Asian Options: Practice Problems and Solutions

This is Section 51 of the Study Guide. See Section 1 here. See Section 2 here. See Section 3 here. See Section 4 here. See Section 5 here. See Section 6 here. See Section 7 here. See Section 8 here. See Section 9 here. See Section 10 here. See Section 11 here. See Section 12 here. See Section 13 here. See Section 14 here. See Section 15 here. See Section 16 here. See Section 17 here. See Section 18 here. See Section 19 here. See Section 20 here. See Section 21 here. See Section 22 here. See Section 23 here. See Section 24 here. See Section 25 here. See Section 26 here. See Section 27 here. See Section 28 here. See Section 29 here. See Section 30 here. See Section 31 here. See Section 32 here. See Section 33 here. See Section 34 here. See Section 35 here. See Section 36 here. See Section 37 here. See Section 38 here. See Section 39 here. See Section 40 here. See Section 41 here. See Section 42 here. See Section 43 here. See Section 44 here. See Section 45 here. See Section 46 here. See Section 47 here. See Section 48 here. See Section 49 here. See Section 50 here. See Section 51 here.

An Asian option "has a payoff that is based on the average price over some period of time" (McDonald 2006, p. 444). An Asian option is thus, path-dependent in that the way in which the underlying stock reached its final price matters for determining the price of the option. An Asian tail refers to settling on the basis of the average price.

Asian options can be based on an arithmetic or a geometric average.

The formula for the arithmetic average of N stock prices is A(T) = (1/N)_i=1^NΣS_ih

The formula for the geometric average of N stock prices is G(T) = (S_h*S_2h*....*S_Nh)^(1/N)

It is always the case that G(T) ≤ A(T)

An Asian average price option occurswhen the average used is the asset price.

An Asian average strike option occurswhen the average used is the strike price. There are four different possible types of Asian options using arithmetic averages and four different types of Asian options using geometric averages. The following formulas hold for the payoffs of the geometric average Asian options. To find the payoffs of arithmetic average Asian options, simply replace G(T) with A(T) in the relevant equation.

Geometric average price call = max[0, G(T) - K]

Geometric average price put = max[0, K - G(T)]

Geometric average strike call = max[0, S_T - G(T)]

Geometric average strike put = max[0, G(T) - S_T]

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 14, pp. 444-449.

Original Practice Problems and Solutions from the Actuary's Free Study Guide:

Problem AO1. At the end of each of the past four months, the stock of Fluctuating Co. had the following prices: $345, $435, $534, and $354. A certain 4-month Asian geometric average price call on this stock has a strike price of $400. It expires today, and its payoff is computed on the basis of the geometric average of the stock prices given above. What is the payoff of this option?

Solution AO1. We first find G(T) = (345*435*534*354)^(1/4) = 410.405543. The payoff of the option is thus Geometric average price call = max[0, G(T) - K] = 410.405543 - 400 = $10.405543

Problem AO2. At the end of each of the past four months, the stock of Fluctuating Co. had the following prices: $345, $435, $534, and $354. A certain 4-month Asian arithmetic average price call on this stock has a strike price of $400. It expires today, and its payoff is computed on the basis of the arithmetic average of the stock prices given above. What is the payoff of this option?

Solution AO2. We first find A(T) = (345 + 435 + 534 + 354)/4= 417. The payoff of the option is thus Arithmetic average price call = max[0, A(T) - K] = 417 - 400 = $17

Problem AO3. A highly peculiar Asian geometric average strike put on the stock of Lucrative Co. has a life of 6 months and is computed based on an average of monthly strikes. The option's payoff is based on the following strike prices: $4 in month 1, $1500 in month 2, $5 in month 3, $1400 in month 4, $23.4 in month 5, $1322 in month 6. At the end of month 6, the stock of Lucrative Co. trades at $44 per share. What is the payoff of this option?

Solution AO3. We first find G(T) = (4*1500*5*1400*23.4*1322)^(1/6) = 104.4598586.

The payoff of the option is thus Geometric average strike put = max[0, G(T) - S_T] = 104.4598586 - 44 = $60.45985865

Problem AO4. The stock price of Predictable Co. is $1 at the end of month 1 and increases by $1 every month without exception. A 99-month Asian arithmetic average price put on Predictable Co. stock has a strike price of $56 and a payoff that is computed based on an average of monthly prices. The option expires at the end of month 99. What is the payoff of this option?

Solution AO4. We recall that 1 + 2 + 3 +... + n = n(n+1)/2.

To find A(T), we need to find (1 + 2 + 3 + ... + 99)/99 = (99*100/2)/99 = 50.

To find the payoff, we use the formula Arithmetic average price put = max[0, K - A(T)] = 56 - 50 = $6

Problem AO5. The stock price of Predictable Co. is $1 at the end of month 1 and increases by $1 every month without exception. A 10-month Asian geometric average price call on Predictable Co. stock has a strike price of $2 and a payoff that is computed based on an average of monthly prices. The option expires at the end of month 10. What is the payoff of this option?

Solution AO5. Here, G(T) = (1*2*3*...*10)^(1/10) = (10!)^(1/10) = 4.528728688

We find the payoff by using the formula Geometric average price call = max[0, G(T) - K] = 4.528728688 - 2 = $2.528728688

See other sections of The Actuary's Free Study Guide for Exam 3F / Exam MFE.