106 research outputs found
A probabilistic solution to the Stroock-Williams equation
We consider the initial boundary value problem \begin{eqnarray*}u_t=\mu
u_x+\tfrac{1}{2}u_{xx}\qquad (t>0,x\ge0),\\u(0,x)=f(x)\qquad
(x\ge0),\\u_t(t,0)=\nu u_x(t,0)\qquad (t>0)\end{eqnarray*} of Stroock and
Williams [Comm. Pure Appl. Math. 58 (2005) 1116-1148] where and the boundary condition is not of Feller's type when . We
show that when belongs to with then the following
probabilistic representation of the solution is valid:
where is a reflecting Brownian motion with drift and is
the local time of at . The solution can be interpreted in terms of
and its creation in at rate proportional to . Invoking the law
of , this also yields a closed integral formula for
expressed in terms of , and .Comment: Published in at http://dx.doi.org/10.1214/13-AOP865 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Three-Dimensional Brownian Motion and the Golden Ratio Rule
Let X =(Xt)t=0 be a transient diffusion processin (0,8) with the diffusion coeffcient s> 0 and the scale function L such that Xt ?8 as t ?8 ,let It denote its running minimum for t = 0, and let ? denote the time of its ultimate minimum I8 .Setting c(i,x)=1-2L(x)/L(i) we show that the stopping time minimises E(|? - t|- ?) over all stopping times t of X (with finite mean) where the optimal boundary f* can be characterised as the minimal solution to staying strictly above the curve h(i)= L-1(L(i)/2) for i > 0. In particular, when X is the radial part of three-dimensional Brownian motion, we find that where ? =(1+v5)/2=1.61 ... is the golden ratio. The derived results are applied to problems of optimal trading in the presence of bubbles where we show that the golden ratio rule offers a rigourous optimality argument for the choice of the well known golden retracement in technical analysis of asset prices.
The British Asian Option
Following the economic rationale of [7] and [8] we present a new class of Asian options where the holder enjoys the early exercise feature of American options whereupon his payoff (deliverable immediately) is the ābest predictionā of the European payoff under the hypothesis that the true drift of the stock price equals a contract drift. Inherent in this is a protection feature which is key to the British Asian option. Should the option holder believe the true drift of the stock price to be unfavourable (based upon the observed price movements) he can substitute the true drift with the contract drift and minimise his losses. The practical implications of this protection feature are most remarkable as not only is the option holder afforded a unique protection against unfavourable stock price movements (covering the ability to sell in a liquid market completely endogenously) but also when the stock price movements are favourable he will generally receive high returns. We derive a closed form expression for the arbitrage-free price in terms of the rational exercise boundary and show that the rational exercise boundary itself can be characterised as the unique solution to a nonlinear integral equation. Using these results we perform a financial analysis of the British Asian option that leads to the conclusions above and shows that with the contract drift properly selected the British Asian option becomes a very attractive alternative to the classic (European) Asian option.British Asian option; American Asian option; European Asian option; fixed/floating strike; arithmetic/geometric average; flexible Asian options; arbitrage-free price; rational exercise boundary; liquid/illiquid market; geometric Brownian motion; the Shiryaev process; optimal stopping, parabolic free-boundary problem; nonlinear integral equation; local time-space calculus
The British Russian Option
Following the economic rationale of [10] and [11] we present a new class of lookback options (by first studying the canonical 'Russian' variant) where the holder enjoys the early exercise feature of American options where upon his payoff (deliverable immediately) is the 'best prediction' of the European payoff under the hypothesis that the true drift of the stock price equals a contract drift. Inherent in this is a protection feature which is key to the British Russian option. Should the option holder believe the true drift of the stock price to be unfavourable (based upon the observed price movements) he can substitute the true drift with the contract drift and minimise his losses. The practical implications of this protection feature are most remarkable as not only is the option holder afforded a unique protection against unfavourable stock price movements (covering thea bility to sell in a liquid market completely endogenously) but also when the stock price movements are favourable he will generally receive high returns. We derive a closed form expression for the arbitrage-free price in terms of the rational exercise boundary and show that the rational exercise boundary itself can be characterised as the unique solution to a nonlinear integral equation. Using these results we perform a financial analysis of the British Russian option that leads to the conclusions above and shows that with the contract drift properly selected the British Russian option becomes a very attractive alternative to the classic European/American Russian option.
Predicting the ultimate supremum of a stable L\'{e}vy process with no negative jumps
Given a stable L\'{e}vy process of index
with no negative jumps, and letting denote its running supremum for , we consider the optimal
prediction problem where
the infimum is taken over all stopping times of , and the error
parameter is given and fixed. Reducing the optimal prediction
problem to a fractional free-boundary problem of Riemann--Liouville type, and
finding an explicit solution to the latter, we show that there exists
(equal to 1.57 approximately) and a strictly increasing
function satisfying ,
and for such that for
every and the following stopping
time is optimal
where is the unique root to a transcendental equation (with
parameters and ). Moreover, if either or
then it is not optimal to stop at when
is sufficiently large. The existence of the breakdown points
and stands in sharp contrast with the Brownian motion
case (formally corresponding to ), and the phenomenon itself may be
attributed to the interplay between the jump structure (admitting a transition
from lighter to heavier tails) and the individual preferences (represented by
the error parameter ).Comment: Published in at http://dx.doi.org/10.1214/10-AOP598 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
The law of the supremum of a stable L\'{e}vy process with no negative jumps
Let be a stable L\'{e}vy process of index
with no negative jumps and let denote its running
supremum for . We show that the density function of can be
characterized as the unique solution to a weakly singular Volterra integral
equation of the first kind or, equivalently, as the unique solution to a
first-order Riemann--Liouville fractional differential equation satisfying a
boundary condition at zero. This yields an explicit series representation for
. Recalling the familiar relation between and the first entry time
of into , this further translates into an explicit
series representation for the density function of .Comment: Published in at http://dx.doi.org/10.1214/07-AOP376 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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