payoff – such as binary (or digital) options

2. Contracts whose payoff does not depend (only) on the value of the underlying at maturity, that is, **Path Dependents Derivatives** – Asian options are part of this category; within this category we distinguish among **weak path dependent derivatives **and** strong path dependent derivatives**

3. Contracts whose underlying is not a single stock or a single stock index but rather a *basket* of different securities – these are called **Multi-Asset Derivatives**

Of course an exotic derivative may belong to more than one category (think for instance of a barrier option whose underlying is a basket of stocks…)

In addition to outlining some features, scope and technicalities of Asian options, in this article we conduct an in-depth investigation of the Delta and the Vega of a simple averaging Asian option and we also briefly describe the behaviour of Gamma. We further look into how increasing the number of the fixings changes the greeks of our option.

*Asian options – an Introduction*

Asian options or average rate options are options whose payoff at maturity is not based on the final price of the underlying asset, but on the average of the prices registered during the life of the option. There exist different categories of Asian options according to the kind of average used in the payoff calculation (hence the distinction between arithmetic and geometric average rate options) and to the time interval between two following registered prices (which can be a day, a week or a month). Also in this case, there exist many variants to the traditional contracts, such as the partial average rate options, in which the prices are registered only in a time interval shorter than the whole life of the option.

Since they payoff of an Asian option depends not only on the value of the underlying at maturity but also on the value taken by the underlying on “fixing dates”, that is, Asian options are path-dependent, no closed form solution exist as long as arithmetic average is used, even though some approximations do exist. For this reason we decided to study sensitivities of this type of options through Montecarlo Simulations.

As far as geometric average is concerned, it is possible to find analytical formulas for pricing Asian call and put options. This is due to the fact that the geometric average of log-normal variables is log-normally distributed in the risk-neutral world.

However, a good approximation for the price of an average option is given in Kemna and Vorst where the geometric average option is priced using Black–Scholes closed formulas, with the growth rate (r – q) set to \left(r – q – \dfrac{\sigma^2}{6} \right)/2, and the volatility set to \dfrac{\sigma}{3}.

Since the payoff of Asian options is based on the average of the underlying asset prices during the term of the product, the uncertainty concerning the fluctuations of the underlying price at maturity decreases. Therefore, the risk exposure to the spot price and volatility is lower for an Asian option compared to a regular European option.

In fact, the volatility of the average of a stock price over time is less than the volatility of the stock. The average is less exposed to sudden crashes or rallies in a stock price, meaning that all is not lost if you own a call and the stock plummets just before expiration.

Also, the higher the number of observations, the lower the price of the option, as shown in the chart below. This is due to the fact that the higher the number of fixings of the average value of the underlying price, the smaller the volatility of the average, ceteris paribus. An Asian option is always worth less than a European equivalent and the higher the more fixing dates there are, the lower the cost of that Asian option.

The averaging periods can be uniform during the life of the option, i.e. the structure can take many forms including weekly, monthly, quarterly or annual averagings from inception to maturity date. These Asian options are said to be averaging-in style.

For instance, a 2-year at-the-money Asian call averaging-in quarterly is a call paying the positive average of the eight underlying returns observed at the end of each quarter throughout the term of the product. Asian options can also be averaging-out style, which means that the average is computed during a specific period near the maturity date. A 2-year 90% Asian call averaging-out monthly during the last year is an out-of-the-money put struck at 90% and for which the average price of the spot is based on the underlying spot closing prices observed during the last 12 months of the product’s life.

In the general case, averaging-in style options are less risky than averaging-out Asian options. Indeed, the uncertainty about future spot prices is lower when the average is computed periodically since the inception date.

In addition to changes in the averaging period and in the way to compute the average price it is worth noting that also the strike price K may not be fixed, that is, the strike price may be floating: this type of Asian option are called average strike option.

An average strike call pays off max (0; S(T) – Savg) and an average strike put pays off max (0; Savg – S(T)). Average strike options can guarantee that the average price paid for an asset in frequent trading over a period of time is not greater than the final price. Alternatively, it can guarantee that the average price received for an asset in frequent trading over a period of time is not less than the final price. It can be valued as an option to exchange one asset for another when S avg is assumed to be lognormal.

As outlined above the most common averaging procedures are the discrete arithmetic averaging, defined by:

A_T = \frac{1}{n}\sum^{n}_{i = 1} S_{t_i}

And the discrete geometric averaging, defined by:

G_T = \left[ \prod_{i = 1}^{n} S_{t_i} \right]^{1/n}

Here St is the asset price at discrete time

In the limit as n ∞, the discrete sampled averages become the continuous sampled averages. The continuous arithmetic average is given by

A_T = \frac{1}{T} \int_0^T{S_t dt}

while the continuous geometric average is defined to be

G_T = \exp{ \left( \frac{1}{T} \int_0^T{S_t dt} \right)}

Because the geometric average of a set of values is always lower than the corresponding arithmetic average, the price of an Asian call with geometric averaging will always be lower than the price of an Asian with arithmetic averaging whilst the opposite happens for Asian puts.

Before introducing our greeks analysis we would like to make some considerations about the behaviour of the price of the Asian option; however we would like to emphasize that the evolution of the price as time passes is highly contract-dependent. What we will comment is the change in the price of a future-style option (that is, an option in which the premium is paid at maturity) written on a future contract. In this way the interest rate (r*) plays no role either on the discounting of the option premium, if paid upfront, and on the expected evolution of the underlying price, since the drift of a future under the risk neutral measure is zero.

Therefore we invite the reader to keep in mind that the statements contained in this article can not be generalized to all Asian options for any level of moneyness. For instance, later in the article, we show that for a deep ITM option, the price of our Asian option is almost flat as time passes. However, this wouldn’t be the case if the premium was paid upfront and there was a positive interest rate r* since the PV of the payoff would increase and this effect would dominate the (almost zero) time value decay.

We start our analysis by studying the behaviour of the price of a “simple” future-stlye Asian option written on a future contract across time; our option is issued at t = 0, has a fixed strike at 100, a life of 1 year and 4 fixings dates (one each 3-month period, ie. at t = 0.25 = 3 months, t = 0.5, t = 0.75 and t = 1 = maturity).

Pretty much as for a “standard” European call option on a non dividend paying stock, the time value decreases as time passes. Since our “reference” option is a future-style option written on a future, we can say that the overall price of the option goes down as time passes (as shown in the chart above). Therefore, for this particular contract we are allowed to say that the Theta (ie, the time decay) of the option is negative (as time passes, the value of the option decreases as time value decreases) for all level of moneyness. However, there is also a characteristic feature of Asian options that should be taken into account: the time decay is larger, especially around the ATMness, when there is a “fixing” – in other words, the passage of a certain time interval Δt which contains a fixing date t* has a greater impact on the price of the option compared to the passage of a Δt time interval in which there are no fixing dates.

This intuitively tells us that the Theta of this Asian option is a non monotonic function – by looking at the above chart this is confirmed by the fact that the reduction in the value of the option between t = 3 months (ie, one moment after the fixing) and t = 4 months is smaller than the decay in the option price between t = 2 months and t = 3 months (ie, the Δt that “contains” the fixing date).

Looking at an ATM Asian call option, we see that the decrease in the price as time passes is a negative function of Implied Volatility, i.e. the higher the IV the faster the loss in the option value due to the passage of time. In other words Theta gets more negative as Implied Volatility rises, meaning that the **Veta** is negative.

*Asian options – the Delta*

The chart of the evolution of the Delta for different level of time and moneyness tells us that the **Charm** or **Delta Bleed** is a function that, between two fixing dates, will have either positive or negative sign depending on the moneyness (if the sian option is ITM the Delta will increase as time passes, ie the **Charm** is positive, whilst the **Charm** is negative for OTM option) – this can be intuitively explained by remembering that the Delta is bounded between some values (0 and 1 for a European call option) and as an options gets deeper ITM and deeper OTM its Delta must get closer to the boundaries of the Delta values (i.e., the more OTM an option becomes the closer its Delta gets to the minimum value that can be taken by the Delta).

However the Delta will always be decreasing, in absolute values, for any Δt that contains at least one fixing date. This means that **Charm** for t = q(n), where q(n) is the *n-th* fixing date will always be negative for a call option and positive for a put option. This feature can be easily explained by the fact that the values between which the Delta of an aAian is bounded are not constant but, rather, they decrease in absolute values as fixing dates pass.

How is it possible to intuitively justify this? If the prices of the underlying goes up by one immediately after the issuance and then stays flat, its impact would be felt across all the K fixing dates of the Asian option whilst if the unit increase in the price of the underlying happens just after the first fixing date, its effect would be “limited” on (K – 1) fixings (since one has already passed). Therefore, given any Asian call, the upper bound of the value of the Delta at anytime during the life of the option will be 1 – ( Q(t) / Q ) with Q(t) being the number of the fixing dates that have already passed and Q being equal to the total number of fixings over the life of the option.

For instance, if an option has two fixings, the maximum value of the Delta for an Asian call after the first fixing date will be 1 – ( 1 / 2 ) = 0.5 given that Q(t) = 1 and Q = 2 since the next fixing date will be the last one. In the same way the maximum Delta value for an option with four fixings after the first fixing date has passed will be 1 – ( 1 / 4 ) = 0.75; for the same option the upper bound of the Delta immediately before maturity would be 0.25.

This happens because the final payoff of an Asian option is calculated as the difference between the average of the price of the underlying in K dates and the strike, meaning that what happens to the underlying after a certain amount of fixings will have an impact only on the remaining fixing dates and thus on the “corresponding” part of the payoff.

For example, if after being flat for 3 of the 4 fixings at 100 the underlying price (the future price in our case) skyrockets to 150, this impressive jump will account only for ¼ of the payoff structure of our option. In other words, the final payoff, assuming strike K = 100, will be given by

\frac {100+100+100+150} {4} - 100 = \frac {450} {4} - 100 = 12.5

That is equivalent to say:

0.25\times100 + 0.25\times100 + 0.25\times100 + 0.25\times150 - 100 = 12.25

If the jump in the price happened just after the first fixing date (and after it the underlying price stayed flat at 150) we would have:

0.25\times100 + 0.25\times150 + 0.25\times150 + 0.25\times150 - 100 = 37.5

The intuition we just outlined is confirmed by the chart above where the evolution of Delta across time for different levels of moneyness is plotted. In fact, the Delta of a deep ITM Asian call option (since K = 100 and the price of the underlying future is flat at 120 – ie, the red line) is very close to its maximum level which is 0.75 after the first fixing (t = 0.25) then it falls after the second fixing date at 0.50 and then finally drops to 0.25 on the third fixing date.

It’s interesting to note that, for the slightly ITM option (yellow line) there are two opposite sign effects on the Delta: in-between two fixings it grows because of the roll-up effect and then it experience a negative jump on each fixing date. These two phenomena also influence the slightly OTM option, however in this case, both the effects have a negative sign on the Delta (that is, the charm is always negative).

In addition to this, it is worth to look at the behaviour of the Delta of the ATM option: it mirrors the shape of the deep ITM Delta however both its levels across time and the jumps on fixings dates are almost equal to one half of the associated values for the deep ITM option. This can be easily understood by recalling that ATM vanilla options are commonly referred to as “50-Delta” options.

Among ATM Asian options, the Delta increases marginally as Implied Volatility IV increases. Also, the time decay between two fixings in the Delta is higher the higher the IV. Therefore we say that, at least for ATM Asian options, the charm is a positive function of the IV and thus the **Vanna** (the change in the Delta due to a change in IV) is also positive across time.

One final consideration about the Delta is that, all other things equal, the Delta is smaller the higher the number of fixing dates. This can be intuitively derived from the idea outlined above that the passage of a fixing date reduces the relevance of future price fluctuations. The higher the number of fixings in the past, the lower the importance of future fixings. In fact, in the chart above all the Deltas have the same value until the first fixing date happens (*one month after issuance, blue line*)

As for simple European options the Vega is positive across all time to maturities and all level of moneyness, meaning that the buyer of an Asian option is Vega long whilst the seller is Vega short.

Regarding the behaviour of Vega, we see that, pretty much in line with European options, Vega decreases as time passes even for ATM options meaning that the derivative of the Vega with respect to the passage of time is negative and it is not a function of the moneyness. We can thus say that the **Veta** is negative. Incidentally, we already saw that the **Veta** was negative by analyzing the effect of the time decay for an increase in Implied Volatility (recalling the Clairaut’s theorem on equality of mixed partials derivatives). However now we can extend our consideration about the **Veta** for any level of moneyness: the effect of the passage of time on the value of the option is a positive function of the Implied Volatility for any value of the underlying price.

But how about the “value” of the **Veta**? Does the moneyness have an impact on the speed at which the Vega decreases as time passes? Chart below shows that the level of moneyness matters for the velocity at which Vega falls as a result of the passage of time: ATM options have the lowest Vega decay (their **Veta** is less negative or smaller in absolute terms) whilst options that start ATM and then they fall and expire slightly ITM and slightly OTM share the same path of Vega decay (i.e., they have a **Veta** of the same magnitude). Compared to the ATM option, their **Veta** accelerate (becomes more negative) faster and then converge into deep ITM and deep OTM to a Vega level of 0.

Finally, from the chart above we can see that for every level of moneyness **Veta** does accelerate if there are fixing dates within the Δt. This implies that the **Veta** is non monotonic but rather peaks on fixing dates and after a fixing it falls.

An interesting feature of Asian options is that they have almost no “Vega convexity”, that is, their **Vomma **(or **Volga**) is 0, as shown in the chart below. Our intuition is that this is the result of the fact that what really matters with Asian options is the average of the underlying and thus the volatility of the average rather than the volatility of the underlying itself.

As it happens for the Delta, the Vega also decreases as the number of fixing dates increases.

However, in contrast with what happens with the Delta, Vega of Asian options with different number of fixings is not the same at issuance: the higher the number of fixing dates the lower the Vega at issuance and the lower the magnitude of the **Veta** (i.e., the slower the roll down – “decay” – of the Vega).

*Asian options – the Gamma*

As far as the Gamma of an Asian option is concerned, there are no general statements that can be made: not only the Gamma convexity (Speed) is a function of moneyness but also the peak for some values of the time does not coincide with the ATM point, that is, K = 100 in our example; also the sign of the Gamma decay (**Zomma**) varies depending not only on the price of the underlying but also on the time.

The way we derived the Gamma of this contract is not straightforward: first of all we looked at the value of Delta for different prices of the underlying. With these data we then conducted a *spline *interpolation in order to get the smallest interpolation error possible. After obtaining the spline we then calculate its derivative with respect to the underlying price. We had to do this because the simple Monte Carlo process gave us unstable results. For this reason, we would like to suggest the reader to use caution when dealing with the chart and the concepts related to the Gamma outlined above.

*Asian options – a Final Remark*

In this article we presented Asian options and some of their sensitivities. However we would like to stress that for these path-dependent options there are a very low number of general statements that can be made regarding the behaviour of the greeks of the contracts – in addition to the usual contract specification that can make a difference for every option, including the vanilla ones, such as whether the option is or not paid upfront, there is a lot of “variance” in the “gamma” (set) of Asian options: for instance, the averaging process can be either arithmetic or geometric with the latter being much more easier because of the existence of closed-form solutions, the strike price may not be fixed and so on… What we are trying to say with this derivative contracts specifications matter a lot!

Furthermore, as previously mentioned, our results are related to one specific, and very easy, contract. Moreover all the results are obtained through simulations. We would thus invite the reader to be cautious with learning by heart what is contained in this article and rather we advise to focus on some common traits of the Asian contract.

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