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That sounds like that I go back to the times before Black-Scholes...

anyone with new idea?

Thanks everyone. Here is my answer - straight forward derivation without risk-neutral measure http://ssrn.com/abstract=2397010

Many build algorithm trading only on MACD.

http://ssrn.com/abstract=2397010

The market consisting of only the OTC market - each option underwriter retails their options to clients by contract.

Any suggestion?:)

I know the real stock price distribution has thick tails, so traditional BS method may encounter unexpected loss and may cost a lot. I want to hedge an option as smooth and cheap as possible. After all, if I underwrite an option, my profit is premium revenue minus hedging cost. If I have to...

However someone told me that SV model is not complete, so forget about it and just stick to BS.

I have heard the rumor that option shall be hedged according to recent volatility. So I guess a SV model outperforms a BS?

I am in a very immature market where even vanilla options are in scarce. To simplify the question, just hedge an European Call on a stock, which is the best?

So basically I can only use Black-Scholes or CEV to hedge them?

Just suppose you have a time grid: t1-t0, t2-t1, t3-t2, ..., tn-tn-1 (tn=T) you want to generate a random motion W on it: W1-0, W2-W1, W3-W2, ..., Wn-Wn-1 (Wn=W) and you want the incremental W's to have the same independent distribution with expectation 0 and variance s*s So you get the total...

I guess you want to generate a random variable on normal or log-normal distribution through a simple random variable on uniform distribution. Let us just think it this way, you line up all your m classmates according to their height, from shortest to tallest. And then you get a quantile...

If you are reading Concepts and Practice of Mathematical Finance, reread 6.108 & 6.109 on page 169

So do you know which model is best for hedging purpose?

Suppose you need to hedge an exotic option without any frequently traded vanilla option as means, and all you have is a money account and stock, then which model would you choose - simple Black-Scholes, CEV modification, Hull White or Heston Stochastic Volatility type, or Jump Diffusion? I came...

Most textbooks stop at Geometric Brownian Motion framework. The advanced books are hard to understand. Can anyone tell me an easy introduction to Heston Model?

Don't be too pushy on yourself, just relax.

Could you tell us which book you quoted?

Thanks for the recommendation. I will list Kerry Back's book as the next book to read after Shreve's.

2.9 Numeraires and Probabilities?

He said he used the stock measure. I guess B/S is a martingale instead?

Thanks for your answer. Would you be more specific on the Margrabe's formula? Any paper or textbook? I think I have found the same conclusion that two risky assets can price an option on their exchange rate.

Yes, that is what I mean - money market numeraire is not the only and essential option to martingale pricing. But I don't know why there is a need to show the uniqueness of a martingale measure of a given numeraire? Or like moretodo just said, there must be a riskless money market numeraire to...

In Shreve's book, we see the discounted stock price can be changed driftless in some equivalent measure. And then by the Martingale Representation theorem, we can discount the expectation of the derivative values at maturity to price the derivative. However, there are other chances of...

Well I guess I have not made the question clear. The traditional argument equates a risk-neutral measure to an equivalent martingale measure (EMM). But I doubt if a debt or bond asset is unnecessary in construction of equivalent martingale measure. Traditional arguments require at least one...

Many thanks.

where did you see them?

Hi guys, I recently read about numeraire changing in Martingale Pricing technique. I heard the "Equivalent Martingale Martingale" dresses this issue and set a standard on whether there is a no-arbitrage price as the probabilistic expectation, called "Market Completeness". So do you have any...

Maybe you are a A student in pure mathematics, but this post has nothing to do with answering the question. pruse has given an excellent answer by pure probability theory, and we all know that. Maybe the way he writes was not so polite, but keep attacking him only tells everyone your envy. Stop...

That is because bond traders quote their prices by accrued interest+clean price. The calculation of accrued interest helps traders quickly filter out the effect of different interest payment days and compare clean prices between bonds. Of course the standard way should be comparing yield to...

I guess the reason why the problem is defined by martingale term instead of simply asking for a solution to an equation is that you can simplify the problem using Ito's lemma - like the way you posted in the end.

I remember there is a proof in John Hull's book.

Previsible process means a random process whose state is determined before another process shows up the result i.e. We can buy the stock before it goes up or down, but we cannot go back to past to buy a stock that goes up today. The event of buying a stock relative to the price change is a...

MACD

If instead of a Bell curve you get the distribution a fat tail distribution, can we conclude it is a summation of normal distributions with different standard deviation? Consider the density p(X)=p(A)/2+p(B)/2 where p(A) = erf(s1) and p(B)=erf(s2) and you can get a mixture of them. Suppose...

I think of an idea to examine whether the underlying price movement follows a GBM. In a standard GBM We have log return like this Ln(Sj/Si) = (a-ss/2)(j-i)+ s(Wj-Wi) & E[(Ln(Sj/Si))^2] = ((a-ss/2)(j-i))^2 +ss(j-i) If we separate a time series of quotes of duration T into equal time...

Thank you for the reply, FKaria. The main reason for the volatility smile, so far as I understand, is that the distribution of the rate of return is not Normal. The heavy tail property of the real distribution in contrast to the theoretical Normal distribution, shows a lower normal standard...

Brilliant! The most traditional way- synthetic option!

I am so glad that finally someone replied my post. Thank you FKaria. I have found that Marc Yor and Geman Yor's book Mathematical Methods in Financial Market has a tedious proof. The reason why I need an analytic solution is I need to delta hedge and create the derivative. I don't know why a...

Hi All. There is formula ready for the lookback call option whose payoff at the maturity is S(T) - Min(S(t), 0<t<T) where S(T) is the terminal underlying price, and S(t) is the underlying price at time t, then the second term means the minimum of underlying price along the duration of the...

How about using B-S framework directly to dynamically hedge an option? Is the tracking error too large to satisfy investors? Can Heston Model reduce the tracking error by a great deal?

Answer to question 1: 0 ~ K2-K1 K2-K1 is the maximum loss at the maturity after shorting the C(K1) and longing the C(K2), which gains C(K1)-C(K2) to set up spontaneously. If C(K1)-C(K2)>K2-K1, setting up the portfolio will give you 0 chance of loss and some chance of winning. Could you...

could you be more specific? P.S. It seems flawed with the quadratic variation drift 1/St=1/So exp{-(r-ss/2)t - sWt} =1/So exp{-r+sst/2 +sBt} Note +sst/2 here who contaminates a Geometric Brownian Motion formula

The argument is really intuitive. If you get the right to buy 1 Euro by only K dollars, it is equivalent to K times the right to sell 1 dollar for 1/K euro. Thus their current value shall be the same Call(So, K) USD = K Put (1/So, 1/K) EUR = KSo Put(1/So, 1/K) USD Call(So, K)= KSo Put(1/So...

Hello everyone, I heard a formula like C(S,K)=P(K,S) proved by Peter Carr. Its simple proof is on foreign exchange market where One Call on Euro denominated by Dollar is somehow equivalent to K Puts on Dollar denominated by Euro, considering the exchange effect, Co(S0,K)=KSo Po(1/So,1/K)...

You said "if" in setting the future rates. How about "elseif"? If the spot rate set in future is only determined by the spot market rate then, you can replicate the rates by swap or whatever. For you are just hedging the future rate, by forward contract on future rates. However, if the spot...

Salih's book is a fine introduction. Its math is simpler than Shreve's. Besides, it proves maths problem friendly for beginners.

Peter Carr's Symmetry

The reflection principle is a fine tool to solve barrier ending density as The distribution for a simple Brownian Motion ending value WT given a single barrier b was hit during (0,T] is N(x-b,sqrt(T)) I doubt if there are two other ways to solve this problem: reach (b,b+db) at some time t...

In martingale assumption, a stopped martingale is still a martingale. So if you stop gambling at a certain loss level. After many times of gambling, your expected profit or loss is still 0, fortunately, in a fair casino. So why do they set stop-loss strategy as an essential standard for...

Technical Analysis Regression Data Mining Chaos Discovery in ascending order of complexity

Here is Milo's solution: https://www.quantnet.com/forum/threads/double-barrier-ctd.8225/

And then the four becomes two since the second and fourth term cancels out in the sum.

\(N'(x)=\frac{e^{-x^2/2}}{\sqrt{2\pi}}\) \(e^{2\alpha m} N'\large(\frac{-m-\alpha T}{\sqrt T}\right)=\frac{1}{\sqrt{2\pi}}exp\large(2\alpha m-\frac{m^2+\alpha^2 T^2+2\alpha mT}{2T} \right)= N'\large(\frac{m-\alpha T}{\sqrt T}\right)\)

Good luck with your job hunt then. I will finish the drifted Brownian motion hitting one barrier post.

I felt unreasonable to use Girsanov to undrift a mean-reverting asset price. Suppose you have a GBM asset and a mean-reverting asset with the same volatility. Do the same option with these two assets as underlying share the same price? Why & Why not?

Good post, Milo. I am surprised by your persistence.

Thank you.

For European Options Call and Put, we have this equation: C-P=S-K/exp(r(T-t)) Thus C-P>S-K C>S-K Since the European Call C has value greater than its intrinsic value S-K, the more valuable American Call is larger than S-K as well. Thus you can always sell an American Call on the market at a...

I mean, there is an example in John Hull's book on dynamic hedging. The historical volatility is plugged in, and a replicating portfolio is constructed right from Black-Scholes. There is cumulative error during a period. However, is it practical to hedge an option on some stock in this way? Or...

Hi Moderator, help me move this post to "recommended reading", thanks.

(c2-c1)/(s2-s1)=Delta So you know...

Does anyone know about a book introducing Variance Gamma Model from scratch? Thanks for any recommendation. :)

It is hard to tell, Zubertrank. The courses have prerequisite conditions. For example, you cannot study 272C without studying 272A and 272B first, which means you have to learn 3 PDE classes altogether.

\[e^{2\alpha m}N'(d2)=e^{-\frac{-4(m)(\alpha T)}{2T}}N'(d2)=N'(d1)\] - completion of squares It is the same way that Shreve used to simplify his complex intgrals. ;)

Simply put, To come up with an option valuation, a quant either computes stochastic probability model or partial differential equation model, but may use Fourier transform method for more complex calculation. Stochastic Calculus tells you why and when to use the two and is the first class you...

Yes it is true.

I guess you are not a US citizen, Pradhu. It is very bad now in quant hiring in US. The best choice for you is learning by yourself. And then it is up to you to go abroad or land a local quant job. I could recommend the books I have used for you. They are not that hard if you learn step by step.

Nano, it depends on your knowledge. Have you learned Stochastic Process? Have you learned Fourier Transform? How well can you solve a Partial Differetial Equation? How much have you mastered Statistics? If you pass the four questions, the only block between you and a quant job is Shreve's book.

Quant is not suggested regarding your corporate finance background. How about Venture Capitalist? Stock Analyst? Passion for Numbers \= Passion for Formulas. You need to be very easy with Calculus and Statistics (including Econometrics) to be a successful quant.

Do you know binomial models, Anna?

Oh thank you Bucks, I just remembered my professor did mention this problem and I found in John Hull 6e Homework 13.17 on this matter. Yes they cancel out.

If the derivatives of N(d1) or N(d2) are not in consideration, the partial derivative on the latter, negation of a share digital option is zero as well, which contradicts reality.

I found the definition of delta with respect to an European option problematic: (\frac{\partial C}{\partial S_t}=\frac{\partial}{\partial S_t}(S_t N(d1)-Ke^{-r(T-t)}N(d2))=N(d1)) while d1 and d2 are actually functions of S. Shall we count in the derivatives of N(d1) and N(d2) ? Or the...

They'd better say Shreve and Glasserman so that they got tuition without teaching.

probability

There are two different expectations on an financial engineer: Price and hedge derivatives as accurate as possible; Make correct bets on the market through statistics and mathematics. The first one is difficult yet accessible to fulfill, but the second one is almost impossible to find...

On numerical methods, you only need an R and some Monte-Carlo books to develop your own algorithm. You can self-study numerical methods on probabilistic approach.

If you want to learn derivative pricing well, here are road map to its knowledge needed Mathematical Finance or Stochastic Calculus - The first and foremost thing you need to understand, and any other knowledge serves it. And from here you can dig into 2 areas: Stochastic Process - the...

I really thank library.nu for their pirated books. Choosing a good book in a particular field is almost impossible without viewing a large amount of books in that field first. Only after reading many books can I buy one truly excellent one.

The world largest scientific e-book site library.nu is closed today, warning that: Library.nu is offline today (joining large sites such as Google and Wikipedia), because the US Senate is considering legislation that would certainly kill us (and most of the internet) forever. The legislation...

Mark Joshi only gaves 2 pages explaining reflection principle in his book Concepts and Practice of Mathematical Finance The problem with most stochastic finance books is that they only briefly discuss the barrier problem from one or two aspects, thus any book is incomplete treating the...

It is just your formula given in post 6 simplified and multiplied by (-1) Try it.

I understand what went wrong regarding negative sign These two are not equal but complimentary: \(P\{\tau _m \le T\}+P\{M(T)<m\}=1\)

Bessel Process and Variance-Gamma ? There are alternative models than Black-Scholes, and it is challenging even today to construct and to calibrate them in financial markets

See, if you differentiate P{M(T)<=m} with respect to T, you just get negative values. To simplify, just make a=0 the driftless case: \(-\frac{m}{2\sqrt{T^3}}N^{'}(m/\sqrt{T})-\frac{m}{2\sqrt{T^3}}N'(-m/\sqrt{T})\)...

completely lost in your symbols. you can use Latex with (tex) (/tex) but [] instead of () However you need to define each variable you use.

This article explains the link between DV01 and Duration. You may have already figured it out before reading it. http://www.closemountain.com/papers/risktransform1.pdf

No it is not change in value expressed in dollars.

It is impossible to determine a commodity forward price function with respect to time, spot price and cost of carry. The convenience yield is determined by the difference between real forward spot spread and cost of carry. The former is suspect to volatile supply and demand.

(MD=\frac{\Delta P/P}{\Delta r}) Check the detailed formula with this. If it does not fit, it is wrong.

No. Shreve's proof is only a sketch. I found the drifted Brownian Motion must be at least Markov to determine the time differential of Radon-Nikodym derivative before multiplying them up.

On page 212, he indeed states (\Theta (u) ) needs to be an adapted process, and the convergence condition comes from 4.3.1, where Ito integral is defined. He further showed an example in 4.3.8 that the integrand in an Ito integral can be stochastic (in fact can jump) but is still a martingale...

Sorry I misinterpret the definition of "adapted" as "previsible". What I meant in post 5 is that if drift is "previsible", then it can be treated as constant in discrete time intervals like the weights in a self-financing portfolio. Therefore my challenge on the theorem still stands. It is...

These lecture notes are also available http://galton.uchicago.edu/~lalley/Courses/390/

"adapted process" might mean that for any given small time interval, we already know the drift before hand, therefore drift is held constant in that interval and not stochastic compared to the volatile part of Brownian Motion. The Girsanov's Transformation originally involves only one Brownian...

To simplify, just the drift of the diffusion.

If the drift of a Brownian motion is constant, Radon-Nikodym derivative could be applied to Girsanov transform the drift to zero, or from zero to a constant. This is no doubt. However, I doubt if it can also be applied to a stochastic drift, say, a mean-reverting stochastic drift. Just...

I got hammered in a seminar last evening. The Morgan-Stanley Doctor said people no longer use simple Black-Scholes setting in Wall Street. They add in Heston Volatility Model to capture the irregular up and downs of price movement. This is a real blow since the whole book of Shreve turns...

http://www.opentradingsystem.com/quantNotes/main.html explicit illustration, large coverage, but poor background theme.

Simplify

It is very odd that Steven Shreve left some Fourier Transform on hitting time proof in his book. And it seems fourier method might be another option in option pricing rather than PDE and martingale approach? Anyone who knows about it?

If we differentiate the Probability of event {M(T)<=m, X(T+r)>m}, or that X(T) breaches the upper barrier m in time interval (T,T+r] , with respect to time increment r, we get the hitting time distribution of X(T): \[\frac{ -\partial P}{\partial T}=\frac{m}{T\sqrt{2\pi T}}e^{-\frac{(m-aT)^2}{2T}}\]

You can change sample time so long as your option price and stock price are observed at the same time. So it is okay to use Monday opening for both prices.

the option value is replicated by a self-financing portfolio, which is set up prior to price changes and can still hedge small random shift of the underlying price in a short period. Both the value and the component of the portfolio is "previsible" about the underlying price change. Therefore...

It depends on how much you want to know. Search "Brownian Motion" in wikepedia and you get the first impression.

The event {M(T+h)<=m} contains the event {M(T)<=m} where h>0 So {M(T)<=m}-{M(T+h)<=m}={M(T)<=m, X(T+r)>m} where 0<r<=m That is where I got the clue.

Sorry for my previous mistake, X(T) shall be M(T) in the equations. \(P \{M(T)\leq m\} = N(\frac{m-aT}{\sqrt{T}})-e^{2am}N(\frac{-m-aT}{\sqrt{T}}),m>0\) \(\frac{\partial}{\partial T}P\{M(T)\leq m\}\)

Take a GRE exam and prove your IQ. Some graduate schools value GRE more than education.

Shreve 7.2.2 Under Probability P, W(T) is a standard Brownian Motion while X(T)=W(T)+aT is a Brownian Motion plus drift. M(T)=Max{X(s); 0<s<T} \(P \{M(T)\leq m\} = N(\frac{m-aT}{\sqrt{T}})-e^{2am}N(\frac{-m-aT}{\sqrt{T}}),m>0\) Can we differentiate the above to get first passage time...

In finance, we don't care about how the drift udt might vary over time, so it is good to have a volatility estimate independent of time drift estimate. True, you can use a R=udt+sdW to estimate s, but at the same time you assume u holds constant among your sample and estimate of s is based on...

Thank you. Thanks for your notation, but I have seen the formula in infinite sum.

O stands for the ball running. initial spaces are cut off so you may not see the trajectory of O clearly, but the O does a random walk verticly and a straight line movement horizontally.

Okay, suppose you are observing a ball running along the time horizon before a right bound T. However it must stays between the line y=a and y=b, once it touches either line, it is finished and no longer runs. _________y=b__________ O O 0- - - - O- - - - - - - >- - - - - --...

This is for single barrier. You need to consider the case that the Wt stopping at a before b, thus reducing the chance stopping at b, I guess.

No, a<W(T)<b and W(t) before W(T) in its path cannot hit either a or b. T is forced stopping time. You can interpret it as Game Over Time - forget about Wt continuing or stopping at barrier a or b, now at time T it stops at whichever value.

Let me clarify the symbols a little bit. The Process（t,W(t)） stops at (Ta, a) (Tb, b) or (T,W(T)) for the first two occasions, W determines the stop, and for the last, t determines the stop The question is: 1) Prob(0=<Ta<u<=T) 2) Prob(0=<Tb<u<=T) 3) Prob( a=<W(T)<=u, u∈[a,b]) Of...

Wt stops when it hits a<0 or b>0 Is there a formula to determin P(Wt=a, t<T) and P(Wt=b, t<T)? Any help would be appreciated

Unfinished Gambler's Ruin Can you answer the question in this post?

Currently my research focuses on double barriers, a rare topic in textbooks.

Thank you. I have downloaded her lecture notes.

Suppose a Standard Brownian Motion Wt without drift. Wt stops when any of the three circumstances below occurs: Wt=b>0 Wt=a<0 t=T Traditional Gambler's Ruin problem focuses only on the T=inf case. However, I want to know P(W ends up at b) and P(W ends up at a) and of course P(W ends at T)...

How about the process with finite time allowed? To preserve a martingale P(ending in upper boundary)+P(ending positive in the end but below upper boundary)=0.5 Am I correct?

For example, an up-and-out call option with Barrier J and Strike K<J. The stock price stops at barrier J at time t <=T. In a risk-neutral measure, J(s=inf{t St=J})*exp(-rs) & S(T)*exp(-rT) is a martingale, correct? Then we have a stop boundary for the martingale S(t)*exp(-rt) a near flat curve...

I guess the deeper mathematics than Shreve involve mainly two areas: One is deeper Calculus like Protter One is deeper probability with martingales like Williams And do I need to understand both?

Say you like party in Boston :D

The reason is that Shreve's book does not provide sufficient knowledge for me to read papers.

I lack measure and probability background. The only thing I know about is statistics at undergraduate level. I do have Models for Probability and Statistical Inference by Stapleton - have not covered convergence part However its focus is on Classical probability instead of stochastic probability...

It seems very short. Haven't tried yet. Currently I have downloaded these books: Arbitrage Theory in Continuous Time by Bjork Mathematics of Financial Market by Elliott and Kopp _They are at the same level as Shreve Continuous Martingales with Brownian Motion by Marc Yor - difficult...

Hello Everyone, I am looking for some books on the foundations of Stochastic Calculus with more mathematics. Because I feel even Shreve's book isn't clear enough on some topics. What makes it worse is that some French book like Mathematical Methods for Financial Markets seems so difficult...

I hate GPA Different Schools have different GPA standard. Plus, teammates can affect your GPA too.

No it does not seem correct. I have to know precisely when the barrier is hit and come up with the present value.

Thank you for clarifying the post. Yes you got what I mean. My question is : is there still a probability approach to price the Instant one touch? I came up with something like forward as the underlying, but the slope barrier for the forward. Obviously, it complicates the path but simplifies...

you mean R(t)=W(T)-W(t) ???

The stock price S does not stop at the barrier, but does the replication portfolio of a barrier option. Shreve illustrate that the barrier option stopped is still a martingale, but I still have not understood that.

So you want to say that a Freezed martingale is still a martingale? If (e^{-rt}S_{t\wedge T_B}) is a martingale, (e^{-rT}S_{t\wedge T_B}) is not.

Textbooks proved this. Wt∧Ta, where Ta = inf{t: Wt=a}, is also a martingale when Wt is a martingale. Does it prove anything? To price barriers? Any explanation? I am totally confused on its usage to price barrier options. Thanks in advance.

Shreve uses (log(St+1)-Log(St))^2 to estimate the volatility.

Consider two digital barrier options: One gives off 1 at expiry so long as the underlying spot reaches some level before expiry The other gives 1 immediately when the underlying spot hits that level Of course these two options should have separate value. I assume the first have a textbook...

Hi Tobi, I have already graduated from MSF in another country. I don't know if I could contact my professor with a discussion. Besides, I don't want to tell anyone other than the committee of a journal about my finding. This is the major issue between me and my professor.

If you mean stochastic probability, Shreve's book can do the job.

since dSdS=σσSSdt Isnt't more straight forward to add up (St+1-St)^2 to estimate the volatility for a Geometric Brownian Motion?

Fluid dynamics is a much more complex field than pricing derivatives. I feel like you are dropping a finance career for an accounting one. Derivative pricing (quantitative finance) is really not that hot given your math background. If you really need to make a fortune through financial market...

Forward you need 0 initial investment.

There is no table like that. Aside from solution to Geometric Brownian Motion, there are solutions to several mean-reverting SDEs. And those are all we have.

In terms of European Option, worse than an expectation numerical integral. In terms of Path-dependent exotic, looks good but with too little sample problem. In terms of American Option, already other methods. When is the application of MC better than other approaches?

My approach to prove existing theorem looks simple but is relatively new. It contains one page or two, and few references. Do you know some journals that accept such paper?

Another option pays the same but lose effect when the underlying hits another level.

Well could you tell me if this is a time dependent boundary or not? An option pays 1 dollar immediately when the underlying hit a predetermined level before expiry.

\[d\frac{S}{S}=\frac{dS}{S}-\frac{SdS}{S^2}-\frac{dS^2}{S^2}+\frac{SdS^2}{S^3}=0\]

\[d\frac{X}{Y}=\frac{d X}{Y}-\frac{Xd Y}{Y^2}-\frac{d X d Y}{Y^2}+\frac{X d Y d Y}{Y^3}\]

Suppose you have a bill with discount rate 4% and 1 month to maturity. The actual discount factor is 4%/12. So the bond is worth 1-4%/12 =PV Another point of view on this bond is (100/PV)=(1+YTM)^(1/12) Then you may solve the YTM. I guess the formula is Not correct.

My problem is a double barrier option between two slope lines (instead of known interval X in (a,b) ） Thank you.

You have to ask your teacher which Utility function and which distribution he assumes in the second problem.

Thank you.(y) How did you input formula BTW?

Hi everyone, I have a peculiar problem here: d(S/S) = d(1) =0 d(S/S) = dS /S+S d(1/S) <product rule> d(1/S) = -dS/ S^2 + (dS)^2/S^3 d(S/S) = dS/S - dS/S + (dS)^2/S^2 =(dS/S)^2 0 =? (dS/S)^2

Alison Etheridge's A Course in Financial Calculus is a strong recommend It has explanations, and very rigorous proofs, also deep concepts, yet unsolvable exercises, plus many typos. If you are bored by Shreve, try Alison. It will give you some challenge and thinking. If you feel it is still...

Thank you guys for your kindly advice. Double Barrier seems to be infinitely harder than One Barrier. The transition density is an infinite series because of infinite reflections between the upper and lower boundary. It was not until late 1990's until mathematicians developed fast series...

Exotic Options

The textbook didn't say about it?

Considering your background, I would really suggest you drop the idea of being a derivative quant. There are too many quants and MFEs nowadays to price financial derivatives. Working in the financial business does not mean you need to know how to price derivatives, which is a small fraction of...

A typical European option pricing via Black-Scholes approach is to solve a Black-Scholes PDE with terminal option payoff f(T,x) Simply put, a Black Sholes PDE involving the first derivative of option value on t and on x (the value of underlying security) and the second derivative on x, with a...

Thank you DStahl. I checked some textbooks - Feyman Kac Theorem guarantees martingale pricing would work for any European Options whose boundary condtion is at maturity. However, the applicability of the martingale approach to path-dependent options remain plausible.

do you mean use the implied volatility of a call to monte-carlo the double barrier?

Yes you got it. Derivative pricing techniques work only well in financial derivatives. Any other field, econometric.

These books are on my shelf. How about Robert Elliot & Ekkehard Kopp Mathematics of Financial Market and Tomas Bjork Arbitrage Theory in Continuous Time?

Partial Differential Equations are generally very difficult to solve. It is not like the integration where you just need to remember under 20 common integrals in order to solve most problems analytically. Sometimes a difficult boundary condition (similar to upper and lower limit of a integral)...

Russell Davidson

There is a freeware similar to matlab, but I have forgot its name, just remembering it starts with "O".

Yes, people do need exotics to gamble in the market. e.g. Some people give up forecasting the price of a stock, but he/she is confident to gamble its volatility. And a convertible might be used to place the bet. A convertible, widely recognized as a special kind of debt, is in fact a very...

Hi everyone, I have a serious doubt: What is the rational for risk-neutral pricing in continuous Geometric Brownian Motion? I know the techniques -Girsanov transformation and discounting expectation But I don't know why Why this can price an option? Why this can price an European...

fufil-satisfy or you could say deceitfully satisfy like no-risk supreme debts, here the "no-risk" is equivalent to "healthy"

Of course there are misuse on fixed-income derivatives. It is like healthy hamburger - unqualified products to fulfil unrealistic customer need

Financial Engineering adds marginal value on financial products - it flavours the underlying to varies taste for investors. metaphor: I don't want a stock, but I want an option derivative on that stock. <=> I don't want a plain ice-cream, but I want a chocolate flavoured ice-cream. You can...

yield to maturity measures the IRR, internal rate of return, and it is the compounded interest rate. You should be able to use YTM to discount all the future cash flows of that bond to its dirty price today.

I need advice on pricing a double knock-out European option whose pay-off at the end is only allowed so long as the underlying has not breached an upper and a lower bound within the option life. e.g. A European Call with strike 20 dollars with both 25 dollar up-and-out and 15 dollar...

I bet Monte-Carlo has far little possible paths than a binomial tree. Say, 2^100 paths only need 100 time points in a tree. Monte-Carlo has a finer ending-time distribution, but too little paths have insufficient representation in a path dependent model. Therefore, I prefer binomial model to...

If you have the analytic solution, I guess you could differentiate it with respect to S

If you mean you want to plot all the paths out, I would suggest you give up. Because there are 2^100 different paths within your 100 -step binomial tree. Simulating paths is a necessary procedure in Monte-Carlo simulation, which however produces far less possible paths than a large binomial...

You don't need to. But Geometric Brownian Motion is the most popular model to reflect stock price movement nowadays.

You have post a serious but often neglected question. Not all distributions can be easily shifted to a risk neutral measure to price an option via martingale approach. It has to be transformed through Radon-Nykodym derivative. The book Brownian Motion Calculus may answer your question on...

Because your steps are not dense enough. Besides, this multiplicative model has error simulating a Brownian motion, try additive model instead, which simulates log(S(t)) and you will never get a negative S(t).

Sure you can. But there are better ways to evaluate an European Option.

Yeah, Brownian Bridge. Build a Brownian midpoint between the initial value and the ending ones, then build 1/4 points between the three. And the list goes on until you feel the path points are as many as you want.

Well, some criticisers on jump-diffusion models simply don't calculate them at all. It is useless when I don't have it.

I prefer excel to matlab to calculate an American option.

A Course in Financial Calculus -the book that saved my education in quant