Consider a European rainbow option that gives the holder the right to buy either $100,000 of an equity index at a strike price of 1000 (asset 1) or $100,000 of a government bond (asset 2) with a strike price of 100% of face value, whichever is worth more at the end of 12 months. On January 15, 2008, the equity index is trading at 950, pays a dividend of 2% annually, and has a return volatility of 22%. Also on January 15, 2008, the government bond is trading at 98, pays a coupon yield of 6%, and has a return volatility of 15%. The risk-free rate is 5%. Using this data, calculate the price and sensitivity of the European rainbow option if the correlation between the rates of return is -0.5, 0, and 0.5.
Since the asset prices in this example are in different units, it is necessary to work in either index points (for asset 1) or in dollars (for asset 2). The European rainbow option allows the holder to buy the following: 100 units of the equity index at $1000 each (for a total of $100,000) or 1000 units of the government bonds at $100 each (for a total of $100,000). To convert the bond price (asset 2) to index units (asset 1), you must make the following adjustments:
Multiply the strike price and current price of the government bond by 10 (1000/100).
Multiply the option price by 100, considering that there are 100 equity index units in the option.
Once these adjustments are introduced, the strike price is the same for both assets ($1000). First, create the RateSpec
:
RateSpec = struct with fields:
FinObj: 'RateSpec'
Compounding: -1
Disc: 0.9512
Rates: 0.0500
EndTimes: 1
StartTimes: 0
EndDates: 733788
StartDates: 733422
ValuationDate: 733422
Basis: 1
EndMonthRule: 1
Create the two StockSpec
definitions.
StockSpec1 = struct with fields:
FinObj: 'StockSpec'
Sigma: 0.2200
AssetPrice: 950
DividendType: {'continuous'}
DividendAmounts: 0.0200
ExDividendDates: []
StockSpec2 = struct with fields:
FinObj: 'StockSpec'
Sigma: 0.1500
AssetPrice: 980
DividendType: {'continuous'}
DividendAmounts: 0.0600
ExDividendDates: []
Calculate the price and delta for different correlation levels.
Price =
111.6683
103.7715
92.4412
Delta =
0.4594 0.3698
0.4292 0.3166
0.4053 0.2512
The output Delta
has two columns: the first column represents the Delta
with respect to the equity index (asset 1), and the second column represents the Delta
with respect to the government bond (asset 2). The value 0.4595 represents Delta
with respect to one unit of the equity index. Since there are 100 units of the equity index, the overall Delta
would be 45.94 (100 * 0.4594 ) for a correlation level of -0.5. To calculate the Delta
with respect to the government bond, remember that an adjusted price of 980 was used instead of 98. Therefore, for example, the Delta
with respect to government bond, for a correlation of 0.5 would be 251.2 (0.2512 * 100 * 10 ).