Michael R. Tehranchi

Can the implied volatility surface move by parallel shifts

This note explores the analogy between the dynamics of the interest rate term structure and the implied volatility surface of a stock. In particular, we prove an impossibility theorem conjectured by Steve Ross.

A characterization of hedging portfolios for interest rate contingent claims

We consider the problem of hedging a European interest rate contingent claim with a portfolio of zero-coupon bonds and show that an HJM type Markovian model driven by an infinite number of sources of randomness does not have some of the shortcomings found in the classical finite-factor models. Indeed, under natural conditions on the model, we find that there exists a unique hedging strategy, and that this strategy has the desirable property that at all times it consists of bonds with maturities that are less than or equal to the longest maturity of the bonds underlying the claim.

Optimal portfolio choice in the bond market

We consider the Merton problem of optimal portfolio choice when the traded instruments are the set of zero-coupon bonds. Working within a Markovian Heath–Jarrow–Morton model of the interest rate term structure driven by an infinite-dimensional Wiener process, we give sufficient conditions for the existence and uniqueness of an optimal trading strategy. When there is uniqueness, we provide a characterization of the optimal portfolio as a sum of mutual funds. Furthermore, we show that a Gauss–Markov random field model proposed by Kennedy [Math. Financ. 4, 247–258(1994)] can be treated in this framework, and explicitly calculate the optimal portfolio. We show that the optimal portfolio in this case can be identified with the discontinuities of a certain function of the market parameters.

A note on invariant measures for HJM models

Abstract.This note analyzes the mean-reverting behavior of time-homogeneous Heath-Jarrow-Morton (HJM) forward rate models in the weighted Sobolev spaces {Hw}w. An explicit sufficient condition is given under which invariant measures exist for the HJM dynamics. In particular, every HJM model with constant volatility and market price of risk has a family of invariant measures parametrized by the distribution of the long rate.

OPTIMAL INVESTMENT FOR ALL TIME HORIZONS AND MARTIN BOUNDARY OF SPACE-TIME DIFFUSIONS

This paper is concerned with the axiomatic foundation and explicit construction of a general class of optimality criteria that can be used for investment problems with multiple time horizons, or when the time horizon is not known in advance. Both the investment criterion and the optimal strategy are characterized by the Hamilton-Jacobi-Bellman equation on a semi-infinite time interval. In the case when this equation can be linearized, the problem reduces to a time-reversed parabolic equation, which cannot be analyzed via the standard methods of partial differential equations. Under the additional uniform ellipticity condition, we make use of the available description of all minimal solutions to such equations, along with some basic facts from potential theory and convex analysis, to obtain an explicit integral representation of all positive solutions. These results allow us to construct a large family of the aforementioned optimality criteria, including some closed form examples in relevant financial models.

Uniform bounds for Black--Scholes implied volatility

In this note, Black--Scholes implied volatility is expressed in terms of various optimisation problems. From these representations, upper and lower bounds are derived which hold uniformly across moneyness and call price. Various symmetries of the Black--Scholes formula are exploited to derive new bounds from old. These bounds are used to reprove asymptotic formulae for implied volatility at extreme strikes and/or maturities.

If

It is shown that if the processes

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B

and

and

f(B)

f(B)