The prerequisite for this course is the first course on pricing interest rate swaps.
Online Finance – Pricing Interest Rate Swaps – The valuation course
And the second more advanced course on interest caps, interest rate floors and other related fixed income options. The same foundations defined here can be used to price range accrual notes as well as commodity linked notes as described in the next two concepts.
- Online Finance – Pricing a Cross Currency Swap – Floating for Floating structure
- Online Finance – Pricing a Cross Currency Swap – Amortizing and Indexed Term sheets
- Online Finance – Interest Rate Options – Pricing Caps & Floors
Commodity Linked Note
These are similar to the accrual range note in that their payment is tied to the performance of a specific physical commodity or commodity index. The procedure followed is similar to that given above, expect that F_{i} now represents the future commodity price. This is calculated using the following formula:
Where S_{0} is the spot price of the commodity today, r is the risk free rate (of the currency in which the price is denominated), q is the convenience yield of the commodity, T is the duration between the payment date and the valuation date.
For instance, let us assume the note ties the payment to the performance of Brent Crude Oil prices.
On the valuation date, the price of Brent is USD 73.47 per barrel. The base volatility of the forward prices is assumed to be 2%. The USD risk free rate is 0.2% and the convenience yield is assumed to be 0%.
The terms for the note are as follows:
Terms and Conditions | |
Notional |
100,000 |
Binary Rate |
12.00% |
Payment |
Annual |
Payment Dates |
1st Day of the year starting January 2011 and ending on the Maturity Date |
Valuation Date |
01/06/2010 |
Maturity Date |
01/01/2014 |
Day count convention |
A/365 |
Lower Bound |
USD 73.00 |
Upper Bound |
USD 74.00 |
The instrument pays 12% annually if the price falls with the range of (73, 74).
Using the forward price formula, above the forward prices at each future payment date is as follows:
Period Start (i) |
Period End (i+1) |
t_{i+1} |
F_{i} |
01/01/2010 |
01/01/2011 |
0.59 |
73.56 |
01/01/2011 |
01/01/2012 |
1.59 |
71.84 |
01/01/2012 |
01/01/2013 |
2.59 |
73.26 |
01/01/2013 |
01/01/2014 |
3.59 |
76.99 |
And the results of the calculation are as follows:
Period Start (i) |
Period End (i+1) |
t_{i} |
t_{i+1} |
ZCt_{i+1} |
d2-Lower Bound |
d2-Upper Bound |
N(d2)-Lower Bound |
N(d2)-Upper Bound |
01/01/2010 |
01/01/2011 |
n/a |
0.59 |
12.150% |
n/a |
n/a |
n/a |
n/a |
01/01/2011 |
01/01/2012 |
0.59 |
1.59 |
12.225% |
-1.05 |
-1.94 |
0.14662 |
0.02622 |
01/01/2012 |
01/01/2013 |
1.59 |
2.59 |
12.349% |
0.13 |
-0.41 |
0.55106 |
0.34024 |
01/01/2013 |
01/01/2014 |
2.59 |
3.59 |
12.418% |
1.64 |
1.21 |
0.94922 |
0.88773 |
‘d_{2}-Lower Bound’ uses the Lower Bound price of 73, as X in the formula, whereas ‘d_{2}-Upper Bound’ used the Upper Bound rate of 74 as X in the formula. In the first row the d_{2} values are “n/a” because the period start date falls before the valuation date.
Period Start (i) |
Period End (i+1) |
Probability of being within range |
Cash flow |
PV of Cash flow |
01/01/2010 |
01/01/2011 |
1.00000 |
7,035.62 |
6,578.17 |
01/01/2011 |
01/01/2012 |
0.12040 |
12,000.00 |
1,203.23 |
01/01/2012 |
01/01/2013 |
0.21081 |
12,000.00 |
1,871.33 |
01/01/2013 |
01/01/2014 |
0.06149 |
12,000.00 |
484.75 |
Price |
10,137.47 |
The probability that the price will fall within the range is equal to ‘N(d_{2})-Lower Bound’ minus ‘N(d_{2})-Upper Bound’. For the first row the probability will be 1 if F_{i} lies within the range and 0 otherwise.
The Cash flow column is equal to the Binary Rate (i.e. the rate at which payments are to be made) * Notional. For the first period, since it is fractional, the cash flow will be multiplied by the remaining tenor in that period to determine the payment, i.e. 12%*100,000*0.59.
The PV of Cash flow = (Cash flow*Probability)/( (1+zct_{i+1})^ t_{i+1} ).
The price is the sum of across all PVs.
For a complete reference to equations and calculator referred to in our course catalog, please see the Derivative Pricing and Financial Risk Equation Glossary.
For topic specific equations, please see the following links:
- Calculating Value at Risk
- Duration, Convexity and Asset Liability Management
- Black Scholes, Derivative Pricing, Binomial Trees
- Calculating Forward Prices and Forward Rates
- Valuation of Interest Rate Swaps and Future Contracts
- Financial Risk, Reward metrics and measures
- Black Formula’s, Valuing Interest Rate Caps and Floors