Christian Fries

891 reputation

bio	website	christian-fries.de
	location	Germany
	age
visits	member for	4 months
	seen	2 hours ago
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See my homepage for some info about me.

Some of my projects:

finmath.net A Java library and spreadsheets with algorithms related to Mathematical Finance (e.g., curve calibration, Monte-Carlo simulation, Bermudan option pricing, American Monte-Carlo).
Obba A middleware to seamlessly use Java/Scala libraries from spreadsheets. Allows to use any Java library as a spreadsheet add-in.
Mathematical Finance (a book on some topics in m.f.)

	891 reputation	bio	website	christian-fries.de	visits	member for	4 months
16 badges		location	Germany		seen	2 hours ago

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28 Comments

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May 13	comment	Quadratic variation quesiton Ito actually tells you that $d(B^2) = BdB + BdB + dt$. So ii) might mean: Proof it using Ito, while iii) means proof it in an elemantary by repeating the proof of Ito for the special case of $B^2$.
May 11	comment	Quadratic variation quesiton Please change the title of this "question" (currently being "Exercise on stochastic calculus" to s.th. meaningful (e.g. use words like "Ito formula" and "Quatratic Variation").
May 6	comment	Is vega of Black-Scholes European type option always positive? I added a remark on general payoffs to my answer. With respect to your comment: assuming that the pay off is positive does not help. Vega is not about level or slope, it is about convexity. It would depend on WHERE your payoff is convex and WHERE it is concave. And how strong convextiy is depending on the underlying.
May 5	comment	What does it mean to adjust for short-run liquidity in finding risk-free rate of return Can you give a reference for this? Did you read this somewhere?
Apr 26	comment	Convexity adjustment for a forward swap rate OK. If this is about swap futures, then he should be more precise...
Apr 26	comment	Convexity adjustment for a forward swap rate The convexity adjustment needed for futures comes from the margining applied to the (undiscounted) future price. In contrast, swaps are collateralized by discounted value, such that a future-like convexity adjustment does not apply. However, if a forward swap rate is paid in an unnatural way (like in a CMS), a convexity adjustment applies.
Apr 26	comment	Convexity adjustment for a forward swap rate The question in this form is incomplete. The swap rate alone does not need any convexity adjustment. You have to specify how this rate is paid.
Feb 19	comment	Upper bound concerning Snell envelope Can you please define the set below ess sup. Is the the set of all stopping times? Also, what is p? Do we have $p \geq 1$.
Feb 9	comment	How to derive the formula of a European Libor call option in a Libor Market Model? The conditional expectation, conditional to $\mathcal{F}_t$, is a random variable. It is $\mathcal{F}_t$-measurable. It is a real number once you look at this random variable at a specific state, namely the states you know at time $t$. The $\log L(t,T)$ is just that state... (maybe you like to look at the picture on page 18 and the interpretation on page 17 and 18 here: books.google.de/books?id=q7c8Pi6QGFQC ) - - The expectation conditional to $\mathcal{F}_{0} = \{ \emptyset, \Omega \}$ is also a random variable, but it is constant (same for every state =$\mathcal{F}_{0}$-measurable)
Feb 9	comment	Why doesn't a simulated delta hedging process go to zero? In addition a big effect comes from the difference in volatitly. The geometric brownian motion with vol 100% (which is by the way quite large) in your BS delta assumption has approx. 100 time the vol of S(t) (since S is normal and S(0) is 100. Another thing which I found strange is that S(t) is evolved backward in time, but I did not dig deeper into your code.
Feb 8	comment	How to derive the formula of a European Libor call option in a Libor Market Model? W.r.t. the Markov property you just have to check: Does the future $T > t$ depend on the past $s < t$ or is the knowledge of your state variables in $t$ enough. In the SDE of the LMM the initial value is L(t), the drift depends on $L(t)$ or $L(\tilde{t})$ for $\tilde{t} > t$ etc., but nothing depends on the past. (There is a little excercise you could try: if you write the LMM as a short rate model, you see that the drift depends not only on $r$, it also depends on that past (to reconstruct the other forward rates). So with respect to $r$ LMM is not Markovian.
Feb 6	comment	What is the relationship between Forward measures and LMM? I believe instead of $dQ^T/dQ=1/(P(0,T)B(T))$ you should write $$\frac{dQ^T}{dQ}_{\vert \mathcal{F_t}} = \frac{P(t,T)}{P(0,T)} \cdot \frac{B(0)}{B(t)}$$ such that it is clear that it is just a change of numberaire from $B$ to $P(T)$.
Feb 6	comment	What is the relationship between Forward measures and LMM? $L(T_{1},T_{2})$ is the continuous time stochastic process of the forward rate. $P(T)$ is the value process of the bond with maturity $T$. I use the following notation: $L(T_{1},T_{2};t) := \frac{P(T_1;t)-P(T_2;t)}{P(T_2;t)}$ where $T_{i}$ denotes maturties and $t$ denote "observation time". If $N = P(T_2)$ is the chosen numeraire, then $L_{1} := L(T_{1},T_{2})$ is a numeraire relative price, hence a martingale.
Feb 6	comment	What is the relationship between Forward measures and LMM? By forward rate I mean forward LIBOR $L(T_{1},T_{2}) = \frac{P(T_{1})-P(T_{2})}{(T_{2}-T_{1}) P(T_{2})}$.
Feb 6	comment	What is the relationship between Forward measures and LMM? @hulk: I believe the "$T$-discounted" in this answer is misleading. The forward bond (without disocunting!) is a martingale. This is the case if you choose $P(t,T_{2})$ as numeraire (not B(t)!).
Feb 6	comment	What is the relationship between Forward measures and LMM? I would stick to the other definition: The forward measure is the measure under which the forward is a martingle. The advantage is that this definition is also valid in a more general context (like multi-curve / OIS-discounting).
Feb 5	comment	What is the relationship between Forward measures and LMM? Yes. If if you have that assumption as an induction start, then (2) will give you that you have a family of forward measures. I update my answer a bit....
Feb 4	comment	Yield Curve construction This is not a question on the methodology of constructing a curve (like for example in quant.stackexchange.com/questions/2982/…). It is a question on where to find free data! Uhhh.
Feb 1	comment	Pricing in HJM framework They just derived the condition for the drift of $f = f(t,T)$ under the equivalent martingale measure w.r.t. $B(t) = \exp( \int f(t,t) \mathrm{d}t)$. So from HJM you now have two options: 1. Choose your model (like LMM) and derive the risk neutral drift for it (in its own formalism) - or - 2. Choose your model by setting $\sigma$ and derive the drift from the HJM drift. - For me, the important role of HJM is that we see how things are linked together. For example: I know how to choose the parameters in LMM to get an LMM dynamic similar to Hull-While ("calibrate LMM to Hull-White") - if needed.
Jan 28	comment	Simulation of GBM One can view his equation as the Euler scheme for log(S). In that case discretization and exact soluation have no difference! The reason why one has to go in discrete steps in the way to generate W(t) from i.i.d. random variables.