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arXiv2011

Pricing a Contingent Claim Liability with Transaction Costs Using Asymptotic Analysis for Optimal Investment

We price a contingent claim liability using the utility indifference argument. We consider an agent with exponential utility, who invests in a stock and a money market account with the goal of maximizing the utility of his investment at the final time T in the presence of positive proportional transaction cost in two cases with and without a contingent claim liability. Using the computations from the heuristic argument in Whalley & Wilmott we provide a rigorous derivation of the asymptotic expansion of the value function in powers of the transaction cost parameter around the known value function for the case of zero transaction cost in both cases with and without a contingent claim liability. Additionally, using utility indifference method we derive an asymptotic expansion of the price of the contingent claim liability. In both cases, we also obtain a "nearly optimal" strategy, whose expected utility asymptotically matches the leading terms of the value function.

q-fin.PR↗

arXiv2011

An application of the method of moments to volatility estimation using daily high, low, opening and closing prices

We use the expectation of the range of an arithmetic Brownian motion and the method of moments on the daily high, low, opening and closing prices to estimate the volatility of the stock price. The daily price jump at the opening is considered to be the result of the unobserved evolution of an after-hours virtual trading day.The annualized volatility is used to calculate Black-Scholes prices for European options, and a trading strategy is devised to profit when these prices differ flagrantly from the market prices.

q-fin.ST↗

arXiv2012

A note on super-hedging for investor-producers

We study the situation of an agent who can trade on a financial market and can also transform some assets into others by means of a production system, in order to price and hedge derivatives on produced goods. This framework is motivated by the case of an electricity producer who wants to hedge a position on the electricity spot price and can trade commodities which are inputs for his system. This extends the essential results of Bouchard & Nguyen Huu (2011) to continuous time markets. We introduce the generic concept of conditional sure profit along the idea of the no sure profit condition of Ràsonyi (2009). The condition allows one to provide a closedness property for the set of super-hedgeable claims in a very general financial setting. Using standard separation arguments, we then deduce a dual characterization of the latter and provide an application to power futures pricing.

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arXiv2011

Fundamental theorems of asset pricing for piecewise semimartingales of stochastic dimension

The purpose of this paper is two-fold. First is to extend the notions of an n-dimensional semimartingale and its stochastic integral to a piecewise semimartingale of stochastic dimension. The properties of the former carry over largely intact to the latter, avoiding some of the pitfalls of infinite-dimensional stochastic integration. Second is to extend two fundamental theorems of asset pricing (FTAPs): the equivalence of no free lunch with vanishing risk to the existence of an equivalent sigma-martingale measure for the price process, and the equivalence of no arbitrage of the first kind to the existence of an equivalent local martingale deflator for the set of nonnegative wealth processes.

q-fin.PR↗

arXiv2011

Indifference Pricing of American Option Underlying Illiquid Stock under Exponential Forward Performance

This work focuses on the indifference pricing of American call option underlying a non-traded stock, which may be partially hedgeable by another traded stock. Under the exponential forward measure, the indifference price is formulated as a stochastic singular control problem. The value function is characterized as the unique solution of a partial differential equation in a Sobolev space. Together with some regularities and estimates of the value function, the existence of the optimal strategy is also obtained. The applications of the characterization result includes a derivation of a dual representation and the indifference pricing on employee stock option. As a byproduct, a generalized Ito's formula is obtained for functions in a Sobolev space.

q-fin.PR↗

arXiv2026

A CDS Option Miscellany

CDS options allow investors to express a view on spread volatility and obtain a wider range of payoffs than are possible with vanilla CDS. We give a detailed exposition of different types of single-name CDS option, including options with upfront protection payment, recovery options and recovery swaps, and also presents a new formula for the index option. The emphasis is on using the Black-76 formula where possible and ensuring consistency within asset classes. In the framework shown here the `armageddon event' does not require special attention.

q-fin.PR↗

arXiv2012

Futures pricing in electricity markets based on stable CARMA spot models

We present a new model for the electricity spot price dynamics, which is able to capture seasonality, low-frequency dynamics and the extreme spikes in the market. Instead of the usual purely deterministic trend we introduce a non-stationary independent increments process for the low-frequency dynamics, and model the large fluctuations by a non-Gaussian stable CARMA process. The model allows for analytic futures prices, and we apply these to model and estimate the whole market consistently. Besides standard parameter estimation, an estimation procedure is suggested, where we fit the non-stationary trend using futures data with long time until delivery, and a robust $L^1$-filter to find the states of the CARMA process. The procedure also involves the empirical and theoretical risk premiums which -- as a by-product -- are also estimated. We apply this procedure to data from the German electricity exchange EEX, where we split the empirical analysis into base load and peak load prices. We find an overall negative risk premium for the base load futures contracts, except for contracts close to delivery, where a small positive risk premium is detected. The peak load contracts, on the other hand, show a clear positive risk premium, when they are close to delivery, while the contracts in the longer end also have a negative premium.

stat.AP↗

arXiv2012

Heat kernel methods in finance: the SABR model

The SABR model is a stochastic volatility model not admitting a closed form solution. Hagan, Kumar, Leniewski and Woodward have obtained an approximate solution by means of perturbative techniques. A more precise approximation was found by Henry-Labordère with the heat kernel expansion method. The latter relies on deep and hard theorems from Riemannian geometry which are almost totally unknown to the professionals of finance, who however are those primarily interested in these results. The goal of this report is to fill this gap and to make these topics understandable with a basic knowledge of calculus and linear algebra.

q-fin.PR↗

arXiv2013

Smiles all around: FX joint calibration in a multi-Heston model

We introduce a novel multi-factor Heston-based stochastic volatility model, which is able to reproduce consistently typical multi-dimensional FX vanilla markets, while retaining the (semi)-analytical tractability typical of affine models and relying on a reasonable number of parameters. A successful joint calibration to real market data is presented together with various in- and out-of-sample calibration exercises to highlight the robustness of the parameters estimation. The proposed model preserves the natural inversion and triangulation symmetries of FX spot rates and its functional form, irrespective of choice of the risk-free currency. That is, all currencies are treated in the same way.

q-fin.PR↗

arXiv2013

The Impact of the Prior Density on a Minimum Relative Entropy Density: A Case Study with SPX Option Data

We study the problem of finding probability densities that match given European call option prices. To allow prior information about such a density to be taken into account, we generalise the algorithm presented in Neri and Schneider (2011) to find the maximum entropy density of an asset price to the relative entropy case. This is applied to study the impact the choice of prior density has in two market scenarios. In the first scenario, call option prices are prescribed at only a small number of strikes, and we see that the choice of prior, or indeed its omission, yields notably different densities. The second scenario is given by CBOE option price data for S&P500 index options at a large number of strikes. Prior information is now considered to be given by calibrated Heston, Schoebel-Zhu or Variance Gamma models. We find that the resulting digital option prices are essentially the same as those given by the (non-relative) Buchen-Kelly density itself. In other words, in a sufficiently liquid market the influence of the prior density seems to vanish almost completely. Finally, we study variance swaps and derive a simple formula relating the fair variance swap rate to entropy. Then we show, again, that the prior loses its influence on the fair variance swap rate as the number of strikes increases.

q-fin.PR↗

arXiv2012

Parameter Estimation using Empirical Likelihood combined with Market Information

During the last decade Levy processes with jumps have received increasing popularity for modelling market behaviour for both derviative pricing and risk management purposes. Chan et al. (2009) introduced the use of empirical likelihood methods to estimate the parameters of various diffusion processes via their characteristic functions which are readily avaiable in most cases. Return series from the market are used for estimation. In addition to the return series, there are many derivatives actively traded in the market whose prices also contain information about parameters of the underlying process. This observation motivates us, in this paper, to combine the return series and the associated derivative prices observed at the market so as to provide a more reflective estimation with respect to the market movement and achieve a gain of effciency. The usual asymptotic properties, including consistency and asymptotic normality, are established under suitable regularity conditions. Simulation and case studies are performed to demonstrate the feasibility and effectiveness of the proposed method.

stat.ME↗

arXiv2012

Robust Hedging of Withdrawal Guarantees (Extended Version)

Withdrawal guarantees ensure the periodical deduction of a constant dollar-amount from a fund investment for a fixed number of periods. If the fund depletes before the last withdrawal, the guarantor has to finance the outstanding withdrawals. We derive a robust hedging strategy which leads to closed form solutions for the guarantee value.

q-fin.PR↗

arXiv2012

A tractable LIBOR model with default risk

We develop a model for the dynamic evolution of default-free and defaultable interest rates in a LIBOR framework. Utilizing the class of affine processes, this model produces positive LIBOR rates and spreads, while the dynamics are analytically tractable under defaultable forward measures. This leads to explicit formulas for CDS spreads, while semi-analytical formulas are derived for other credit derivatives. Finally, we give an application to counterparty risk.

q-fin.PR↗

arXiv2012

Perturbative Expansion of FBSDE in an Incomplete Market with Stochastic Volatility

In this work, we apply our newly proposed perturbative expansion technique to a quadratic growth FBSDE appearing in an incomplete market with stochastic volatility that is not perfectly hedgeable. By combining standard asymptotic expansion technique for the underlying volatility process, we derive explicit expression for the solution of the FBSDE up to the third order of volatility-of-volatility, which can be directly translated into the optimal investment strategy. We compare our approximation with the exact solution, which is known to be derived by the Cole-Hopf transformation in this popular setup. The result is very encouraging and shows good accuracy of the approximation up to quite long maturities. Since our new methodology can be extended straightforwardly to multi-dimensional setups, we expect it will open real possibilities to obtain explicit optimal portfolios or hedging strategies under realistic assumptions.

q-fin.CP↗

arXiv2012

Short-time asymptotics for marginal distributions of semimartingales

We study the short-time asymptotics of conditional expectations of smooth and non-smooth functions of a (discontinuous) Ito semimartingale; we compute the leading term in the asymptotics in terms of the local characteristics of the semimartingale. We derive in particular the asymptotic behavior of call options with short maturity in a semimartingale model: whereas the behavior of \textit{out-of-the-money} options is found to be linear in time, the short time asymptotics of \textit{at-the-money} options is shown to depend on the fine structure of the semimartingale.

math.PR↗

arXiv2012

Minimax Option Pricing Meets Black-Scholes in the Limit

Option contracts are a type of financial derivative that allow investors to hedge risk and speculate on the variation of an asset's future market price. In short, an option has a particular payout that is based on the market price for an asset on a given date in the future. In 1973, Black and Scholes proposed a valuation model for options that essentially estimates the tail risk of the asset price under the assumption that the price will fluctuate according to geometric Brownian motion. More recently, DeMarzo et al., among others, have proposed more robust valuation schemes, where we can even assume an adversary chooses the price fluctuations. This framework can be considered as a sequential two-player zero-sum game between the investor and Nature. We analyze the value of this game in the limit, where the investor can trade at smaller and smaller time intervals. Under weak assumptions on the actions of Nature (an adversary), we show that the minimax option price asymptotically approaches exactly the Black-Scholes valuation. The key piece of our analysis is showing that Nature's minimax optimal dual strategy converges to geometric Brownian motion in the limit.

q-fin.CP↗

arXiv2013

Pricing for Large Positions in Contingent Claims

Approximations to utility indifference prices are provided for a contingent claim in the large position size limit. Results are valid for general utility functions on the real line and semi-martingale models. It is shown that as the position size approaches infinity, the utility function's decay rate for large negative wealths is the primary driver of prices. For utilities with exponential decay, one may price like an exponential investor. For utilities with a power decay, one may price like a power investor after a suitable adjustment to the rate at which the position size becomes large. In a sizable class of diffusion models, limiting indifference prices are explicitly computed for an exponential investor. Furthermore, the large claim limit is seen to endogenously arise as the hedging error for the claim vanishes.

q-fin.PR↗

arXiv2012

A parsimonious model for intraday European option pricing

A stochastic model for pure-jump diffusion (the compound renewal process) can be used as a zero-order approximation and as a phenomenological description of tick-by-tick price fluctuations. This leads to an exact and explicit general formula for the martingale price of a European call option. A complete derivation of this result is presented by means of elementary probabilistic tools.

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