Q. How does AD&Co interpolate input rates? Does it produce a smooth forward curve?
A. We accept up to 14 input yield-curve points. Then, we select more points and construct a cubic spline. After that, we derive a forward curve that matches the splined bond (or swap) rates and is linear between them.

Cubic spline is the smoothest theoretical function that passes through given points and is bound to zero second derivatives at each of the edges. Spline is continuous up to its second derivative. Hence, the short forward curve will be continuous up to its first derivative. In theory, AD&Co’s linear interpolation of forward rate between spline nodes makes the forward curve “smooth, in general;” linear granulation segments are still somewhat noticeable.

Although a very smooth short forward curve is required only for special instruments (FRAs, for example) and not for MBS, AD&Co can compile its interest rate library with an arbitrary topology of input yield curve, interpolation grid, and pricing lattice. For example, we can “seed” interpolation points semiannually or even monthly; linear granulation becomes invisible and the forward curve lives up to expectation. By the same token, AD&Co can expand the input yield-curve points if it is absolutely necessary for a client. We have seen and met these requests.

Q. AD&Co’s lattice claims to be arbitrage-free. Why do we need to “fudge” rates?
A. The lattice is constructed such that the rate nodes and transitional probabilities are computed to preclude arbitrage. That is, if we price a static bond (or swap) operating backward on this lattice, we recover the exact price. On the other hand, when running Monte-Carlo, a limited sample of paths is employed and they, collectively, just approximate the arbitrage-free law. AD&Co offers a simple and efficient option to “fudge” a limited set of short rate paths so that the averaged discount factors match bond prices exactly. We recommend using this option for most practical cases, especially when the selected number of paths is not large.

Q. A firm is using a 3rd-party OAS system that has no volatility calibration tool. Can AD&Co help?
A. Yes. Take a look at our weekly Market Analysis where we post calibration results for all single-factor rate models we support. Alternatively, consider licensing our Excel, Bloomberg-connected, tool called IR_GUI. It allows you to load market data and calibrate a model as often as needed. Calibration results can then be plugged into the vendor system that supports the same model (Black-Karasinski, Squared Gaussian, or Hull-White).

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