Asian Options I

Today I’m going to introduce the first exotic option that I’ve looked at in my blog, the Asian option. There is quite a lot to be said about them so I’ll do it over a few posts. I’ll also be updating the pricers to price these options in a variety of ways.

The simplest form of Asian option has a payoff similar to a vanilla option, but instead of depending on the value of the underlying spot price at expiry, it instead depends on the average the underlying spot price over a series of dates specified in the option contract. That is, the value of the option at expiry C(T) is the payoff

    \[C(T) = \Big( {1\over N}\sum^{N}_{i=0} S_{t_i} - K\Big)^+\]

where K is the strike, N is the number of averaging dates, and S_t is the value of the spot that is realised at each of these dates.

First of all, what does it mean that this is an exotic option? From the market, we will be able to see the prices of vanilla options at or near each of the dates that the contract depends on. As I discussed in my post on Risk-Neutral Valuation, from these prices we can calculate the expected distribution of possible future spot prices at each date – shouldn’t we be able to use this to calculate a unique price of the Asian option consistent with the vanilla prices?

The answer, unfortunately, is no. Although we know the distribution at each time, this doesn’t tell us anything about the correlations between the price at time t_1 and time t_2, which will be critical for the Asian option. To illustrate this, let’s consider a simple Asian that only averages over two dates, t_1 and t_2, and we’ll take t_2 to be the payoff date for ease, and try to calculate its price via risk-neutral valuation. The price at expiry will be

    \[C(t_2) = \Big( {1\over 2}\big(S_{t_1} + S_{t_2}\big) - K\Big)^+\]

To calculate the price at earlier times, we use the martingale property of the discounted price in the risk-neutral measure

    \[C(0) = N(0)\cdot{\mathbb E}^{RN} \Bigl[ \ {C(t_2) \over N(t_2)}\ |\ {\cal F}_0\ \Bigr]\]

and expanding this out as an integral, we have

    \[C(0) = \delta_{0,t_2} \int \int \Big( {1 \over 2} \big( S_{t_1} + S_{t_2} \big) - K \Big)^+\ p(S_{t_1},S_{t_2})\ dS_{t_1} dS_{t_2}\]

where \delta denotes the discount factor to avoid confusion with the pdfs inside the integral. From the market, we know both p(S_{t_1}) and p(S_{t_2}), which are respectively the market-implied probability distributions of the spot price at times t_1 and t_2 calculated from vanilla call prices using the expression we derived before

    \[p(S_{t}) = {1 \over \delta_{0,t}}{\partial^2 C(t) \over \partial K^2}\]

But, we don’t know the JOINT distribution p(S_{t_1},S_{t_2}) which expresses the effect of the dependence of the two distributions. We know from basic statistics that

    \[p(S_{t_2},S_{t_1}) = p(S_{t_2} | S_{t_1}) \cdot p(S_{t_1})\]

and since the spot at the later time depends on the realised value of the spot at the earlier time p(S_{t_2} | S_{t_1}) \neq p(S_{t_2}), so we don’t have enough information to solve the problem from only the market information available from vanilla options.

What is this telling us? It is saying that the market-available information isn’t enough to uniquely determine a model for underlying price movements with time. There are a variety of models that will all re-create the same vanilla prices, but will disagree on the price of anything more exotic, including our simple Asian option. However, observed vanilla prices do allow us to put some bounds on the price of the option. For example, a careful application of the triangle inequality tells us that

    \[\Big( {1\over 2}\big(S_{t_1} + S_{t_2}\big) - K \Big)^+ \leq {1\over 2} \Big( S_{t_1} - K_1 \Big)^+ + {1\over 2} \Big( S_{t_2} - K_2 \Big)^+\]

for any K_1 + K_2 = 2K; the expressions on the right are the payoff at expiry of call options on S_{t_1} and S_{t_2} respectively, and we can see the prices of these on the market, which allow an upper bound on the asian price. This means that, while we can’t uniquely price the option using the market information alone, we CAN be sure that any model consistent with market vanilla prices will produce an asian price that is lower than the limit implied by this inequality.

In order to go further, we need to make a choice of model. Our choice will determine the price that we get for our option, and changing model may well lead to a change in price – analysing this ‘model risk’ is one of a quant’s main jobs. A very simple choice such as Black Scholes [but note that BS might not be able to re-create an arbitrary vanilla option surface, for example if there is a vol smile at some time slices] will allow us to solve the integral above by numerical integration or by Monte Carlo (this option form still doesn’t have a closed form solution even in BS).

In the coming posts I’m going to develop the Monte Carlo pricer on the blog to price asian options in BS, and look at a number of simplifying assumptions that can be made which will allow us to find closed form approximations to the price of Asian options.

 

Finally, a brief note on the naming and usage of Asian options. Opinion seems divided – some people believe they are called Asian options because they’re neither European nor American, while some people say it’s because they were first traded in Japan. They are often traded so that the days on which the averaging is done are the business days of the last month of the option’s life. As the look up period is spread out over a period, they are less susceptible to market manipulation on any particular date, which can be important in less liquid markets, although the theoretical prices of Asian options like this are usually very similar in price to vanilla options at any of the look up dates. More exotic Asians might average over the last day of a month for a year. This average is much less volatile than the underlying over the same period, so Asian options are much cheaper than a portfolio of vanillas, but it’s easy to imagine that they might well match the risk exposure of a company very well, since companies trading in two or more currency areas will probably be happy to hedge their average exchange rate over a period of time rather than the exchange rate at any one particular time.

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