10

Amazingly, there are several different methods for computing bond forward price – the underlying ideas are the same (forward price = spot price - carry), but the computational details differ a bit based on market convention. Let's start with the basics. Assume between now ($t_0$) and the forward settlement date $t_2$, the bond makes a coupon payment at time ...


9

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 ...


8

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 ...


7

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 ...


7

[Short answer] No closed-form formula in general. You need to resort to numerical methods. Monte Carlo is preferred by most practitioners but you could also use Finite Difference schemes (and sometimes even Fourier inversion techniques depending on the model used and the instruments to be priced). [Long answer] One usually distinguishes between 2 classes ...


6

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 ...


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

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 ...


5

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 ...


4

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 $N(ln(S_0)+(\mu-\frac{1}{2}\sigma^2)T, \sigma^2 T)$ ...


4

I think you are confused by the definitions and interpretations of $\psi(x,y,z,t)$ and $\phi(x,y,z,t)$. The quantity $\phi(x,y,z,t)$ is a probability density function. Infinitesimally, it represents the probability of transitioning from an initial state $[S_t,r_t,q_t]=[S_0,r_0,q_0]$ at $t=0$ to a state $[S_t,r_t,q_t]=[x,y,z]$ at $t>0$. As such, $\phi(x,y,...


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 $\exp(...


3

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 ...


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, ...


3

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 ...


3

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

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


2

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 ...


2

There is a much better pricing formula which is an accurate approximation. Anecdotally I believe that the difference between this and the "offical" CDSW calculator on Bloomberg will be within about 0.5% or less of the notional, especially if the CDS curve is flat. For a \$1 notional of short-protection contract with coupon $C$, market spread $S$ and $T$ ...


2

You have two main papers that show this result: In a finite framework and in a somewhat simplified continuous framework, see Harrison & Pliska (1981), Corollary on page 228, Proposition 2.9 and Proposition 3.31. If you are familiar with French, this webpage of HEC Montreal has some slides that go through the finite case proofs $-$ look for documents ...


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

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

Do these work for you? P34 of http://web.mit.edu/junpan/www/SVJ.pdf P1360 of http://www.darrellduffie.com/uploads/pubs/DuffiePanSingleton2000.pdf P2045 of http://www.math.ku.dk/~rolf/bakshi.pdf


2

For an American option, you have the right to exercise at any intermediate time. Then, at time $T-1$, if you exercise your option, you obtain the payoff $X_{T-1}$. However, if you wait to exercise at the maturity $T$, your value is $\frac{1}{1+r}\mathbb{E}^Q\left(X_T \mid \mathscr{F}_{T-1} \right)$. Your option value at time $T-1$ is the maximum of these ...


2

The answer is yes. In fact, there always exist a 'Black Scholes like' formula. Easy to show too. If the risk neutral distribution of the price has cumulative density $P$ and probability density $p$, then $$ E(S-K)^+=E((S-K)\ 1_{S>K})=E(S\ 1_{S>K})-K\ E(1_{S>K}) $$ The second expectation is just $P(K)$, ie the probability that the option ends up in ...


2

Any time that a contract is cancellable by either party, it will be cancelled. That's because it is always to one party's advantage to cancel rather than carry on. The exception is that the contract is worth exactly zero, which has effectively zero probability. Therefore, the value of the contract is whatever will get paid out on cancellation. I.e. The ...


2

There's no such thing as an undated repo. For example , what would happen if the underlying security matures? Term repos of Treasuries of one week to one year are reasonably common. The pricing models incorporate some generic factors such as the shape of the Fed Funds curve, plus some security specific factors such as the likelihood of the underlying ...


2

Typically structures like this are traded as notes. They will be sold at a face value of 100%, where that is normally the combination of a zcb (ie 1y usd, say 97.5%), expected coupon (say +10%), short Knock In put (also knocked out by the autocall feature, say -8%), and some profit for the issuer (in this case, 100%-97.5%-10%+8%=0.5%). Sometimes these are ...


1

Supply and demand... If you want an event that produce a change in the value of a currency, just look at the ruble. As Russia, gets more and more isolated and inflation spins out of control the ruble lose its value against other currencies.


1

In the Merton jump diffusion model, the stock price process consists of a continuous part and a discrete part (this one represents the jumps). While deriving the PDE for the riskless portfolio and imposing the riskless evolution, the discrete part can't be instantaneously hedged. In fact, you can assume that the effects of jumps can be nullified on average, ...


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