76 research outputs found
Robust pricing and hedging of double no-touch options
Double no-touch options, contracts which pay out a fixed amount provided an
underlying asset remains within a given interval, are commonly traded,
particularly in FX markets. In this work, we establish model-free bounds on the
price of these options based on the prices of more liquidly traded options
(call and digital call options). Key steps are the construction of super- and
sub-hedging strategies to establish the bounds, and the use of Skorokhod
embedding techniques to show the bounds are the best possible.
In addition to establishing rigorous bounds, we consider carefully what is
meant by arbitrage in settings where there is no {\it a priori} known
probability measure. We discuss two natural extensions of the notion of
arbitrage, weak arbitrage and weak free lunch with vanishing risk, which are
needed to establish equivalence between the lack of arbitrage and the existence
of a market model.Comment: 32 pages, 5 figure
Vanna-volga pricing
The vanna-volga method, also called the traders rule of thumb is an empirical procedure that can be used to infer an implied-volatility smile from three available quotes for a given maturity. It is based on the construction of locally replicating portfolios whose associated hedging costs are added to corresponding Black-Scholes prices to produce smile-consistent values. Besides being intuitive and easy to implement, this procedure has a clear financial interpretation, which further supports its use in practice. --
A Forward Equation for Barrier Options under the Brunick&Shreve Markovian Projection
We derive a forward equation for arbitrage-free barrier option prices, in
terms of Markovian projections of the stochastic volatility process, in
continuous semi-martingale models. This provides a Dupire-type formula for the
coefficient derived by Brunick and Shreve for their mimicking diffusion and can
be interpreted as the canonical extension of local volatility for barrier
options. Alternatively, a forward partial-integro differential equation (PIDE)
is introduced which provides up-and-out call prices, under a Brunick-Shreve
model, for the complete set of strikes, barriers and maturities in one solution
step. Similar to the vanilla forward PDE, the above-named forward PIDE can
serve as a building block for an efficient calibration routine including
barrier option quotes. We provide a discretisation scheme for the PIDE as well
as a numerical validation.Comment: 20 pages, Quantitative Finance Volume 16, 2016 - Issue
Vanna-Volga methods applied to FX derivatives : from theory to market practice
We study Vanna-Volga methods which are used to price first generation exotic
options in the Foreign Exchange market. They are based on a rescaling of the
correction to the Black-Scholes price through the so-called `probability of
survival' and the `expected first exit time'. Since the methods rely heavily on
the appropriate treatment of market data we also provide a summary of the
relevant conventions. We offer a justification of the core technique for the
case of vanilla options and show how to adapt it to the pricing of exotic
options. Our results are compared to a large collection of indicative market
prices and to more sophisticated models. Finally we propose a simple
calibration method based on one-touch prices that allows the Vanna-Volga
results to be in line with our pool of market data
On the monotonicity principle of optimal Skorokhod embedding problem
In this paper, we provide an alternative proof of the monotonicity principle
for the optimal Skorokhod embedding problem established by Beiglb\"ock, Cox and
Huesmann. This principle presents a geometric characterization that reflects
the desired optimality properties of Skorokhod embeddings. Our proof is based
on the adaptation of the Monge-Kantorovich duality in our context together with
a delicate application of the optional cross-section theorem and a clever
conditioning argument
The joint law of the extrema, final value and signature of a stopped random walk
A complete characterization of the possible joint distributions of the
maximum and terminal value of uniformly integrable martingale has been known
for some time, and the aim of this paper is to establish a similar
characterization for continuous martingales of the joint law of the minimum,
final value, and maximum, along with the direction of the final excursion. We
solve this problem completely for the discrete analogue, that of a simple
symmetric random walk stopped at some almost-surely finite stopping time. This
characterization leads to robust hedging strategies for derivatives whose value
depends on the maximum, minimum and final values of the underlying asset
Robust Hedging with Proportional Transaction Costs
Duality for robust hedging with proportional transaction costs of path
dependent European options is obtained in a discrete time financial market with
one risky asset. Investor's portfolio consists of a dynamically traded stock
and a static position in vanilla options which can be exercised at maturity.
Both the stock and the option trading is subject to proportional transaction
costs. The main theorem is duality between hedging and a Monge-Kantorovich type
optimization problem. In this dual transport problem the optimization is over
all the probability measures which satisfy an approximate martingale condition
related to consistent price systems in addition to the usual marginal
constraints
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