Interview Questions VII: Integrated Brownian Motion

 
For a standard Weiner process denoted W_t, calculate

    \[{\mathbb E} \Bigl[ \Bigl( \int^T_0 W_t dt \Bigr)^2 \Bigr]\]

This integral is the limit of the sum of W_t at each infinitessimal time slice dt from 0 to T, it is called Integrated Brownian Motion.




We see immediately that this will be a random variable with expectation zero, so the result of the squared expectation will simply be the variance of the random variable, which is what we need to calculate. Here I show two ways to solve this, first by taking the limit of the sum and second using stochastic integration by parts

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1) Limit of a sum

(1)   \begin{align*} I &= \int^T_0 W_t dt \nonumber \\ &= \lim_{n \to \infty}{\frac T n}\sum^n_{i=1} W_{\frac {Ti} n} \nonumber \end{align*}

This sum of values along the Wenier process is not independent of one-another, since only the increments are independent. However, we can re-write them in terms of sums of these independent increments

(2)   \begin{align*} W_{\frac {Ti} n} &= \sum^i_{j=1} ( W_{\frac {Tj} n} - W_{\frac {T(j-1)} n} ) \nonumber \\ &= \sum^i_{j=1} \overline{W}_{\frac {Tj} n} \nonumber \end{align*}

where \overline{W}_{\frac {T(j-1)} n} \sim {\mathbb N}(0,{\frac T n}) are the individual independent increments of the brownian motion. Substituting into our previous equation and reversing the order of the summation

(3)   \begin{align*} \lim_{n \to \infty}{\frac T n}\sum^n_{i=1} W_{\frac {Ti} n} &= \lim_{n \to \infty}{\frac T n} \sum^n_{i=1} \sum^i_{j=1} \overline{W}_{\frac {Tj} n} \nonumber \\ &= \lim_{n \to \infty}{\frac T n} \sum^n_{j=1} \sum^n_{i=j} \overline{W}_{\frac {Tj} n} \nonumber \\ &= \lim_{n \to \infty}{\frac T n} \sum^n_{j=1} (n-j+1) \overline{W}_{\frac {Tj} n} \nonumber \\ \end{align*}

which is simply a weighted sum of independent gaussians. To calculate the total variance, we sum the individual variances using the summation formula for \sum_1^N n^2

(4)   \begin{align*} {\mathbb V}[I] &= \lim_{n \to \infty} {\frac {T^2} {n^2}} \sum^n_{j=1} (n-j+1)^2 {\frac T n} \nonumber \\ &= \lim_{n \to \infty} {\frac {T^3} {n^3}} \Bigl( {\frac 1 3} n^3 + O(n^2) \Bigr) \nonumber \\ &= {\frac 1 3} T^3 \end{align*}

which is the solution.

2) Stochastic integration by parts

The stochastic version of integration by parts for potentially stochastic variables X_t, Y_t looks like this:

    \[X_T \cdot Y_T = X_0 \cdot Y_0 + \int^T_0 X_t dY_t + \int^T_0 Y_t dX_t\]

Re-arranging this gives

    \[\int^T_0 Y_t dX_t = \Bigl[ X_t \cdot Y_t \Bigr]^T_0 - \int^T_0 X_t dY_t\]

Now setting Y_t = -W_t and X_t = (T-t) we have

(5)   \begin{align*} \int^T_0 W_t dt &= \Bigl[ -W_t \cdot (T-t) \Bigr]^T_0 + \int^T_0 (T-t) dW_t \nonumber \\ &= \int^T_0 (T-t) dW_t \nonumber \end{align*}

We recognise this as a weighted sum of independent gaussian increments, which is (as expected) a gaussian variable with expectation 0 and variance that we can calculate with the Ito isometry

(6)   \begin{align*} {\mathbb E}\Bigl[ \Bigl( \int^T_0 (T-t) dW_t \Bigr)^2 \Bigr] &= {\mathbb E}\Bigl[ \int^T_0 (T-t)^2 dt \Bigr] \nonumber \\ &= \Bigl[ T^2t - Tt^2 + {\frac 1 3} t^3 \Bigr]^T_0 \nonumber \\ &= {\frac 1 3} T^3 \end{align*}

which is the solution.

Credit Valuation Adjustment (CVA)

Now that we’ve covered a basic model for the default of firms and the pricing of Credit Default Swaps, we’re ready to consider the implication of your counterparty’s credit risk on the price of a derivative contract signed with them – this is called the ‘Credit Valuation Adjustment’ or CVA, and is the amount that one should change the value of an uncollatorised credit-risk-free derivative to reflect the counterparty’s risk of default.

When we have priced derivatives under Black-Scholes assumptions, we’ve implicitly assumed that the contract runs to expiry. However, if our counterparty defaults before the end of the contract, this won’t happen (typically, in the event of default a liquidator will attempt to collect everything the firm is owed from counterparties, and use the finite resources collected to pay back creditors as much as possible).

If we hold a single derivative contract against a defaulting counterparty and it’s out-of-the-money (ie. the value is negative – we owe them money), we will still be required to hold the position. On the other hand, if we’re in-the-money, we won’t necessarily get all of our contract value back – as with CDS contracts, we assume that of the true value of the contract, we will only receive a fraction R, the ‘Recovery Rate’. If we gain nothing if the value is negative, but lose something when the value is positive, then to calculate the pricing impact we need to calculate expected value of the contract at a future point if it is positive, and ignore the price impact coming from the chance that it is negative, which we call the Expected Positive Exposure (EPE).

For a contract depending on underlier S with value C(S,t) at future time t, we can calculate this by integrating across all possible underlier values that generate a positive contract value at that time

    \[EPE(t) = \int_{C(s) \textgreater 0} C(S,t) \cdot p(S) dS\]

this will be larger for contracts whose values have a higher volatility, which are more likely to have a high positive or negative value at some point in the future.

Our loss due to a counterparty default event at time t is the discounted expectation of our loss due to a default accounting for the finite recovery value

    \[(1 - R) \cdot EPE(t) \cdot P(0,t)\]

where P(0,t) is the discount factor to time t as usual.

In order to calculate the CVA for a whole contract, we need to sum the product of the loss-given-default at t time and the probability of default at time t

    \[CVA = (1-R) \int_0^{T_f} EPE(t) \cdot P(0,t) \cdot S'(t) \; dt\]

where T_F is the time of the final cashflow due to the contract and S'(t) is the probability of default at time t as discussed in the previous post on firm default models, and if this is uncorrelated to the value of the contract, this integral is relatively straightforward to calculate. Default probabilities can be calibrated from observed CDS prices on the counterparties, most major counterparties (typically large investment banks) will have CDS contracts available on the market to use for calibration.

It’s worth mentioning that CVA is a pricing problem – we’re treating the pricing adjustment as the price of an exotic derivative and hedging it with a portfolio of CDS contracts. This means we’re calculating an actual dollar value, and when a trader quotes a derivative price to a counterparty he should also quote a ‘CVA adjustment’ to represent the counterparty’s chance of default. Internally, the trader will often have to pass this straight through to a ‘CVA Desk’, which internally manages the risk the bank faces due to the risk of default of any given counterparty – and will then reimburse the trader if she faces a loss due to such an event!

It’s worth noting that many simple products like ZCBs, forwards and FRAs have model-independent prices, so we can calculate their value independent of any assumptions about the market’s future behaviour. However, we’ve had to make some assumptions about the underlying volatility in the market to calculate EPEs, so the CVA for these products is model-dependent – we need to make some assumptions about the model driving the market and can’t enforce a price today purely by no-arbitrage considerations (although it will be informed by the prices of other model-dependent products whose prices we can see).

We’ve also discussed only the CVA adjustment from an individual contract, which is already quite computationally complex (in a future post, I’ll calculate this for some simple products in a simple model). If we have a portfolio of products in place against a specific counterparty, these should all be considered together and may lead to some offsetting exposures that reduce the CVA adjustment on a portfolio basis (although even this isn’t always the case, depending upon ‘netting agreements’ in the Credit Support Annex – CSA – that you have in place with the given counterparty…).

Further, CVA only applies to non-collaterised positions. Many positions (like futures) require the deposit of margin in an escrow account, exactly to try and cover for the risk of counterparty default. This leads instead to ‘gap risk’ (ie. that the margin might not be sufficient) but also to the related ‘Funding Valuation Adjustment’ (or FVA, essentially the cost of funding the margin you provide).