# TARF PSR PFE Exposure calculation model

Pre-settlement Risk (PSR) & Potential Future Exposure (PFE) are calculated to assess counterparty credit risk for derivative transactions. PSR calculates the risk of a counterparty default at a static point in time while PFEs assess the risk over the life time of the transaction. The measures are used to set counterparty and credit limits to mitigate against these risks.

**Pre-settlement risk (PSR)** is the risk that a counterparty to a transaction does not honour his/her end of the deal and defaults on the transaction prior to the settlement date. The other party to the transaction will need to cover the position in the open market most likely at adverse prices/rates.

**Potential Future Exposure** (**PFE**) is defined as the maximum expected credit exposure over a specified period of time calculated at some level of confidence (i.e. at a given quantile).

In this series of posts, we look at TARF PSR PFE calculations for FX derivatives. We will begin with:

- A call option on USD (put option on Yen),
- A long forward contract to buy USD at a given strike at a pre-specified time in the future in exchange for yen, and
- A target redemption forward, i.e. a series of forward contracts that terminate when the cumulative positive intrinsic value reaches a specified target value.

**Call Option – Pre-settlement risk (PSR)**

We are looking at a one year European call option on the USD dollar. The buyer has the right to exercise the option to buy USD 10,000 at the strike rate of USDJPY = 110. The current spot rate (S_{0}) is 111.21 and annualized volatility (σ) is 7.88%. The JPY risk free rate (r_{d}) is 0.128% and USD risk free rate (r_{f}) = 1.732%. The option’s start date (and the current calculation date) is 30-06-2017.

The buyer will exercise the right to buy USD, if and only if, the spot exchange rate at maturity exceeds the strike rate, i.e. if USD appreciates against the yen or alternatively JPY depreciates against the dollar relative to the strike rate.

PSR will be the highest for the option buyer when the derivative is in the money and conversely out of the money for the counterparty. If the counterparty fails to meet his obligation and defaults on the transaction, the buyer of the option will either need to replace the derivative as soon as possible or cover the long USD position in the open market at expiry. Given that the USD has depreciated against JPY, he will need to pay more units of JPY to purchase a single unit of USD at expiry, as compared to the original strike exchange rate; or he must pay a higher premium to replace the option now.

Let us assume that the counterparty defaults immediately after the call option is purchased and the premium remitted.

To calculate the pre-settlement risk of an immediate default, we calculate the current value of the position or the replacement cost at the time of default. In this instance, the current value of the position is simply the call premium.

However, if there is a delay in covering the position, the replacement cost may not always equal the current value of the position as of the default date. We may use a VaR based approach to determine the current replacement cost of the position to account for delays in re-covering the initial transaction. PSR is this case is the current value of the option plus the maximum increase in value of the option that can be experienced over a specified period (say, the 10 holding days it could take to replace the option) at a given confidence level.

For the PSR calculation, we begin by capturing the historical series of risk free rates for the domestic (JPY) and foreign (USD) currencies. We have used the JPY LIBOR and the USD LIBOR as proxies for these rates. Monthly rates for the year 2013 – June 2017 were obtained for www.global-rates.com.

Assuming constant time to expiry, spot rate & annualized volatility, the value of the call option is calculated using the Black Scholes call option pricing formula for each rates data point, to obtain a monthly series of call option values.

A return series is then generated by taking the natural log of consecutive data point option values. The maximum of the return series gives the worst case monthly VaR% (i.e. the increase in the value of the option) under the VaR histogram approach. The worst case increase over 10 holding days is calculated as:

Worst case monthly VaR % x Sqrt (12 x 10/365) x Current Value.

Alternatively, a Variance Covariance (VCV) approach may be used to determine the worst case increase:

- Calculate the monthly volatility of the option value returns, i.e. the standard deviation of the return series
- Calculate the z-value for a given confidence level, e.g. 99%
- Calculate the monthly VaR % = (1) x (2)
- Worst case increase over 10 holding days = (3) x Sqrt (12 x 10/365) x Current Value

The PSR for the call option is its Current Value plus the Worst case increase over 10 holding days.

**Call Option – Potential Future Exposure**

To determine the maximum credit exposure over the tenor of the call option we need to first create a Monte Carlo simulation model for the spot exchange rate.

We use a twelve step model to simulate monthly spot rates for the year. The risk free rates term structure and implied volatility are assumed to remain unchanged from what they were on the current valuation date.

We next calculate the value of the option for the current valuation date and on each subsequent month end using the Black Scholes formula with the corresponding simulated spot:

‘t’ is the time elapsed, while ‘T’ represents the time till maturity, i.e. Tenor of option – time elapsed.

For this particular simulation run, the plot of future option values across times till maturity is depicted below:

1000 simulations are run and the future options values for each time step are captured in a data warehouse.

For a given future date, the PFE is calculated as the upper first percentile (99% confidence interval) of the simulated values for a given future date; MPFE is the maximum simulated value across all runs, and Average is the average option value or expected exposure across all runs.

In summary, the PSR & PFE values for the call option are as follows:

1-month PFE is the potential future exposure 1 month into the future. 9 months into the future the PFE stands at 139,885.78, and so on.

If we were to stress implied volatility, i.e. assume that implied volatility increased by say 3 times the existing levels (applied as a constant for the life time of the option) the PSR & PFE values would also increase:

*Note: The invariant nature of the interest rate term structure and implied volatility at future data points are issues that need to be reviewed and considered when refining the PFE model.*