Financial Math Archive – “Tech Corner”

This briefing note covers concepts used in the valuation of certain derivative products. Quanto adjustments, Convexity adjustments and more. These are useful when modelling more complex financial products.

1. Quanto Adjustments

2. Convexity Adjustments

3. Asset Swapping

4. Funding Costs

1. Quanto Adjustments

When a derivative has a payoff in a currency other than the domestic currency of it's underlying, the forward price of the underlying must be adjusted. This adjustment is called the quanto adjustment.

Example: a swap dealer is asked to receive Sterling Libor (GBP) in Japanese Yen. the "quantoing" of the Sterling Libor is done via the following formula:

Sterling Libor + ( -v1 * v2 * corr )

where,

v1 = Sterling Libor volatility

v2 = JPY/GBP fx volatility (yen/sterling)

corr = correlation between v1 & v2

t = time in years to Sterling Libor setting date

The quanto adjustment is made because the swap dealer has risk to sterling rates going lower or yen rates going higher. The swapper will hedge this quanto risk by paying fixed in yen and receiving fixed in sterling, weighted by the current fx rate. The risk is that as the yen fx sells off relative to sterling, and Sterling Libor rates push up, the swapper finds himself overhedged in sterling swaps, with the position having moved against him. This +ve correlation is "bad", and makes the trade less valuable to have on.

Quanto adjustment for a Forward Price:

Method 1:

P forward = Pspot*EXP((r-d)t)

Adjusted P fwd = Pfwd*EXP(-v1*v2*corr*t)

where,

r = the domestic interest rate

d = dividend yield

Method 2:

Adjusted P fwd = Pspot*EXP((r-d-v1*v2*corr)*t)

2. Convexity Adjustments

When a rate which is applicable to one period is applied to an interest calculation period with a different tenor, the rate must be adjusted.

Example: A swap dealer is asked to enter into a 1 year Canadian interest rate swap, starting in one year. The swapper will receive the 1 year BA rate plus or minus a spread, and pay the 30 year CAD Swap Rate for the 1 year period, set in advance and paid in arrears.

The first step in pricing this swap is to forecast both the 1 year rate and the 30 year rate, starting in one years time.

The swap dealer has an exposure to 30 year swaps. He is at risk to the 30 year rate (set in one year) rising. He will hedge this risk by shorting some amount of 1+30 year swap. The problem is that the hedge has the negative convexity of a 30 year instrument, whereas the pay side of the original swap has the convexity characteristics of a 1 year instrument.

The "cost" of this negative convexity is captured in the following adjustment:

adjustment = (F x 100)^2 x (V x 100)^2 x T x gamma

2 x delta x 10,000

where,

F = 1+30 forecasted forward rate= 6.36%

V = volatility of 1+30 forward = 11.19%

delta = delta of $100 of 30 year forward swap = 0.14

gamma = gamma of $100 of 30 year forward swap = 0.000327

The adjustment is approximately 0.05%, and therefore the convexity adjusted 30 year rate used would be 6.41%

3. Asset Swap - an example

This example takes a premium priced bond, and cleans it up to a " par floater".

BOND

Ford 7.25% 15JAN03

YIELD: 7.10% (30 basis points over the swap rate)

PRICE: 100.296

PRICE + ACCRUED = 101.866441 (SETTLES 03OCT00)

Customer buys $100MM par amount of bond. The Swap rate is 6.80%. When the customer enters the swap, he effectively pays away the fixed cashflows of the bond, and receives floating (Libor based) cashflows from the swap dealer.

SWAP

NOTIONAL: USD 100,000,000

START: 03OCT00

MATURITY: 15JAN03

CUSTOMER PAYS: 7.25%, FULL FIRST COUPON

CUSTOMER RECEIVES: 3 MONTH LIBOR + 30, SHORT FIRST COUPON

CUSTOMER RECEIVES: USD$ 1,866,441 ON START DATE (The swap dealer "upfronts" the customer the amount over par that is required to buy the bond).

4. Funding Costs

Sometimes, swap market makers may be required to charge for loans which are sometimes a component of off-market transactions. These loans should be priced at a similar level to where the customer would go to the market to borrow, or at a level comparable to where their debt would asset swap to.

Lets assume the 5 year swap rate is 5.50% semi-annual.

In an asset swap, the customer pays the swap market maker 7.50% semi on $100 for 5 years, and receives BA's flat. In addition, the customer receives an upfront payment of $8.64.

This swap can be broken up into 2 components:

1. an at-market swap - 5.5% semi versus BA's flat on $100

2. a loan - where the customer receives $8.64 at the start of the deal, and pays 2% semi on the $100. This is similar to a mortgage, where the semi-annual payments of 2%*$100/2 represent repayment of the loan (principal and interest).

Using a spreadsheet or an HP calculator, both these components have an NPV of zero (are at-market) when discounting is done using the swap curve (the BA curve), here 5.5%.

The Swap market maker may require a spread to BA's when lending money. If BA's+25 is required, then the discount rate for the loan value would be 5.75%. Using the HP17b:

PV= -8.64

FV= 0

N= 10

if I%R = 5.50, PMT = 0.99999, or 2.000% semi on $100

if I%R = 5.75, PMT = 1.00642, or 2.013% semi on $100

This adds 1.3 basis points to original swap, requiring the customer to pay 7.513% semi.

This information is purely illustrative, using simplified examples for educational purposes only. For more information, please contact me at edmurphyconsulting@gmail.com