Vectorized Functions to Estimate the Turnbull Wakeman Approximation for Continuous Asian Options
Asian options
Asian options are options for which the payoff depends on the average price (5 ) of the underlying asset over some period of time in the life of the underlying asset. Because of this fact, Asian options have lower volatility and hence are cheaper than European or American options. Typically, they are written on assets which have low trading volumes (thin trading) and high volatility' of the underlying price or on assets where there is the potential for its price to be manipulated. It is more difficult to manipulate the average value of an asset over a period of time than it is to manipulate it just at the expiration of an option. Such assets include electricity, crude oil, bunker prices, freight rates and futures/forward contracts on freight rates, amongst others. Thus, averaging of the spot price smoothens out the option payoff, in comparison to using the spot price on a particular date as is the case with conventional options. They were introduced in 1987 when Banker's Trust Tokyo office used them for pricing average options on crude oil contracts; hence the name "Asian" options.
If the payoff of a call option is calculated as max( S — X, 0) and the payoff of a put as max(X — S, 0), whereS is the average value of the underlying asset calculated over a predetermined averaging period, they are called average price call and put Asian options, respectively. If the payoff of a call option is calculated as max(S — S , 0) and the payoff of a put as max(S — S, 0) they are called average strike call and put Asian options, respectively. That is, the strike prices (X) of these options are an average of the spot prices of the underlying asset over a period of time. Average strike options can guarantee that the average price paid (received) for an asset in frequent trading, over a period of time, is not greater (less) than the final price. Asian options can be either European style or American style.
There are no known analytical pricing formulas when options are defined in terms of arithmetic averages. This is because the distribution of the arithmetic average of a set of lognormal distributions does not have analytical properties and therefore, the lognormal assumptions collapse. Therefore, Asian options are either based on the average price of the underlying asset, or alternatively, they are the average strike type. There are four commonly used pricing models to price Asian options: (i) the Kemma and Vorst (1990) geometric closed-form; (ii) the Turnbull and Wakeman (1991) arithmetic form approximation; (iii) the Levy (1992) arithmetic form approximation; and (iv) the Curran's (1992) approximation. These are discussed in turn.
Model 1: The Kemma and Vorst model
Kemma and Vorst (1990) proposed a closed-form pricing solution10 to geometric averaging price options, as the geometric average of the underlying prices follows a lognormal distribution, whereas under average rate options, this condition collapses." The prices of the geometric averaging Asian calls and puts are:
where, N(d) = cumulative standard normal distribution function; a = standard deviation of spot price; r = risk-free rate of interest; and q = dividend yield.
Given the price of the geometric Asian (VB ) derived from the closed-form solution of Kemma and Vorst (1990), we can price the arithmetic Asian (V л) as follows:
where, V* is the estimated value of the arithmetic Asian through simulation and V* is the estimated value of the geometric Asian through simulation. However, it should be noted that geometric average price Asian options are hardly ever traded in commodity markets and not at all in the freight market.
Model 2: The Turnbull and Wakeman model
As there are no closed-form solutions to arithmetic averages due to the inappropriate use of the lognormal assumption under this form of averaging, an analytical approximation was proposed by Turnbull and Wakeman (1991), which suggests that the distribution under arithmetic averaging is approximately lognormal. The prices of the calls and puts under this approximation are:
where, T, = time remaining until maturity; о = volatility; b = adjusted mean; and Mj and M, are the first and second moments of the arithmetic average, under risk neutrality.
If the averaging period has not yet begun, then T = T — t, while if the averaging period has begun T, = T. If the averaging period has begun, the strike price is adjusted as:
where, T = original time to maturity; X = original strike price; T, = remaining time to maturity; and S = average asset price. It should be noted that when the cost-of-carry is zero (like in the case of freight options), Equations (3.40) and (3.41) then become similar to Equations (3.31) and (3.32) under the Black (1976) model, as the underlying asset is a futures/forward.
Model 3: The Levy arithmetic rate approximation
Levy' (1992) extends the Turnbull—Wakeman analytical approximation by' arguing that Asian options should not be estimated on a continuous basis but rather on a discrete time. The prices of the calls and puts under this approximation are:
where, S = arithmetic average of the know asset price fixings.
The aforementioned analytical approximations by Turnbull and Wakeman (1991) and Levy (1992) can be computed using Monte Carlo simulation, which gives relatively accurate prices for path-dependent Asian options.
Model 4: The Curran approximation
Curran (1992) derives an approximation for arithmetic Asian options based on a geometric conditioning approach. By taking the natural logarithm both of the geometric distribution as well as of the price of the underlying asset at each point in time the underlying asset price can then be conditioned on the geometric distribution and integrate accordingly. Thus, the expected payoffs of Asian options can be calculated conditional on the risk-neutral distribution of the geometric average of the underlying asset. The prices of the calls and puts under this approximation are:
where, S = initial asset price; X = strike price of option; r = risk-free rate; b = cost-of- carry (zero in the case of non-storable commodities like freight); T = time to expiration in years; t( = time to first averaging point; At = time between averaging points; n = number of averaging points; cr = volatility of asset; and N(x) = the cumulative normal distribution function.
Other exotic options
The complex hedging needs of modern corporate and financial institutions often need more than the plain vanilla - traditional hedging products. Innovations in option pricing by financial engineers allow these institutions to customise hedging instruments in order to meet their market exposure. Therefore, several types of option contracts, known as exotic options, have been developed for hedging and investment purposes. Exotic options are tailor made to meet the user's hedging or investment requirements and their payoffs and structures can be quite complex. Exotic options are mostly traded in OTC markets. In the pricing models of the plain vanilla options most of the variables (i.e. spot market price, strike price, risk-free rate, volatility of the underlying asset, time to expiry, etc.) are assumed to be constant during the life of the options, whereas in the case of exotic options, the values of these variables can be changed, depending on the kind of exotic option. Besides Asian options, other types of exotic options include - see also Hull (2017): [1]
- • Non-standard American options: American option with extended features: (i) early exercise is restricted to certain dates - Bermudian option; (ii) early exercise is allowed only for a certain part of the life of the option; and (iii) the strike price may change during the life of the option.
- • Packages: Portfolio consisting of European put and call options.
- • Rainbow options: Options which involve two or more underlying assets.
- • Shout options: European style option where the holder "shouts" to the writer at one time during its life. The holder receives the payoff depending on the shout price or the price at the maturity, whichever makes the profit greater.
Source: https://ebrary.net/176106/business_finance/asian_options
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