I calculated like: mu=0 =>dV_t / V_t =sigma*dW_t?Then it follows V_t=V_0*exp(-0.5*sigma^2)*t+W_t , with W_t having mean zero=> E(V_T)=V_0*exp(-0.5*sigma^2) or am i wrong?Imo this is due to the fact that a relative change of 10% up and afterwards downcauses a value lesser than the initial value.Or do you use again a measure which gives you E(V_T)=V_0?Still a little bit confused, Felix

you got it almost - a bracket error:it should be: (note the curly brackets)V_t=V_0*exp{(-0.5*sigma^2)*t+W_t},and then use that if X is normally distributed with mean m and standard deviation v, i.e. X~N(m,v), then E[exp{X}] = exp{m + 0.5 v^2}.PhilippPS: there was a small slip in my post, too: I forgot the second V_t in dV_t = V_t* sigma dW_t

oops, today something is really wrong with me. Of course I meant to writeV_t=V_0*exp{(-0.5*sigma^2)*t+sigma*W_t},

Yeah, sorry for my bracket error 2, forgot those ofcourse I understand this now.And that V_t+1=V_t*sigma*W_t +V_t (discretized) is a decreasing process on average(sigma>0) is caused due to the discretization (since the use of V_t isn't correct here)?Thx for your answers, Phillip, i see my errors now more clearly

"And that V_t+1=V_t*sigma*W_t +V_t (discretized) is a decreasing process on average(sigma>0) is caused due to the discretization (since the use of V_t isn't correct here)?"the discretized process is also a martingale (i.e. not decreasing on average).

I see that E(V_t+1)=V_t,but if i simulate this process with W_t ~ N(0,1) [delta_t=1],the process is declining as far as i tried it...e.g. let's say if you got two random numbers: let's say sigma*W_t = +/- 0.5Let's say we have a price of V_t= 10.First sigma*W_t=+0.5 => V_(t+1) = 15 or an increase of 50%Second sigma*W_(t+1)=-0.5=> V_(t+2) = 7.5 or an decrease of 50%First -0.5, then +0.5 yields the sameIf you say both +/- 0.5 have the same probability as has any other pair ofvalues, the process should decrease on average.Very very heuristic, but it still makes me believe the process decreases onaverage, so where am i wrong?Jetzt ist mein Pulver aber wirklich verschossen

The lognormal distribution has very fat tails. From below, the process is bounded by zero, but not from above. Once in a while, you'll have extremely large final values V_T. And that will be enough to make E[V_T]=V_0.I encountered this phenomenon when I simulated a lognormal random walk for CDS rates which have very high volatility (e.g. 80-100%). Over a 5-year horizon, you essentially either get something close to zero (in most of the cases), or an extremely large value (in the other case). The mean was still at V_0, though, because one large realisation cancelled all the other realisations close to zero. numerical example:If you substitute T=4, sigma=100%, V_0=1, W(T)=sigma* sqrt(T)* epsilon = 2* epsilon, into the closed-form solution, (where epsilon is N(0,1)-distributed) you get V_T=V_0*exp{(-0.5*sigma^2)*t+sigma*W_t} = exp{2(epsilon-1)},The exponent will be smaller than zero in N(1) ~~ 84% of the cases. So in 6 out of seven cases you'll see a decrease. But if you have a large absolute value in epsilon, things look different: let's say epsilon=+-3. For epsilon =-3 we get V_T=0.0003. For epsilon =+3, we get V_T=54.6. And 54.6 can balance a lot of zero-realisations. Bottom line: The most freqent occurrence is not always the expected value.PS: Of course with such high volatilities you should simulate using the exact solution over one time-step: V_(t+dt)=V_t*exp{(-0.5*sigma^2)*dt+sigma*(W_(t+dt)-W_t)},

Accepted Though it still annoyes me, that, in 6 out of 7 cases (on average), the asset value will sink.

Hi, I am doing a dissertation on Capital Structure Arbitrage with CDS, is there any jounal or research(for literature review) on this subject? Thank you.

I don't really see the best way to compute the delta in the credit grades model . Could someone give me any information about it .

Why dont u just shock the stock price 1% with a constant asset vol...note the effect on the Spread...convert that spread effect into CDS equivalent % points...then the ratio of CDS% movement to 1% = delta

1)If i shock the stock price 1% , it will have an effect on the asset vol (Asset/Stock relation in structural model) , so i 'm not sure of what you mean by "with constant asset vol" .2)If i do what you say : Shock on stock and then take the ratio (%change on spread/% shock stock) ; what is the meaning of this delta in term of hedging : if i have a position of (delta shares) it's not realy delta-neutral on my CDS Market Value ?Thanks

1) If u shock the stock 1% and keep the asset vol constant at say, 25%...then yes it will have an effect on the implied equity vol (although small), but thats what you would expect as you can use a structural model to define skew as a fn of leverageSure, if you want to assume some asset vol skew as well, then incorporate the avol change for stock price change and judge the delta2)Think u misunderstood me...I never said delta = ratio (%change on spread/% shock stock), but meant (%CDS price (or mkt value)/% shock stock)

In fact I guess you are both saying quite the same thing as Credit delta=Credit duration * (%change spread)/(% change stock)=(%change CDS)/(% change stock).The good thing (I think) with that is if you already got a pricing tool for CDS that does not link spread to equity you should have your Merton-like credit model running on its own and nevetheless calculating credit delta.Am I right?