VaR for interest rate option portfolios

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Gaia
Posts: 6
Joined: Sat Jan 24, 2009 1:26 am

VaR for interest rate option portfolios

Postby Gaia » Mon Jan 26, 2009 3:55 pm

I have a portfolio of interest rate derivatives, say plain vanilla swaps, caps and floors, and I have an O-Garch(1,1) covariance matrix for 13 points of the euro interest rate curve. Now I want to compute 1-day and 10-days VaR and CVaR (or Expected Shortfall) of my portfolio.
I'm thinking about using historical bootstrapping technique to generate the P&L distribution for the portfolio returns, this should not create problem for 1-day horizon and I will check if it creates problems with 10-days horizon, since bootstrapping destroy any mean reversion property. My data (from which I will bootstrap) are historical returns of interest rates-I have an interest rate curve composed by 13 points, starting from overnight rate up to 25-year rate. Thus I generate scenarios for interest rates, then in each scenario I price my portfolio of interest rate derivatives and calculate the difference between each simulated portfolio value and the original portfolio value to have a P&L distribution of portfolio returns from which to calculate VaR and CVaR. My questions are:

a. What is the input volatility to be used in each scenario to price caps and floors? Should I simulate the volatility in each scenario or should I keep constant for each scenario my initial estimate of the covariance matrix of interest rates? What do practitioners do in practice? Does anyone have any reference on this?

b. Suppose I have in my portfolio some exotic that must be priced by simulation. Does this imply that in each scenario I must simulate the price of the exotic? That is, suppose I price an exotic on a Hull White tree; is it correct that when I calculate VaR with historical simulation, in each scenario I have to build the tree to price the exotic?

c. Should all these issues suggest me to assume that returns on interest rates are normally distributed and then use parametric VaR and CVaR? Basically, does anyone know with which model I am less wrong?

Thank you very much!
Gaia

coalexander
Posts: 815
Joined: Sun Sep 28, 2008 10:30 pm

Re: VaR for interest rate option portfolios

Postby coalexander » Tue Jan 27, 2009 11:38 am

Hi Gaia

Lots of questions, so my answers are repeated after them for ease of reading this reply.

Gaia: .... I will check if it creates problems with 10-days horizon, since bootstrapping destroy any mean reversion property.
COA: Note that using the GARCH model in the bootstrap, as in the filtered historical simulation method described in Section IV.3.3.4 will not destroy the mean-reversion property. I attach an extract from Section IV.3.3 (Title of Section: Improving the Accuracy of Historical VaR) and an excel example that does filtered historical simulation.

Gaia: My data (from which I will bootstrap) are historical returns of interest rates-I have an interest rate curve composed by 13 points, starting from overnight rate up to 25-year rate. Thus I generate scenarios for interest rates, then in each scenario I price my portfolio of interest rate derivatives and calculate the difference between each simulated portfolio value and the original portfolio value to have a P&L distribution of portfolio returns from which to calculate VaR and CVaR. My questions are:

a. What is the input volatility to be used in each scenario to price caps and floors? Should I simulate the volatility in each scenario or should I keep constant for each scenario my initial estimate of the covariance matrix of interest rates? What do practitioners do in practice? Does anyone have any reference on this?
COA: You need historical data on cap IMPLIED volatilities as well as on interest rates. The O-GARCH matrix is used only for the simulation of interest rates (with volatility and correlation clustering). Indeed it is generally true, for any option VaR, that you need data on two types of risk factors – both the underlying price and the implied volatility. All this is explained (it takes 60 pages!) in Chapter IV.5.

Gaia: b. Suppose I have in my portfolio some exotic that must be priced by simulation. Does this imply that in each scenario I must simulate the price of the exotic? That is, suppose I price an exotic on a Hull White tree; is it correct that when I calculate VaR with historical simulation, in each scenario I have to build the tree to price the exotic?
COA: Yes. Although because of the difficulty and the time this takes, we often use a Taylor series to map a whole portfolio using its Greeks rather than full-revaluation. See Section IV.5.5.7. This is quicker and easier than full revaluation of every product, but it is open to serious model risk. See Chapter IV.6.

Gaia: c. Should all these issues suggest me to assume that returns on interest rates are normally distributed and then use parametric VaR and CVaR? Basically, does anyone know with which model I am less wrong?
COA: Analytic VaR models for options portfolios based on normally distributed factors are highly inaccurate. Some methods are outlined, and their limitations discussed in Section IV.5.3. The question of the model risk inherent in each different approach is discussed in detail in Chapter IV.6, and throughout the empirical examples of Vol IV I highlight this very important issue.

Hope this helps!

Carol

Gaia
Posts: 6
Joined: Sat Jan 24, 2009 1:26 am

Re: VaR for interest rate option portfolios

Postby Gaia » Wed Jan 28, 2009 4:41 am

Carol,

thank you very much, your answers are always so clear and useful!

Gaia


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