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P2.T5.414. Exotic options: shout, Asian, and exchange

Nicole Seaman

Director of FRM Operations
Staff member
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AIMs: Identify and describe the characteristics and pay-off structure of the following exotic options: shout, Asian, and exchange

Questions:

414.1. At the beginning of the year, Acme Industry Company takes a long position (purchases) in an exotic shout option with a maturity of one year where the underlying asset is the price of oil. The strike price is $100.00. At the end of September, when the shout option has a remaining life of three months, the price of oil is $112.00 and Acme executes a "shout." The riskfree rate is 4.0% per annum with continuous compounding. Let c(S, K, σ, r, T) denote the Black-Scholes-Merton option value of a call option with current asset price (S), strike price (K) and maturity(T). At the time that the shout is exercised (+ 0.75 years into the life of the option with 0.25 years remaining until expiration), what is the value of the long position?

a. $12.00
b. $12.00 + BSM_c(112, 100, σ, 0.04, 0.25)
c. $12.00*exp(-4.0%*3/12) + BSM_c(100, 100, σ, 0.04, 0.25)
d. $12.00*exp(-4.0%*3/12) + BSM_c(112, 112, σ, 0.04, 0.25)

414.2. Consider an Asian option with a maturity of one year and where the average asset price is computed as the arithmetic average of four end-of-quarter prices, given by: $19.00 (end of March), $27.00 (end of June), $25.00 (end of September), and $17.00 (end of December; and the final asset price at option expiration). Which of the following Asian options offers the highest payoff at expiration?

a. An average strike call
b. An average strike put
c. An average price call with strike of $20.00
d. An average price put with strike of $20.00

414.3. Let c(S, K, σ, q, r, T) denote the standard Black-Scholes-Merton (BSM) price function of a European call option with current asset price (S), strike (K), volatility (σ), riskfree rate (r) and maturity (T). Consider an exotic exchange option which gives the holder the right but not the obligate to pay an asset (U), worth U(T) at expiration, in exchange for receiving another asset (V), worth V(T) at expiration. How is the BSM modified to value the exchange option?

a. K is replaced by U(T)
b. Riskfree rate (r) is replaced by dividend yield on asset (U)
c. Volatility (σ) is replaced by volatility of the difference between returns on (U) and (T)
d. All of the above

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