A model calculation is made of the near-field diffusion from a small extended surface line source... more A model calculation is made of the near-field diffusion from a small extended surface line source into a reversed uniform shear flow. A solution is found by the method of matched asymptotic expansions, under the assumption of small Peclet number. Analytic expressions are deduced for the special cases of very narrow and very wide reversed flow layers. In the first case the reversed flow is found to hinder the diffusion, while in the latter it assists. Solutions for intermediate widths are presented in terms of numerically-determined coefficient functions.
We extend the calculation by Turfus and Shubert (2016) of analytic pricing of CoCo bonds under a ... more We extend the calculation by Turfus and Shubert (2016) of analytic pricing of CoCo bonds under a Hull-White short rate model for equity conversion intensity to a beta blend short rate model which encompasses both Hull-White (normal) and Black-Karasinski (lognormal) models. The asymptotic expansion assumes that the volatility of the conversion intensity is small. Results are calculated up to second order accuracy in the volatility. This is considered adequate for most practical purposes.
‘Freedom and Belonging’ is a collection of essays and articles written for the website Societal V... more ‘Freedom and Belonging’ is a collection of essays and articles written for the website Societal Values over a number of years, reflecting the authors' attempt to engage with the major issues of the day, such as terrorism, populism and a resurgent nationalism, migration, the environment, identity politics, the possibilities and threats of information technology and automation, but against a backdrop of the timeless concerns of philosophy, such as peace and conflict, freedom, morality, the law, the political order and, perhaps even more fundamentally, the search for the eternal verities and virtues among the chaos of human events.
We seek analytic formulae for the pricing of extinguishable cap-floor Libor quanto flows which pa... more We seek analytic formulae for the pricing of extinguishable cap-floor Libor quanto flows which pay capped and/or floored foreign currency Libor conditional on survival of a named debt issuer. We assume that the foreign currency in which the Libor is paid and the credit default risk (observed for debt denominated in the domestic currency) are potentially correlated with the exchange rate between foreign and domestic currency; and further that this exchange rate may drop by a fixed relative amount in the event of a credit default event. The stochastic credit intensity model is assumed to be Black-Karasinski and the interest rate model to be beta-blend, a family of models which encompasses within its scope both the Black-Karasinski and the Hull-White cases. The solutions are obtained as perturbation expansions which are valid in the limit of the foreign interest rate and the credit default intensity being small. A Green's function solution is found for the governing PDE which is asymptotically valid under this assumption. This is calculated to second order and used to calculate to first order accuracy the price of extinguishable cap-floor Libor quanto flows, in particular taking account of the assumed correlations. When the interest rate model is Hull-White a composite expansion is proposed which has the advantage of greater consistency with the known exact formula for the conditional bond price in the limit of zero correlation.
We consider calibration of the Black-Karasinski short rate model to a given interest rate (or cre... more We consider calibration of the Black-Karasinski short rate model to a given interest rate (or credit intensity) term structure, conducting an asymptotic analysis in the limit of low rates. We base our analysis on the perturbation expansion approach proposed by Turfus and Shubert (2016). We calibrate the model so as to give consistent representation of the probability distribution of T-maturity zero coupon bond prices at all times up to maturity.
We extend the Hull-White short rate model to include the integrated short rate as a separate inde... more We extend the Hull-White short rate model to include the integrated short rate as a separate independent variable and incorporate credit default risk, governed by a Black-Karasinski model, into cash flows. We derive an analytic representation of the associated pricing kernel and apply it to the pricing of risky compounded interest rate payments, including with caps and floors. We illustrate our results numerically applying them to the pricing of extinguishing fix-float swaps.
We consider an extension of the Hull-White short rate model which incorporates smile and skew, ef... more We consider an extension of the Hull-White short rate model which incorporates smile and skew, effectively through a quadratic dependence of the diffusion on the short rate. We derive an asymptotic representation of the pricing kernel for this new model in semi-analytic form, using this to obtain accurate, easily computed, asymptotic formulae for zero coupon bonds and LIBOR options. Unlike comparable models such as SABR, ours appears to be usable even for pricing options with long times to maturity. Further, the same calibrated model can be used for all possible maturities and LIBOR tenors.
This year marks the thirtieth anniversary of the publication of the seminal short rate model of B... more This year marks the thirtieth anniversary of the publication of the seminal short rate model of Black and Karasinski [1991]. We look back over the early career of its co-originator Piotr Karasinski and record his story of how the model came to be developed, going on to review the impact it had subsequently, and the reasons behind the revival of interest which has been evident in recent years.
We consider the Heston stochastic volatility model for equity options and use operator formalism ... more We consider the Heston stochastic volatility model for equity options and use operator formalism and perturbation expansion techniques to derive a formal analytic expression for the Green's function (pricing kernel). This is expanded explicitly as a perturbation series in powers of the vol of vol up to second order and used to obtain an asymptotically accurate pricing formula for vanilla equity options, replicating results obtained previously by Ito and Malliavin techniques.
We consider the Hull-White short rate model and extend the known closed-form pricing kernel to in... more We consider the Hull-White short rate model and extend the known closed-form pricing kernel to include the integrated short rate as a separate independent variable, applying it to cap/floor pricing.
We present an analytic pricing kernel for a two-factor Black-Karasinski (lognormal) short rate mo... more We present an analytic pricing kernel for a two-factor Black-Karasinski (lognormal) short rate model as a rapidly convergent perturbation expansion valid in the limit of low rates. Even the leading order expansion is found to be extremely accurate in most circumstances. We use this expansion to derive analytic formulae for conditional bond prices and thus for zero rates and forward rates. The model is equally applicable for the modelling of credit spreads and satisfies the important requirement of guaranteeing positive implied default probabilities. We suggest how these results could be used for interest rate and credit spread scenario generation in risk capital calculations and provide some representative scenario calculations.
We present a Green's function solution (aka pricing kernel) for a generic multi-factor short ... more We present a Green's function solution (aka pricing kernel) for a generic multi-factor short rate model based on correlated state variables of Ornstein-Uhlenbeck type, whose drift is assumed to be an affine function of the state variables (and of time). The solution is obtained in general as a perturbation expansion valid in the limit of low rates (an assumption almost invariably satisfied). We exhibit explicit solutions in the case where the interest rate model is of Hull-White and of Black-Karasinski type. We observe that the theory is equally applicable to the modelling of stochastic credit default intensity governed by a Black-Karasinski model as it is to interest rate modelling. It is not difficult either to extend it to multi-asset pricing problems.
A model calculation is made of the near-field diffusion from a small extended surface line source... more A model calculation is made of the near-field diffusion from a small extended surface line source into a reversed uniform shear flow. A solution is found by the method of matched asymptotic expansions, under the assumption of small Peclet number. Analytic expressions are deduced for the special cases of very narrow and very wide reversed flow layers. In the first case the reversed flow is found to hinder the diffusion, while in the latter it assists. Solutions for intermediate widths are presented in terms of numerically-determined coefficient functions.
We extend the calculation by Turfus and Shubert (2016) of analytic pricing of CoCo bonds under a ... more We extend the calculation by Turfus and Shubert (2016) of analytic pricing of CoCo bonds under a Hull-White short rate model for equity conversion intensity to a beta blend short rate model which encompasses both Hull-White (normal) and Black-Karasinski (lognormal) models. The asymptotic expansion assumes that the volatility of the conversion intensity is small. Results are calculated up to second order accuracy in the volatility. This is considered adequate for most practical purposes.
‘Freedom and Belonging’ is a collection of essays and articles written for the website Societal V... more ‘Freedom and Belonging’ is a collection of essays and articles written for the website Societal Values over a number of years, reflecting the authors' attempt to engage with the major issues of the day, such as terrorism, populism and a resurgent nationalism, migration, the environment, identity politics, the possibilities and threats of information technology and automation, but against a backdrop of the timeless concerns of philosophy, such as peace and conflict, freedom, morality, the law, the political order and, perhaps even more fundamentally, the search for the eternal verities and virtues among the chaos of human events.
We seek analytic formulae for the pricing of extinguishable cap-floor Libor quanto flows which pa... more We seek analytic formulae for the pricing of extinguishable cap-floor Libor quanto flows which pay capped and/or floored foreign currency Libor conditional on survival of a named debt issuer. We assume that the foreign currency in which the Libor is paid and the credit default risk (observed for debt denominated in the domestic currency) are potentially correlated with the exchange rate between foreign and domestic currency; and further that this exchange rate may drop by a fixed relative amount in the event of a credit default event. The stochastic credit intensity model is assumed to be Black-Karasinski and the interest rate model to be beta-blend, a family of models which encompasses within its scope both the Black-Karasinski and the Hull-White cases. The solutions are obtained as perturbation expansions which are valid in the limit of the foreign interest rate and the credit default intensity being small. A Green's function solution is found for the governing PDE which is asymptotically valid under this assumption. This is calculated to second order and used to calculate to first order accuracy the price of extinguishable cap-floor Libor quanto flows, in particular taking account of the assumed correlations. When the interest rate model is Hull-White a composite expansion is proposed which has the advantage of greater consistency with the known exact formula for the conditional bond price in the limit of zero correlation.
We consider calibration of the Black-Karasinski short rate model to a given interest rate (or cre... more We consider calibration of the Black-Karasinski short rate model to a given interest rate (or credit intensity) term structure, conducting an asymptotic analysis in the limit of low rates. We base our analysis on the perturbation expansion approach proposed by Turfus and Shubert (2016). We calibrate the model so as to give consistent representation of the probability distribution of T-maturity zero coupon bond prices at all times up to maturity.
We extend the Hull-White short rate model to include the integrated short rate as a separate inde... more We extend the Hull-White short rate model to include the integrated short rate as a separate independent variable and incorporate credit default risk, governed by a Black-Karasinski model, into cash flows. We derive an analytic representation of the associated pricing kernel and apply it to the pricing of risky compounded interest rate payments, including with caps and floors. We illustrate our results numerically applying them to the pricing of extinguishing fix-float swaps.
We consider an extension of the Hull-White short rate model which incorporates smile and skew, ef... more We consider an extension of the Hull-White short rate model which incorporates smile and skew, effectively through a quadratic dependence of the diffusion on the short rate. We derive an asymptotic representation of the pricing kernel for this new model in semi-analytic form, using this to obtain accurate, easily computed, asymptotic formulae for zero coupon bonds and LIBOR options. Unlike comparable models such as SABR, ours appears to be usable even for pricing options with long times to maturity. Further, the same calibrated model can be used for all possible maturities and LIBOR tenors.
This year marks the thirtieth anniversary of the publication of the seminal short rate model of B... more This year marks the thirtieth anniversary of the publication of the seminal short rate model of Black and Karasinski [1991]. We look back over the early career of its co-originator Piotr Karasinski and record his story of how the model came to be developed, going on to review the impact it had subsequently, and the reasons behind the revival of interest which has been evident in recent years.
We consider the Heston stochastic volatility model for equity options and use operator formalism ... more We consider the Heston stochastic volatility model for equity options and use operator formalism and perturbation expansion techniques to derive a formal analytic expression for the Green's function (pricing kernel). This is expanded explicitly as a perturbation series in powers of the vol of vol up to second order and used to obtain an asymptotically accurate pricing formula for vanilla equity options, replicating results obtained previously by Ito and Malliavin techniques.
We consider the Hull-White short rate model and extend the known closed-form pricing kernel to in... more We consider the Hull-White short rate model and extend the known closed-form pricing kernel to include the integrated short rate as a separate independent variable, applying it to cap/floor pricing.
We present an analytic pricing kernel for a two-factor Black-Karasinski (lognormal) short rate mo... more We present an analytic pricing kernel for a two-factor Black-Karasinski (lognormal) short rate model as a rapidly convergent perturbation expansion valid in the limit of low rates. Even the leading order expansion is found to be extremely accurate in most circumstances. We use this expansion to derive analytic formulae for conditional bond prices and thus for zero rates and forward rates. The model is equally applicable for the modelling of credit spreads and satisfies the important requirement of guaranteeing positive implied default probabilities. We suggest how these results could be used for interest rate and credit spread scenario generation in risk capital calculations and provide some representative scenario calculations.
We present a Green's function solution (aka pricing kernel) for a generic multi-factor short ... more We present a Green's function solution (aka pricing kernel) for a generic multi-factor short rate model based on correlated state variables of Ornstein-Uhlenbeck type, whose drift is assumed to be an affine function of the state variables (and of time). The solution is obtained in general as a perturbation expansion valid in the limit of low rates (an assumption almost invariably satisfied). We exhibit explicit solutions in the case where the interest rate model is of Hull-White and of Black-Karasinski type. We observe that the theory is equally applicable to the modelling of stochastic credit default intensity governed by a Black-Karasinski model as it is to interest rate modelling. It is not difficult either to extend it to multi-asset pricing problems.
We present a new model for pricing contingent convertible (CoCo) bonds which facilitates the calc... more We present a new model for pricing contingent convertible (CoCo) bonds which facilitates the calculation of equity, credit and interest rate risk sensitivities. We assume a lognormal equity process and a Hull-White (normal) short rate process for the conversion intensity with a downward jump in the equity price on conversion. We are able to derive an approximate solution in closed form based on the assumption that the conversion intensity volatility is asymptotically small. The simple first order approximation is seen to be accurate for a wide range of market conditions, although particularly for longer maturities higher order terms in the asymptotic expansion may be needed.
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