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At least 523 records · Page 29Linked to original sources
arXiv2013

An analytic multi-currency model with stochastic volatility and stochastic interest rates

We introduce a tractable multi-currency model with stochastic volatility and correlated stochastic interest rates that takes into account the smile in the FX market and the evolution of yield curves. The pricing of vanilla options on FX rates can be performed effciently through the FFT methodology thanks to the affinity of the model Our framework is also able to describe many non trivial links between FX rates and interest rates: a second calibration exercise highlights the ability of the model to fit simultaneously FX implied volatilities while being coherent with interest rate products.

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arXiv2013

Utility maximisation and utility indifference price for exponential semi-martingale models with random factor

We consider utility maximization problem for semi-martingale models depending on a random factor $ξ$. We reduce initial maximization problem to the conditional one, given $ξ=u$, which we solve using dual approach. For HARA utilities we consider information quantities like Kullback-Leibler information and Hellinger integrals, and corresponding information processes. As a particular case we study exponential Levy models depending on random factor. In that case the information processes are deterministic and this fact simplify very much indifference price calculus. Then we give the equations for indifference prices. We show that indifference price for seller and minus indifference price for buyer are risk measures. Finally, we apply the results to Geometric Brownian motion case. Using identity in law technique we give the explicit expression for information quantities. Then, the previous formulas for indifference price can be applied.

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arXiv2013

US Corporate Bond Yield Spread : A default risk debate

According to theoretical models of valuing risky corporate securities, risk of default is primary component in overall yield spread. However, sizable empirical literature considers it otherwise by giving more importance to non-default risk factors. Current study empirically attempts to provide relative solution to this conundrum by presuming that problem lies in the subjective empirical treatment of default risk. By using post-hoc estimator approach of Lubotsky & Wittenberg (2006), we construct an efficient indicator for risk of default, by using sample of 252 US non-financial corporate data (2000-2010). On average, our results validate that almost 48% of change in yield spread is explained by default risk especially in recent financial crisis period (2007-2009). Hence, our results relatively suggest that potential problem lies in the ad-hoc measurement methods used in existing empirical literature.

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arXiv2013

Instantaneous mean-variance hedging and instantaneous Sharpe ratio pricing in a regime-switching financial model, with applications to equity-linked claims

We study hedging and pricing of unattainable contingent claims in a non-Markovian regime-switching financial model. Our financial market consists of a bank account and a risky asset whose dynamics are driven by a Brownian motion and a multivariate counting process with stochastic intensities. The interest rate, drift, volatility and intensities fluctuate over time and, in particular, they depend on the state (regime) of the economy which is modelled by the multivariate counting process. Hence, we can allow for stressed market conditions. We assume that the trajectory of the risky asset is continuous between the transition times for the states of the economy and that the value of the risky asset jumps at the time of the transition. We find the hedging strategy which minimizes the instantaneous mean-variance risk of the hedger's surplus and we set the price so that the instantaneous Sharpe ratio of the hedger's surplus equals a predefined target. We use Backward Stochastic Differential Equations. Interestingly, the instantaneous mean-variance hedging and instantaneous Sharpe ratio pricing can be related to no-good-deal pricing and robust pricing and hedging under model ambiguity. We discuss key properties of the optimal price and the optimal hedging strategy. We also use our results to price and hedge mortality-contingent claims with financial components (equity-linked insurance claims) in a combined insurance and regime-switching financial model.

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arXiv2013

The Small-Maturity Heston Forward Smile

In this paper we investigate the asymptotics of forward-start options and the forward implied volatility smile in the Heston model as the maturity approaches zero. We prove that the forward smile for out-of-the-money options explodes and compute a closed-form high-order expansion detailing the rate of the explosion. Furthermore the result shows that the square-root behaviour of the variance process induces a singularity such that for certain parameter configurations one cannot obtain high-order out-of-the-money forward smile asymptotics. In the at-the-money case a separate model-independent analysis shows that the small-maturity limit is well defined for any Ito diffusion. The proofs rely on the theory of sharp large deviations (and refinements) and incidentally we provide an example of degenerate large deviations behaviour.

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arXiv2013

A note on pricing of contingent claims under G-expectation

In this paper, we study the pricing of contingent claims under G-expectation. In order to accomodate volatility uncertainty, the price of the risky security is supposed to governed by a general linear stochastic differential equation (SDE) driven by G-Brownian motion. Utilizing the recently developed results of Backward SDE driven by G-Brownian motion, we obtain the superhedging and suberhedging prices of a given contingent claim. Explicit results in the Markovian case are also derived.

math.PR↗
arXiv2013

Volatility Swap Under the SABR Model

The SABR model is shortly presented and the volatility swap explained. The fair value for a volatility swap is then computed using the usual theory in financial mathematics. An analytical solution using confluent hypergeometric functions is found. The solution is then verified using Rama Cont's functional calculus.

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arXiv2013

Barrier Options under Lévy Processes: a Simple Short-Cut

In this paper we present a very simple way to price a class of barrier options when the underlying process is driven by a huge class of Lévy processes. To achieve our goal we assume that our market satisfies a symmetry property. In case of not satisfying that property some approximations can be obtained.

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arXiv2014

An Iterated Azéma-Yor Type Embedding for Finitely Many Marginals

We solve the $n$-marginal Skorokhod embedding problem for a continuous local martingale and a sequence of probability measures $μ_1,...,μ_n$ which are in convex order and satisfy an additional technical assumption. Our construction is explicit and is a multiple marginal generalisation of the Azema and Yor (1979) solution. In particular, we recover the stopping boundaries obtained by Brown et al. (2001) and Madan and Yor (2002). Our technical assumption is necessary for the explicit embedding, as demonstrated with a counterexample. We discuss extensions to the general case giving details when $n=3$. In our analysis we compute the law of the maximum at each of the n stopping times. This is used in Henry-Labordere et al. (2013) to show that the construction maximises the distribution of the maximum among all solutions to the $n$-marginal Skorokhod embedding problem. The result has direct implications for robust pricing and hedging of Lookback options.

math.PR↗
arXiv2013

Interest-Rate Modelling in Collateralized Markets: Multiple curves, credit-liquidity effects, CCPs

The market practice of extrapolating different term structures from different instruments lacks a rigorous justification in terms of cash flows structure and market observables. In this paper, we integrate our previous consistent theory for pricing under credit, collateral and funding risks into term structure modelling, integrating the origination of different term structures with such effects. Under a number of assumptions on collateralization, wrong-way risk, gap risk, credit valuation adjustments and funding effects, including the treasury operational model, and via an immersion hypothesis, we are able to derive a synthetic master equation for the multiple term structure dynamics that integrates multiple curves with credit/funding adjustments.

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arXiv2013

Robust price bounds for the forward starting straddle

In this article we consider the problem of giving a robust, model-independent, lower bound on the price of a forward starting straddle with payoff $|F_{T_1} - F_{T_0}|$ where $0<T_0<T_1$. Rather than assuming a model for the underlying forward price $(F_t)_{t \geq 0}$, we assume that call prices for maturities $T_0<T_1$ are given and hence that the marginal laws of the underlying are known. The primal problem is to find the model which is consistent with the observed call prices, and for which the price of the forward starting straddle is minimised. The dual problem is to find the cheapest semi-static subhedge. Under an assumption on the supports of the marginal laws, but no assumption that the laws are atom-free or in any other way regular, we derive explicit expressions for the coupling which minimises the price of the option, and the form of the semi-static subhedge.

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arXiv2014

Efficient Solution of Backward Jump-Diffusion PIDEs with Splitting and Matrix Exponentials

We propose a new, unified approach to solving jump-diffusion partial integro-differential equations (PIDEs) that often appear in mathematical finance. Our method consists of the following steps. First, a second-order operator splitting on financial processes (diffusion and jumps) is applied to these PIDEs. To solve the diffusion equation, we use standard finite-difference methods, which for multi-dimensional problems could also include splitting on various dimensions. For the jump part, we transform the jump integral into a pseudo-differential operator. Then for various jump models we show how to construct an appropriate first and second order approximation on a grid which supersets the grid that we used for the diffusion part. These approximations make the scheme to be unconditionally stable in time and preserve positivity of the solution which is computed either via a matrix exponential, or via P{á}de approximation of the matrix exponent. Various numerical experiments are provided to justify these results.

q-fin.CP↗
arXiv2013

Hedging of Game Options under Model Uncertainty in Discrete Time

We introduce a setup of model uncertainty in discrete time. In this setup we derive dual expressions for the super--replication prices of game options with upper semicontinuous payoffs. We show that the super--replication price is equal to the supremum over a special (non dominated) set of martingale measures, of the corresponding Dynkin games values. This type of results is also new for American options.

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arXiv2013

A variation of the Canadisation algorithm for the pricing of American options driven by Lévy processes

We introduce an algorithm for the pricing of finite expiry American options driven by Lévy processes. The idea is to tweak Carr's `Canadisation' method, cf. Carr [9] (see also Bouchard et al [5]), in such a way that the adjusted algorithm is viable for any Lévy process whose law at an independent, exponentially distributed time consists of a (possibly infinite) mixture of exponentials. This includes Brownian motion plus (hyper)exponential jumps, but also the recently introduced rich class of so-called meromorphic Lévy processes, cf. Kyprianou et al [16]. This class contains all Lévy processes whose Lévy measure is an infinite mixture of exponentials which can generate both finite and infinite jump activity. Lévy processes well known in mathematical finance can in a straightforward way be obtained as a limit of meromorphic Lévy processes. We work out the algorithm in detail for the classic example of the American put, and we illustrate the results with some numerics.

math.PR↗
arXiv2013

On the pricing and hedging of options for highly volatile periods

Option pricing is an integral part of modern financial risk management. The well-known Black and Scholes (1973) formula is commonly used for this purpose. This paper is an attempt to extend their work to a situation in which the unconditional volatility of the original asset is increasing during a certain period of time. We consider a market suffering from a financial crisis. We provide the solution for the equation of the underlying asset price as well as finding the hedging strategy. In addition, a closed formula of the pricing problem is proved for a particular case. The suggested formulas are expected to make the valuation of options and the underlying hedging strategies during financial crisis more precise.

q-fin.PR↗
arXiv2013

On option pricing in illiquid markets with jumps

One of the shortcomings of the Black and Scholes model on option pricing is the assumption that trading of the underlying asset does not affect the price of that asset. This assumption can be fulfilled only in perfectly liquid markets. Since most markets are illquid, this assumption might be too restrictive. Thus, taking into account the price impact in option pricing is an important issue. This issue has been dealt with, to some extent, for illiquid markets by assuming a continuous process, mainly based on the Brownian motion. However, the recent financial crisis and its effects on the global stock markets have propagated the urgent need for more realistic models where the stochastic process describing the price trajectories involves random jumps. Nonetheless, works related to markets with jumps are scant compared to the continuous ones. In addition, these previous studies do not deal with illiquid markets. The contribution of this paper is to tackle the pricing problem for options in illiquid markets with jumps as well as the hedging strategy within this context, which is the first of its kind to the best knowledge.

q-fin.PR↗
arXiv2014

A new method to obtain risk neutral probability, without stochastic calculus and price modeling, confirms the universal validity of Black-Scholes-Merton formula and volatility's role

A new method is proposed to obtain the risk neutral probability of share prices without stochastic calculus and price modeling, via an embedding of the price return modeling problem in Le Cam's statistical experiments framework. Strategies-probabilities $P_{t_0,n}$ and $P_{T,n}$ are thus determined and used, respectively,for the trader selling the share's European call option at time $t_0$ and for the buyer who may exercise it in the future, at $T; \ n$ increases with the number of share's transactions in $[t_0,T].$ When the transaction times are dense in $[t_0,T]$ it is shown, with mild conditions, that under each of these probabilities $\log \frac{S_T}{S_{t_0}}$ has infinitely divisible distribution and in particular normal distribution for "calm" share; $S_t$ is the share's price at time $t.$ The price of the share's call is the limit of the expected values of the call's payoff under the translated $P_{t_0,n}.$ It coincides for "calm" share prices with the Black-Scholes-Merton formula with variance not necessarily proportional to $(T-t_0),$ thus confirming formula's universal validity without model assumptions. Additional results clarify volatility's role in the transaction and the behaviors of the trader and the buyer. Traders may use the pricing formulae after estimation of the unknown parameters.

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