Year End Turn Rates

The year-end turn-rate (or the turn-of-the-year rate) is the interest rate for the period between the last business day of the year until the first good business day in the next year. For example, in the US the 2014 turn rate is the interest rate for the period between 31-Dec-2014 until 2-Jan-2015. The rate can […]

Lamperti Transform

Let \(X_t\) be an Ito process given by $$dX_t=f(X_t,t)dt+\sigma(X_t,t)dw_t$$ The Lamperti transformation is, $$Z_t=\phi(X_t,t)=\int\frac{1}{\sigma(x,t)}dx$$ and has a unit diffusion; that is, \(dZ_t=(…)dt + dw_t\). This transformation is fairly well used in finance, and particularly useful with separable volatility functions both in analytic and numerical contexts. However I am not sure the name is well known […]

SABR Calibration: A simple, explicit initial guess

The SABR model is widely used, particularly in the interest rate world, to help manage the volatility smile. Depending on 4 parameters, \(\alpha\), \(\beta\), \(\rho\) and \(\nu\), often \(\beta\) is considered a fixed constant whilst the other 3 parameters are calibrated to liquid market prices. There tends to be two types of calibration algorithms; local […]

Arbitrage-Free SABR: Finite Difference Techniques

In January a new approach to the SABR model was published in Wilmott magazine, by Hagan et al., the original authors of the well-known SABR model. They review some of the weaknesses of the model, and concentrate on the problem of negative probabilities induced by the original approximation formula, especially at low strikes. To solve […]

Fed Fund Swap Nuances

A few people have recently asked me about the details of USD Fed Fund Swaps. I collected some of my answers together in this post for convenience. Fed Fund Swap and OIS Swap differences Fed Fund Swaps and OIS swap are easily confused at first glance, both are swaps involving the Federal Funds rate, both […]

Computing ATM implied volatility analytically

I recently had to compute ATM implied bpvol (or normal volatility) as well as Black volatility from an ATM option premium. Immediately, I looked for a library function which, more often than not, are written for the general case and make use of a solving routine. To my later irritation, I had completely forgotten that […]

Arbitrage free SABR and near negative rates

This week in London the Thalesians hosted a presentation by Pat Hagan titled ‘Arbitrage free SABR‘. Knowing the popularity of Pat’s presentations, I had planned to arrive early to ensure I had a good seat, unfortunately I mis-calculated how long it took to get from the City to Canary Wharf, and so I arrived on […]

CloudBees: The extra member of the team

In football, the home team supporters energise their players by showing appreciation of good plays and creating an intimidating and distracting atmosphere for the opposing team. This advantage can be so great that is likened to having an extra player in the team, referred to as the ‘twelfth player‘ in the team. In many traditional […]

Two Curves Upfront

At the end of last year Peter Caspers released a short paper Normal Libor in Arrears. I could not help but wonder if this was not the result of the update to a library which had previously used Black’s model for the convexity correction. In the summer of 2011, like many others, I had to […]