SERVING THE QUANTITATIVE FINANCE COMMUNITY

BerndSchmitz
Topic Author
Posts: 242
Joined: August 16th, 2011, 9:48 am

### Discounting of a trade collateralized in foreign currency

Hi everybody,

I was trying to show in the lines of "funding beyond discounting" that the PV of a domestic claim $V_T$ collateralized in foreign currency is given by
$V_t = E^Q \left[ e^{- \int_t^T r_{cfor}(s) ds} \frac{X_t}{X_T} V_T \right]$
where $r_{cfor}$ is the foreign collateral rate, $X_t = X_t^{dom/for}$ is the FX rate and $Q$ the measure associated with my domestic funding rate $r_F$ (or the corresponding bank account).

My thoughts are that at every point in time t I receive/pay $V_t$ as collateral, which has to be converted into foreign currency $\tilde{V}_t = \frac{V_t}{X_t}$ (probably I receive/pay $\tilde{V}_t$ straight away), earns $r_{cfor}$ and has to be converted back into domestic currency with $X_{t + \delta t}$.

However, I'm struggling to put that into the equation
$V_t = E^Q \left[ e^{- \int_t^T r_F(s) ds} V_T + \int_t^T e^{- \int_t^s r_F(u) du} (r_F(s) - ...) V_t ds \right]$

Does anybody have a solution or hint for me?

Thanks,
Bernd

BerndSchmitz
Topic Author
Posts: 242
Joined: August 16th, 2011, 9:48 am

### Re: Discounting of a trade collateralized in foreign currency

Does nobody have any idea?

Does anybody disagree with my valuation formula $V_t = E^Q \left[ e^{- \int_t^T r_{cfor}(s) ds} \frac{X_t}{X_T} V_T \right]$?

cenperro
Posts: 31
Joined: October 7th, 2014, 1:33 pm

### Re: Discounting of a trade collateralized in foreign currency

I think I disagree.. unless the FX is deterministic.
no FX vol should be involved in that calculation

BerndSchmitz
Topic Author
Posts: 242
Joined: August 16th, 2011, 9:48 am

### Re: Discounting of a trade collateralized in foreign currency

I guess you are right as I have found a similar conclusion in several sources (e.g. in Henrard's "Ineterst Rate Modelling in the Multi-Curve Framework"). Naively I thought that I can simply use $\frac{1}{X_t} e^{\int_t^T r_{cfor}(s) ds} X_T$ as the domestic collateral account (convert one unit of the domestic currency per spot FX rate into the foreign currency, let it grow at the foreign collateral rate and convert it back at the future FX rate) and plug this into the formula. Does anybody know why this rather heuristic approach produces the wrong result (i.e. what error do I implicitly make)?

savr
Posts: 48
Joined: January 21st, 2013, 3:28 pm

### Re: Discounting of a trade collateralized in foreign currency

The error is that Q should be taken to be the foreign risk-neutral measure.
(Suggest moving to Student Forum)

cenperro
Posts: 31
Joined: October 7th, 2014, 1:33 pm

### Re: Discounting of a trade collateralized in foreign currency

No need for Q to be the foreign risk-neutral measure. What is not correct is considering that the discount rate for the foreign collateralized trade is the  foreign rate...I do not have "funding beyond discounting" here so I do not know how the calculation is tackled there, but, when faced with foreign collateralization is good to think of all the transactions involved and how you hedge the FX risk embedded in the remuneration of the collateral account. With that in mind it is 'easy' to see how the collateral effect reduces (under certain hypothesis) to discounting with your domestic funding rate adjusted with the cross currency basis. In fact, if there is not basis, collateralization in one currency or the other results in the same value.

savr
Posts: 48
Joined: January 21st, 2013, 3:28 pm

### Re: Discounting of a trade collateralized in foreign currency

@cenperro
What do you mean "No need"? Both identities (Q=dom RN, Q=for RN) can't be simultaneously true when rates and FX are dependent, you have to choose one.

cenperro
Posts: 31
Joined: October 7th, 2014, 1:33 pm

### Re: Discounting of a trade collateralized in foreign currency

I mean that there is no need to involve a foreign Risk neutral measure in the calculation (with all the implications of changing measure for the underlying assets). If the investor is an USD investor facing and EUR collateralized trade (but written on USD underlyings: e.g: an USD swap) all calculations can be done in an USD (domestic) risk neutral and we just need to be careful with the 'discounting' rate e^{\int_t^T (r^x(s) } within the F-K solution of the valuation PDE. The right r^x in there is ~(usd rate plus a basis adjustment. THe foregin rate (EUR rate) is not involved). This can be seen via hedging arguments.
Could you use a foreign measure? for sure you could..you can always change to whatever measure you want but I do not think that helps in this case and could only complicate the problem

savr
Posts: 48
Joined: January 21st, 2013, 3:28 pm

### Re: Discounting of a trade collateralized in foreign currency

The right r^x in there is ~(usd rate plus a basis adjustment.
Agreed. But the OP's formula starts with a foreign collateral rate discount factor. And while the ratio of Xt/XT that follows contributes its drift to the effective discount rate, yielding as total the rate you correctly describe, it also leaves behind a smelly martingale part, which should not be there if Q is the domestic RN.

cenperro
Posts: 31
Joined: October 7th, 2014, 1:33 pm

### Re: Discounting of a trade collateralized in foreign currency

yes..as was not sure what she/he exactly meant by r_cfor did not want to say that. But you are right and the initial expression is 'fishy'  :-)