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7

Note first that this key equation is only assumed to hold true under some extra assumptions. Typically those assumptions are taken to be about absence of arbitrage, though it is possible to weaken them somewhat if you are willing to consider portfolio arguments or collectively agreeable objective function. Anyway, the argument is this: if all the risk can ...


6

You can find a simple proof in the discrete time case at http://kalx.net/ftapd.pdf. I'm not sure what you are trying to derive with your Ito calculus, but here is a rigourous derivation of the Black-Sholes/Merton PDE: http://kalx.net/dsS2011/bms.pdf. The Black-Scholes '73 derivation is not mathematically correct. The modern approach does not use so called ...


6

Well, that's still a very general question. A few elements of answer : Bonds pay interest on a regular basis, semiannual for US treasury and corporate bonds, annual for others such as Eurobonds, and quarterly for others. You need to distinguish between fixed coupon bonds, zero coupon bonds, bonds with an amortization schedule, floating rate notes based on ...


5

To simplify notations, let $a:= -1.96\sigma$ and $b := \mu - 0.5\sigma^2$. The development in the book could be justified if both $a\sqrt{t}$ and $bt$ are small (close to zero), and if we have that $|a\sqrt{t}| > |bt|$. Recall that $\exp (x+y)= \exp(x)\exp(y)$, $\exp(x)\approx 1 + x,\quad \text{if } x\approx 0$. Then, using these properties we have ...


5

Have you looked into "noise trader" models? This seems like a market that is mostly noise. A few betters may have some information on or real knowledge of who might win, but certainly nothing like equity markets where there are a lot of people who really know the ins and outs of the firms they're trading. The classic model is Pete Kyle's, which should give ...


5

You cannot deduce the real-world probabilities from the option prices. It may seem strange, but here is a simple example which might help you to understand. Suppose that everyone in the market agrees on the real-world probabilities, and that they are not changing for any external reason. Then suppose that the investment board of a large pension fund ...


5

There are many different ways a pricing model can be better : It can allow to reproduce the observed market price (Fit criterion) It takes into account a specific recognized behaviour of the underlying S, say the forward smile dynamic. If you write a product whose value is mostly derived from said behaviour, you dont want to miss that aspect. (Don't fill ...


4

In the way that you have posed the question, I would say that we are here discussing a derivative-pricing model rather than a predictive model. That's an important distinction because a predictive model would be assessed by its ability to generate money. In contrast, I think of derivative pricing as a fancy way of doing interpolation/extrapolation on ...


4

You cannot get "true probabilities" (empirical distribution) from the BS model. Option price is required initial investment, which is risk neutral expectation of payout. “True probabilities” are irrelevant in Black-Scholes. However, you can estimate the risk-neutral probability distribution (i.e. implied risk-neutral density) of the stock returns through ...


3

I could not find any such detailed documentation after some weeks of looking (not non-stop obviously). It is appallingly documented. I do understand fully what it does though so am happy to field some questions on it if you like. In a nutshell, I can tell you it is a standard reduced-form credit model under a constant hazard rate (i.e. homogeneous Poisson ...


3

I wanted to add this side note to Quantelbex' answer: Both factors in $\exp(a\sqrt t)\exp(b t)$ go to one as $t$ goes to zero, but for small $t$, the $\exp(b t)$ term approaches one faster. For $t=\frac {a^2}{b^2}$ both factors are the same, if $t$ is smaller than $\frac {a^2}{b^2}$, we have $\exp(a\sqrt t) > \exp(bt)$. Thus the approximation that ...


3

The true probabilities underlying the B-S equation are actually postulated. The pricing process is assumed to follow the stochastic process $d S_t =\mu S_t d t + \sigma S_t dW_t$, where $W_t$ is the Wiener process. It means that (for simplicity, let's talk about European call) $\ln S_T$ is distributed as $N(ln(S_0)+(\mu-\frac{1}{2}\sigma^2)T, \sigma^2 T)$ ...


3

Might be a bit overlapping with nicolas' answers, but here it goes: Id say you would have to look at the prediction-power of the model at hand. What if you do a backtest where you set a time t in the future? Set a price range for the stock at time t, and check with market data how often the price have been within the range. Then, for each model calculate ...


3

This is very standard financial theory. The answer to the question is given in the first chapter of Duffie's Dynamic Asset Pricing Theory and in Cochrane's Asset Pricing. The latter is more elementary, but you have to read more to get to the answer.


3

Since I did not get any comments to my latest update, and since I find it quite convincing, I hereby post my solution as an answer. maybe I can prove that Q exists assuming a lognormal distribution of $S_t$. Assuming $dS_t = \mu S_t dt + \sigma S_t dW_t$ By Itô, $d(e^{-rt} S_t) = -r e^{-rt} S_t dt + e^{-rt} S_t dS_t$. Replacing with the definition of ...


3

The only requirement if you are risk neutral is the property of martingale on your discounted stock price $M_t=e^{-rt} S_t$. But if you apply Itô $d( S_t\cdot e^{-rt} e^{rt})=d(M_t\cdot e^{rt})=r_tM_te^{rt}dt + ..dW_t=r_tS_tdt+..dW_t$ you see see that under the risk free probability, the asset price must have $r_t$ as yield and to answer to your question, ...


2

The first think you have to ask is ¿¿What price??? Monetary price or equity price?? All answers,the ones I read, related to monetary price, but are equity price really risk free???? One of the biggest problem with Black Scholes (personal opinion) is that they consider the behave of equity price as monetary price: Solve this ODE: S(t)'/dt= r*S(0), this tell ...


2

Increased volatility towards the event start is definitely from increased order flow. There are some papers specifically on "prediction markets", the ones with practical applications are on market making which I suspect is generally a loss-making operation conducted by the exchanges themselves when a market is opened. Given the short-time periods and small ...


2

US market uses the STREET convention. UK market uses the DMO convention. EUR market uses the ICMA convention (Germany uses also a lot MOOSMULLER convention). The main difference between these conventions are: - the way the number of days is calculated for the discount factors - the day count convention used - the calendar used in case of adjsuted ...


2

I am rather a fan of mathematical/statistical software for doing numerical finance (R/Matlab). But returning to your question: The commercial software UNRISK is based on mathematica, a computer algebra system. Usually you can use the Unrisk functions right in mathematica and price financial derivatives there. There also exists Jave interfaces if you want ...


2

And, as suggested by everyone on and offline, the winner is... The Handbook of Fixed Income Securities by Frank J. Fabozzi.


1

That's a great question and it is what I always wanted to try to do. I guess I found a solution using PDE approach. Change of numeraire would be more intuitive indeed, but I am not very good in stochastic calculus. The idea is as follows: 1) Let's consider portfolio $\Pi = V(X,Y,t) - \Delta_X X - \Delta_Y Y$. I will found $\Delta_X$ and $\Delta_Y$ such ...


1

It used to be (and maybe still is) Standard Securities Calculation Methods ..., a 2 volume set, but it is no longer available. You might find it in a library. It is available for $$$ at http://www.sifma.org/research/bookstore.aspx, or maybe you can find a used copy. TIPS, Inc., http://www.tipsinc.com/prods.htm, has an app and software implementing the books. ...


1

It is true that you cannot infer the real World probabilities from the BSM formula directly. It is also equally true that the "right value" of the option in the real world is obtained by replacing the risk free rate with the expected return of the stock. Another example of this is simply to look at the real world price of a forward on the stock. If ...


1

A simpler solution I found is to discount the differences between current spread and original spread: $MV_{CDS}=T \cdot (s_0 - s_t )\cdot \sum_{i=1}^{T}{e^{-r\cdot i }}$


1

I'm not expert. However, it seems clear that you're generating an upper bound on the seller value. You have to model the risk of default, as well as any convenantal terms for structured default, to generate an expected payout rate, and deduct that from the DCFs, to get a more realistic value. If the terms include a swap put model that separately. To set a ...



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