Sorry for this rookie question. I read all posts related to this issue on forum but can not get an idea.How can I price a double barrier option, discrete monitoring or continuous monitoring?As we know the one barrier option there is analytic formula on, say, John Hull`s book, or Haug`s book. Is there similar analytic formula (or a simple numerical method) to price a double barrier option? I assume the volatility and interest rate to be constant, at this step.Cheers

The GBM case with continuous monitoring is solvable, as well as some more complicated processes.For GBM, with [$]X_t = \log S_t[$], it amounts to solving for a Green function [$]G(t,x,y)[$] which is the probability densityto start at x and reach y at t without ever exiting the interval, say [$](0,b)[$]. This can be reduced to solving the PDE problem[$]G_t = \frac{1}{2} G_{xx}[$] with the conditions [$]G(t,0,y) = G(t,b,y) = 0[$] and [$]G(0,x,y) = \delta(x-y)[$] with [$](x,y) \in (0,b)[$] The solution is easily found by a spectral expansion[$] G(t,x,y) = \sum_{n=1}^{\infty} c_n \, e^{-n^2 \pi^2 t/(2 b^2)} \sin( \frac{n \pi x}{b})[$],which you can easily check solves the PDE and the boundary conditions.Finding the [$]c_n = c_n(y)[$] comes from the initial condition, left to you.There are a zillion simple numerical methods for this one too. In Mathematica, for example, you could just give the problem to NDSolve. Or set up a simple explicit lattice backwards recursion.Anyway, it is standard, so if you get stuck a little persistent googling will turn up complete answers.Once you have the appropriate Green function, then to value a payoff [$]w(x)[$] just take [$] e^{-r T} \int_0^b G(T,x,y) w(y) \, dy[$].For put or call payoffs, the integral can be done term-by-term. You'll see this once you get the [$]c_n(y)[$].For the 'more complicated processes' I allude to above, I actually devote two chapters of my forthcoming book'Option Valuation under Stochastic Volatility II' to this topic. It gets quite interesting for jump-diffusions,where the process can exit by either a diffusive touch of the barrier(s) or by a jump in either direction. Conditional on a jump exit, with probability one, the jump will overshoot the barrier and the overshoot densities play a role in the solution.Excerpt from the book below shows a sample path that exits below with a jump: More than you wanted to know, I'm sure, but couldn't resist plugging the book.

For the discretely monitored case: You can easily get closed-form solutions using higher-order binary options. The continuously monitored case can be also approached with the method of images. The main references are (there is some overlap between the papers):Buchen, Peter W. (2001) "Q-Options and Dual-Expiry Exotics", Working Paper, School of Mathematics and Statistics, University of SydneyBuchen, Peter W. (2001) "Image Options and the Road to Barriers", Risk Magazine, Vol. 14. No. 9, pp. 127-130Skipper, Max and Peter W. Buchen (2003) "The Quintessential Option Pricing Formula", Working Paper, School of Mathematics and Statistics, University of SydneyBuchen, Peter W. (2004) "The Pricing of Dual-Expiry Exotics", Quantitative Finance, Vol. 4, No. 1, pp. 101-108Konstandatos, Otto (2008) "Pricing Path Dependent Exotic Options: A Comprehensive Mathematical Framework", VDM VerlagAnd as an additional reading:Buchen, Peter W. and Otto Konstandatos (2005) "A new Method of Pricing Lookback Options", Mathematical Finance, Vol. 15, No. 2, pp. 245-259Buchen, Peter W. and Otto Konstandatos (2009) "A new Approach to Pricing Double-Barrier Options with Arbitrary Payoffs and Exponential Boundaries", Applied Mathematical Finance, Vol. 16, No. 6, pp. 497-515I really like this framework as it gives you a unified setting in which you can value a lot of different payoffs within the Black and Scholes setting.Here are the basic steps when pricing double barrier options with discrete monitoring. I try to follow the notation introduced by Buchen, Konstandatos and Skipper. Assume we consider a double knock-out option with a lower barrier at [$]B_L[$] an upper barrier at [$]B_U[$], a digital payoff of [$]X[$] at maturity and no rebate.Let [$]\left( \tau_i \right)_{i = 1}^n[$] be the times-to-maturity at the [$]n[$] monitoring dates with [$]\tau_i < \tau_{i + 1}[$]. For [$]\tau \in \left[ 0, \tau_1 \right)[$], the option value [$]\tilde{V}_1(S, \tau)[$] satisfies the initial value problem[$]\mathcal{L} \left\{ \tilde{V}_1 \right\}(S, \tau) = 0 \qquad \text{for } (S, \tau) \in \mathbb{R}_+\times \left[ 0, \tau_1 \right) [$],[$]\tilde{V}_1(S, 0) = X \mathrm{1} \left\{ S > B_L \right\} \mathrm{1} \left\{ S < B_U \right\} = X \left( \mathrm{1} \left\{ S > B_L \right\} - \mathrm{1} \left\{ S > B_U \right\} \right)[$].Here, [$]\mathcal{L}[$] is the Black and Scholes forward operator. The solution can be expressed in terms of bond binary options as[$]\tilde{V}_1(S, \tau) = X \left( \mathcal{B}_{B_L}^+(S, \tau) - \mathcal{B}_{B_U}^+(S, \tau) \right)[$].For [$]\tau \in \left[ \tau_1, \tau_2 \right)[$], the option value [$]\tilde{V}_2(S, \tau)[$] satisfies the initial value problem[$]\mathcal{L} \left\{ \tilde{V}_2 \right\}(S, \tau) = 0 \qquad \text{for } (S, \tau) \in \mathbb{R}_+\times \left[ \tau_1, \tau_2 \right) [$],[$]\tilde{V}_2 \left( S, \tau_1 \right) = \tilde{V}_1 (S, \tau_1) \mathrm{1} \left\{ S > B_L \right\} \mathrm{1} \left\{ S < B_U \right\} = \tilde{V}_1 (S, \tau_1) \left( \mathrm{1} \left\{ S > B_L \right\} - \mathrm{1} \left\{ S > B_U \right\} \right)[$].The solution can be expressed in terms of second order bond binary options as[$]\tilde{V}_2(S, \tau) = X \left( \mathcal{B}_{B_L B_L}^{+ +} \left( S, \tau - \tau_1, \tau \right) - \mathcal{B}_{B_L B_U}^{+ +} \left( S, \tau - \tau_1, \tau \right) - \left( \mathcal{B}_{B_U B_L}^{+ +} \left( S, \tau - \tau_1, \tau \right) - \mathcal{B}_{B_U B_U}^{+ +} \left( S, \tau - \tau_1, \tau \right) \right) \right)[$].Now continue applying this recursively, and you get a closed-form solution in terms of a [$]n[$]-dimensional normal CDF. If this is practically useful is another question..

Thanks, Alan and LocalVolatility, I think I got the idea(hopefully!).And another related issue: If I can price the double barrier option correctly (Ikeda&Kuminato, FD, MC, etc), what is the most practical way to find the greeks(especially delta)? I am considering a structured product with a double barrier option inside, the only way to hedging is delta-hedge(there is no standard European options available in the market)Cheers

I would use the analytical formulas (spectral or images), which are infinite sums -- and just differentiate term-by-term

QuoteOriginally posted by: AlanI would use the analytical formulas (spectral or images), which are infinite sums -- and just differentiate term-by-termand likely add closed form approximation adjustments for discrete monitoring, if not very liquid market + continuous monitoring in option contract...

LesleYLyu, you will also find early draft of my paper here, some also on double barriershttp://www.espenhaug.com/BarrierTransformations.pdfand yes infinite sum, but the trick I used in end of paper to increase accuracy seems to work well (think I described it in more detail somewhere else, but long time ago.) (And with limited testing, I only traded a few double barriers, but many single barriers).run the "models" against each other...(more polished version in my book Models on Models)

Thanks Alan and Collector,I am trying to work with a mature system of pricing and hedging in my company. The pricing and hedging modules should be kind of separate. Because the only way of hedging is delta hedge, basically I only need deltas for each value of stock price. The idea is:Pricing model(e.g. BS, Local vol, MC)---->Calculate Delta---->Delta hedging engineHence I would not prefer to derive a function to calculate delta, but using for example, finite difference for a certain pricing model.Is this idea practical? i.e., if I calculate the barrier options values(from say, Haug`s book, the infinite sum formula), then calculate delta for a given stock price using finite difference, is this value will be very difference from analytic formula?Cheers

the difference approx. can be made accurate enough for practical purposes if you are careful Some questions for you:-what are the underlyings?-how are you planning on handling jumps in the pricing module?-how will you test the effectiveness of the proposed hedging program?

QuoteOriginally posted by: Alanthe difference approx. can be made accurate enough for practical purposes if you are careful Some questions for you:-what are the underlyings?-how are you planning on handling jumps in the pricing module?-how will you test the effectiveness of the proposed hedging program?Underlying is a stock indexI will not consider jumps in pricing module at this step...Is this a big problem? The stock index has no European options, I can not use static hedge.I will use Monte Carlo and test the effectiveness from , say, the distribution of replication costs (the mean of the cost should = option price)cheers

QuoteOriginally posted by: LesleYLyuI will not consider jumps in pricing module at this step...Is this a big problem?Probably. As a general rule, pure diffusion models have a difficult time describing the market.One way to check is try your same process models, pricing and hedging program against say SPX vanilla options.Not barrier options but vanilla. Don't compare against Monte Carlos, but against real historical option prices. The hedging errors for that might be sobering, esp. in stress environments. For example, attemptingto replicate SPX put options purely by delta-hedging was a famous program run by a firm called Leland O'Brien Rubinstein Assoc.It ended badly in 1987. If your program is economically important, you might want to stress test it underthat scenario.

QuoteOriginally posted by: AlanQuoteOriginally posted by: LesleYLyuI will not consider jumps in pricing module at this step...Is this a big problem?Probably. As a general rule, pure diffusion models have a difficult time describing the market.One way to check is try your same process models, pricing and hedging program against say SPX vanilla options.Not barrier options but vanilla. Don't compare against Monte Carlos, but against real historical option prices. The hedging errors for that might be sobering, esp. in stress environments. For example, attemptingto replicate SPX put options purely by delta-hedging was a famous program run by a firm called Leland O'Brien Rubinstein Assoc.It ended badly in 1987. If your program is economically important, you might want to stress test it underthat scenario.Thanks Alan. You mean I should try pricing and hedging barrier options(because it has discoutinuous behavior near barreir) froma model with jump, say, variance gamma, or BSM with jump?Since I am a OTC option writer, I am only considering option replication costs. Until now I am only using BS, pure diffusion. Do you think there will be a big change in replication cost if I switch to a jump model?Cheers

QuoteOriginally posted by: AlanQuoteOriginally posted by: LesleYLyuI will not consider jumps in pricing module at this step...Is this a big problem?Probably. As a general rule, pure diffusion models have a difficult time describing the market.One way to check is try your same process models, pricing and hedging program against say SPX vanilla options.Not barrier options but vanilla. Don't compare against Monte Carlos, but against real historical option prices. The hedging errors for that might be sobering, esp. in stress environments. For example, attemptingto replicate SPX put options purely by delta-hedging was a famous program run by a firm called Leland O'Brien Rubinstein Assoc.It ended badly in 1987. If your program is economically important, you might want to stress test it underthat scenario.BTW, two questions in your reply:1. How to measure the "hedging errors?"2. You mentioned replicate SPX vanilla options purely by delta-hedging may be a disaster, then in case we can not do static hedge, what will you suggest?Any reference articles, books are truly welcome!Cheers

QuoteOriginally posted by: LesleYLyuQuoteOriginally posted by: AlanQuoteOriginally posted by: LesleYLyuI will not consider jumps in pricing module at this step...Is this a big problem?Probably. As a general rule, pure diffusion models have a difficult time describing the market.One way to check is try your same process models, pricing and hedging program against say SPX vanilla options.Not barrier options but vanilla. Don't compare against Monte Carlos, but against real historical option prices. The hedging errors for that might be sobering, esp. in stress environments. For example, attemptingto replicate SPX put options purely by delta-hedging was a famous program run by a firm called Leland O'Brien Rubinstein Assoc.It ended badly in 1987. If your program is economically important, you might want to stress test it underthat scenario.BTW, two questions in your reply:1. How to measure the "hedging errors?"2. You mentioned replicate SPX vanilla options purely by delta-hedging may be a disaster, then in case we can not do static hedge, what will you suggest?Any reference articles, books are truly welcome!Cheers1. Premium received minus replication cost2. For short vanilla puts, indeed disaster is possible. For short barrier options, your losses should be more controlled. So, my point is not that disaster awaits for those barrier options -- it's that the distribution of hedging errors is likely much widerthan you would estimate from your Monte Carlos based on a diffusion model. The only way to even start to know is to test the proposed hedging plan against empirical data --not models! -- and then think about scenarios not present in the empirical data. The reason there are no books is probably telling about the ad hoc nature and wishful/sloppy thinking in industry hedging practices ...