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Hey how to calculate Expected positive exposure in the case of interest rate swap? Assume that I simulate $M$ interest rate paths for time grid $0=t_0\le t_1 \le ... \le t_N = T.$ What is the procedure now to calculate value of a swap for each trajectory and each time step using Longstaff-Schwarz LSM? I know that in the case of american option we can go backward but I dont understand how to do it in the case of IR swap. Its easy to calculate cash flows at each time $t_1,...,t_N$ but what next?

Math122
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3 Answers3

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The expected positive exposure

The expected positive exposure of a swap (or any other type of asset) at a given date $t_i$ is the expectation of the positive part of its value at that date (as that's what you stand to lose if the counterpart defaults, if the value is negative, you lose nothing). This is computed by taking the average over the $M$ paths of your Monte Carlo simulation: $$ EPE(t_i) = \mathbb{E}\left[ \max(V(t_i), 0) \right] \approx \frac{1}{M} \sum_{\omega=1}^M \max(V(t_i, \omega), 0) $$

So, now the question is how to get a grid of your swap values at all paths and dates?

Valuing the swap at all Monte Carlo dates and paths

There are two possibilities:

1. Closed-form formula

You have a closed-form formula giving you the swap price from your interest rates, which is the case. So, here the procedure is simple and you don't even need LSM:

  • at each date $t_i$, you simply take the $M$ simulated interest rates scenarios at that date and plug each one into your formula to get $M$ swap prices;
  • you take the average of the positive parts to get the expected positive exposure at that date $t_i$.
2. LSM

You don't have any closed-form formula to price the swap. In this case, you have to remember that the swap's value at each node $(t_i, \omega)$ of your Monte Carlo is in fact a conditional expectation of its discounted future flows under the risk-neutral measure: $$ V(t_i, \omega)= \mathbb{E} \left[ \sum_{t > t_i} D(t_i, t)Flow(t) \mid (t_i, \omega) \right] $$

(by $\mathbb{E} \left[ \ast \mid (t_i, \omega) \right]$ I mean the expectation of $\ast$ conditional on the state of world, in your case the values of interest rates, being the one in your date $t_i$ and path $\omega$)

Remark that the flows falling after $t_{i+1}$ are actually equal to the swap value at $t_{i+1}$ (everything discounted to $t_i$): $$ \sum_{t > t_{i+1}} D(t_i, t)Flow(t) = D(t_i, t_{i+1}) V(t_{i+1}) $$ Leading to this expression: $$ V(t_i, \omega)= \mathbb{E} \left[ \sum_{t_i < t \leq t_{i+1}} D(t_i, t)Flow(t) \mid (t_i, \omega) \right] + \mathbb{E} \left[ D(t_i, t_{i+1}) V(t_{i+1}) \mid (t_i, \omega) \right] $$

The first term is usually straightforward to compute (it falls in the previous section).

For the second term, you can see the similarity to American options, where you need to compute the continuation value, which is a conditional expectations and that you approximate using a LS.

Here, you can approximate this term using a regression of the discounted values of your swap $Y = D(t_i, t_{i+1}) V(t_{i+1})$ on some regressor $X$ depending on your interest rates values (e.g. zero-coupon bond price, annuity, etc.).

By starting from $t_N$ and moving backwards, you will get your swap values on all dates and paths.

byouness
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  • Great answer! I have still some questions:
    1. You write that value of a swap at time $t_i$ is a conditional expectation of its discounted future flows under the risk-neutral measure, so we don't take into account cash-flow from time $t_i$ during calculation of the value of a swap at this time? So the value of a swap at maturity is $0$?
    2. What do you mean by "The first term is usually straightforward to compute (it falls in the previous section)"? It will be the expected value of a cash-flow from the next time step, why is it straightforward co calculate?
     – Math122 May 13 '21 at 18:45
  • Since $V(t_i, \omega)= \mathbb{E} \left[ \sum_{t > t_i} D(t_i, t)Flow(t) \mid (t_i, \omega) \right]$ then we don't have to divide the problem into the sum of two components but just do the regression of all futures cash-flows againt the current interest rate level? You write that the regressor should depend on interest rates values, but can we use the interest rates themselves? Or we have to calculate at each point $(t_i,\omega)$ value of a bond from the formula $\text{exp}{-\frac{T}{N}\times \sum_{k = i}^{N-1} r(t_i,\omega)}$?
    • – Math122 May 13 '21 at 18:51
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  • It's just a matter of convention, are you valuing the swap before the flows are settled or not :) You can replace all $>, <$ by $\geq, \leq$ and vice versa if you want the flows of a given date to be included in the value at that date.
  • Usually, $Flow(t) = N \delta R(t)$ where $N$ is the notional and $\delta$ the day count fraction of the flow's period. So, the expectation will be: $N \delta F(t_i, t) P(t_i, t)$ where $F(t_i, t)$ is the forward rate seen from $t_i$ and $P(t_i, t)$ is the ZC bond price at $t_i$ for maturity $t$, so you can get its value explicitely without regression.
    • – byouness May 13 '21 at 19:01
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    2 (continued) Of course, you could also value your whole swap like this at each $(t_i, \omega)$ but the question asks how to value it using LSM. – byouness May 13 '21 at 19:02
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  • The idea is to start from $V(t_N)$ where the value of the swap is zero and the regression is trivial and = 0, then move backwards and use the flows falling between $t_{N-1}$ and $t_{N}$ to get a vector of $V(t_{N-1}$, and repeat... you don't want to recompute all the future flows at each date $t_i$ but rather use what you computed up to $t_{i+1}$ to get the value at $t_i$:

    • $$V(t_i) = \mathbb{E} \left[ D(t_i, t_{i+1}) V(t_{i+1} \right] \approx f(X(t_i))$$

    – byouness May 13 '21 at 19:08