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http://ssrn.com/abstract=2397010

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?

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...

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...

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...

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...

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)...

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...

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?

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...

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

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...

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...

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.

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?

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...

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

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)...

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...

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.

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...

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

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?

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

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...

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...

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...