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Text of the page (random words):
stant volatility process it does not meet the requirements for a general ito process 5 with stochastic volatility using an estimator computed over the whole time interval 0 t may not be a good approximation for the volatility at the time that the barrier is hit a possible way around this is for the adjustment 3 applied at time t i to only depend on a volatility estimator computed from samples near the time this would be possible although it is not clear what is the best way to select these times besides an important point to note is that we do not need a good estimate of the volatility since that is not the goal here as explained in the previous post adjustment 3 works because it corrects for the expected overshoot when the barrier is hit specifically at the first time for which m i k the overshoot is r x t i k if there was no adjustment then the overshoot is positive and the leading order term in the discrete barrier approximation error is proportional to 𝔼 r the positive shift added to x t i is chosen to compensate for this giving zero expected overshoot to leading order and reducing the barrier approximation error the same applies to any similar adjustment as long as there is sufficient freedom in choosing m i then it should be possible to do it in a way that has zero expected overshoot taking this to the extreme it should be possible to compute the adjustment at time t i using only the sampled values x t i 1 and x t i figure 1 barrier overshoot consider adjustments of the form for θ ℝ 2 ℝ by model independence if this adjustment applies to a process x then it should equally apply to the shifted and scaled processes x a and bx for constants a and b 0 equivalently θ satisfies the scaling and translation invariance 6 this restricts the possible forms that θ can take lemma 1 a function θ ℝ 2 ℝ satisfies 6 if and only if for constants p c proof write θ 0 u as the sum of its antisymmetric and symmetric parts by scaling invariance the first term on the right is proportional to u and the second is proportional to u hence for constants p and c using translation invariance as required i will therefore only consider adjustments where the maximum of the process across the interval t i 1 t i is replaced by 7 according to 3 the barrier condition sup t t x t k is replaced by the discrete approximation max i m i k there are various ways in which 7 can be parameterized but this form is quite intuitive the term px t i 1 p x t i 1 is an interpolation of the path of x and c x t i x t i 1 represents a shift proportional to the sample deviation across the interval replacing the σ δt term of the simple shift 3 the purpose of this post is to find values for p and c giving a good adjustment improving convergence of the discrete approximation figure 2 adjusted barrier overshoot the discrete barrier condition m i k given by 7 can be satisfied while the process is below the barrier level giving a negative barrier overshoot r x t i k as in figure 2 as we will see this is vital to obtaining an accurate approximation for the hitting probability continue reading model independent discrete barrier adjustments print opens in new window print tweet share on reddit opens in new window reddit email a link to a friend opens in new window email like loading george lowther stochastic calculus leave a comment 9 july 23 12 july 23 discrete barrier approximations it is quite common to consider functions of real time stochastic process which depend on whether or not it crosses a specified barrier level k this can involve computing expectations involving a real valued process x of the form 1 for a positive time t and function f ℝ ℝ i am using the notation 𝔼 a s to denote the expectation of random variable a restricted to event s or 𝔼 a 1 s one example is computing prices of financial derivatives such as barrier options where t represents the expiration time and f is the payoff at expiry conditional on hitting upper barrier level k a knock in call option would have the final payoff f x x a for a contractual strike of a knock out options are similar except that the payoff is conditioned on not hitting the barrier level as the sum of knock in and knock out options is just an option with no barrier both cases involve similar calculations alternatively the barrier can be discrete meaning that it only involves sampling the process at a finite set of times 0 t 1 t n t then equation 1 is replaced by 2 naturally sampling at a finite set of times will reduce the probability of the barrier being reached and so if f is nonnegative then 2 will have a lower value than 1 it should still converge though as n goes to infinity and the sampling times become dense in the interval if the underlying process x is brownian motion or geometric brownian motion possibly with a constant drift then there are exact expressions for computing 1 in terms of integrating f against a normal density see the post on the reflection principle for more information however it is difficult to find exact expressions for the discrete barrier 2 other than integrating over high dimensional joint normal distributions so it can be useful to approximate a discrete barrier with analytic formulas for the continuous barrier this is the idea used in the classic 1997 paper a continuity correction for discrete barrier options by broadie glasserman and kou freely available here we may want to compute the continuous barrier expectation 1 using monte carlo simulation this is a common method but involves generating sample paths of the process x at a finite set of times this means that we are only able to sample at these times so necessarily are restricted to discrete barrier calculations as in 2 i am primarily concerned with the second idea this is a very general issue since monte carlo simulation is a common technique used in many applications however as it only represents sample paths at discrete time points it necessarily involves discretely approximating continuous barrier levels you may well ask why we would even want to use monte carlo if as i mentioned above there are exact expressions in these cases in answer such formulas only hold in very restrictive situations where the process x is a brownian motion or geometric brownian motion with constant drift more generally it could be an ito process of the form 3 where b is standard brownian motion this describes x as a stochastic integral with respect to the predictable integrands σ and μ which represent the volatility and drift of the process strictly speaking these are linear volatility and drift terms rather than log linear as used in many financial models applied to nonnegative processes such as stock prices this is simply the choice made here since this post is addressing a general mathematical problem of approximating continuous barriers and not restricting to such specific applications if the volatility and drift terms in 3 are not constant then the exact formulas no longer hold this is true even if they are deterministic functions of time in practice these terms are often stochastic and can be rather general in which case trying to find exact expressions is an almost hopeless task even though i concentrate on the case with constant volatility and drift in any calculations performed here this is for convenience of exposition the idea is that as long as σ is piecewise continuous then locally it is well approximate as constant and the techniques discussed here should still apply in addition to considering general ito processes 3 the ideas described here will apply to much more general functions of the process x than stated in 1 in the financial context this means more general payoffs than simple knock in or knock out options for example autocallable trades involve a down and in put option but additionally contain a discrete set of upper barriers which cause the trade to make a final payment and terminate they may also allow the issuer to early terminate the trade on a discrete set of dates furthermore trades can depend on different assets with separate barriers on each of them or on the average of a basket of assets or have different barrier levels in different time periods the list of possibilities is endless but the idea is that each continuous barrier inside a complex payoff will be approximated by discretely sampled barrier conditions for efficiency we may also want to approximate a discrete barrier with a large number of sampling times by one with fewer the methods outlined in the post can also be used for this in particular the simple barrier shift described below could be used by taking the difference between the shift computed for the times actually sampled and the one for the required sample times i do not go into details of this but mention it now give an idea of the generality of the technique figure 1 discrete barrier approximation error let s consider simply approximating a continuous barrier in 1 by the discrete barrier in 2 this will converge as the number of sampling times t i increases but the problem is it converges very slowly we can get an idea of the order of the error when the sampling times have a δt spacing which with equally spaced times is given by δt t n this is as shown in figure 1 above when the process first hits the continuous barrier level it will be on average about δt 2 before the next sampling time if x behaves approximately like a brownian motion with volatiity σ over this interval then it will have about 50 chance of being above k at the next discrete time on the other hand it will be below k with about 50 probability in which case with will drop a distance proportional to σ δt below on average this means that if the continuous barrier is hit there is a probability roughly proportional to σ δt that the discrete barrier is not hit so the error in approximating a continuous barrier 1 by the discrete case 2 is of the order of σ δt which only tends to zero at rate 1 n continue reading discrete barrier approximations print opens in new window print tweet share on reddit opens in new window reddit email a link to a friend opens in new window email like loading george lowther stochastic processes 17 comments 1 july 23 11 july 23 extending filtered probability spaces in stochastic calculus it is common to work with processes adapted to a filtered probability space as with probability space extensions it can sometimes be necessary to enlarge the underlying space to introduce additional events and processes for example many diffusions and local martingales can be expressed as an integral with respect to brownian motion but sometimes it may be necessary to enlarge the space to make sure that it includes a brownian motion to work with also in the theory of stochastic differential equations finding solutions can sometimes require enlarging the space extending a probability space is a relatively straightforward concept which i covered in an earlier post extending a filtered probability space is the same except that it also involves enlarging the filtration it is important to do this in a way which does not destroy properties of existing processes such as their distributions conditional on the filtration at each time let s consider a filtered probability space an enlargement is firstly an extension of the probability spaces it is a map from ω to ω measurable with respect to and and preserving probabilities so ℙ π 1 e π e for all in addition it is required to be measurable for each time t 0 meaning that for all consequently any adapted process x t lifts to an adapted process x t π x t on the larger space defined by x t ω x t π ω as with extensions of probability spaces this can be considered in two steps first we extend to the filtered probability space on ω with induced sigma algebra consisting of sets π 1 e for and to the filtration this is essentially a no op since events and random variables on the original filtered probability space are in one to one correspondence with those on the enlarged space up to zero probability events next the sigma algebras are enlarged to and this is where new random events are added to the event space and filtration such arbitrary extensions are too general for many uses in stochastic calculus where we merely want to add in some additional source of randomness consider for example a standard brownian motion b defined on the original space so that for any times s t b t b s is normal and independent of does it necessarily lift to a brownian motion on the enlarged space the answer to this is no it need not be the case that b t b s is independent of for an extreme case consider the situation where and π is the identity so there is no enlargement of the sample space if the filtration is is extended to the maximum consider what happens to our brownian motion the increment b t b s is measurable so is not independent of it in fact conditioned on the entire path of b is deterministic it is definitely not a brownian motion with respect to this new filtration similarly martingales submartingales and supermartingales will not remain as such if we pass to this enlarged filtration the idea is that if for random variables x y defined on our original probability space then this relation should continue to hold in the extension it is required that this is exactly relative independence of and over recall that two sigma algebras and are relatively independent over a third if for all and the following properties are each equivalent to this definition for all bounded measurable random variables x and measurable y for all bounded measurable x for all bounded measurable x this leads us to the idea of a standard extension of filtered probability spaces definition 1 an extension of filtered probability spaces is standard if for each time t 0 the sigma algebras and are relatively independent over continue reading extending filtered probability spaces print opens in new window print tweet share on reddit opens in new window reddit email a link to a friend opens in new window email like loading george lowther the general theory of semimartingales leave a comment 17 june 23 17 april 24 probability space extensions and relative products according to kolmogorov s axioms to define a probability space we start with a set ω and an event space consisting of a sigma algebra f on ω a probability measure ℙ on this gives the probability space ω f ℙ on which we can define random variables as measurable maps from ω to the reals or other measurable space however it is common practice to suppress explicit mention of the underlying sample space ω the values of a random variable x ω ℝ are simply written as x rather than x ω for ω ω it is intuitively thought of as a real number which happens to be random rather than a function for one thing we usually do not really care what the sample space is and instead only care about events and their probabilities and about random variables and their expectations this philosophy has some benefits frequentl...