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13

Let $$dS_t = \mu S_t dt + \sigma S_t dW_t + S_{t^-} dJ_t$$ where $$J_t = \sum_{j=1}^{N_t} (V_j - 1)$$ is a compound Poisson process, with $V_j$ i.i.d. jump sizes (positive random variables) whose statistical properties are not relevant for what needs to be proven and $N_t$ a standard Poisson process of intensity $\lambda$. The processes $W_t$, $N_t$ and ...

5

For one thing, what you have written is incorrect. Black-Scholes uses the standard normal CDF: $$N(d_1) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{d_1} e^{-x^2/2} \, dx$$ So $N'(d_1) = \frac{e^{-d_1^2/2} }{\sqrt{2 \pi}}$ and using the chain rule we get $$\frac{\partial N(d_1)}{\partial V} = N'(d_1) \frac{\partial d_1}{\partial V} = \frac{e^{-d_1^2/2} }{\... 5 if Y=1 the stock price doesn't change since it's a percentage change not an absolute, so we have to subtract one when drift compensating. See my book Concepts etc for my discussion. 4 If the company was risk free the lender would always get back the promised amount L at maturity. So the lender would be holding a risk free bond. But companies are not risk free, there is a chance that they won't be able to repay the full amount L. This can be modeled as a risk free bond plus a "thing" which will have negative value if the company ... 3 The solution that you provided in your question is conditional on the number of jumps being equal to some fixed n. To get the option price, you need to take the probability weighted sum over all values of n \in \mathbb{N}. Starting from the standard risk-neutral pricing formula, you use the tower law to condition on the total number of jumps until ... 3 This is an optimal control problem. Consider a self-financing strategy \pi := (\pi_s)_{s\in[t,T]} over the horizon [t,T] consisting in, over each infinitesimal period of time [t,t+dt[, investing a fraction \pi_t of the current wealth in a risky asset S_t and placing the remaining part in the risk free asset B_t. Given the following dynamics$$ ...

3

You're using a wrong tool for the job. Write your Monte Carlo in a faster language (Java would probably suffice, if not than C++ which is standard for such things). Then you will be able to efficiently generate more than 1000 paths. In fact, doing Monte Carlo derivatives pricing with 1000 paths is worthless. Your results are, most probably, very inaccurate. ...

3

We construct a locally risk-free self-financing portfolio $X_t$, at time $t$, with $\Delta_t^1$ share of debt and $\Delta_t^2$ share of equity. That is, \begin{align*} X_t = \Delta_t^1 D_t + \Delta_t^2 E_t. \end{align*} Then, \begin{align*} dX_t &=\Delta_t^1 dD_t + \Delta_t^2 dE_t\\ &=\Delta_t^1\bigg[\Big(\frac{\partial D_t}{\partial t} + \mu A_t\...

2

detailed description of the solution of this problem using Excel is in the second chapter of the book Credit Risk Modeling using Excel and VBA Gunter Löffler

2

I'll decompose your big question into smaller questions and answer them in (hopefully) simple terms. 1. What is meant by the risk neutral measure? This is how I understand the risk-neutral measure (commonly denoted by $\mathbb{Q}$): It is the probability measure under which the current value of all financial assets at a time, say $t$, are equal to the ...

2

$\lambda$ is the intensity of the number of jumps per unit of time. If you call $N_t$ the number of jumps up to time $t$ then $E[dN_t]=\lambda dt$ is the expected number of jumps in the interval $(t,t+dt)$ For more details you can check the wiki page https://en.m.wikipedia.org/wiki/Poisson_point_process

2

Your statement should be correct, the weights into the risky asset are not bounded between $0$ and $1$. Essentially, by setting $r=0$ you omit the term which shows that your weights always sum up to one, simply by choosing the weight for the risk-free asset to be $1-\pi^*$. In other words, obtaining $\pi^*>1$ simply implies you go short in the risk-free ...

2

Use Ito for jumps $$dS_t = \frac{\partial S_t}{\partial t} dt + \frac{\partial S_t}{\partial W_t}dW_t + \frac{1}{2}\frac{\partial^2 S_t}{\partial W_t^2} dt + \frac{\partial S_t}{\partial N_t}d N_t$$ The first part is pretty straight forward $$\frac{\partial S_t}{\partial t} dt = S_t(\mu - \frac{1}{2}\sigma^2)$$ \frac{\partial S_t}{\partial W_t}dW_t ... 2 Since d_1 = d_2 + \sigma\sqrt{\tau}, you need to know the volatility of your asset value process. You typically estimate it from equity prices (see e.g. Hull's book). 1 At the terminal date, the value will be: 1.44, 0.84, 0.84, 0.49 in the four states: UU, UD, DU, and DD, respectively. The probability of an up move is: (1.1-0.7)/(1.2-0.7)=0.8 So the probability of the four terminal states are: 0.64, 0.16, 0.16. 0.04. Easy to verify that the value is 1 at time zero: \frac{1}{1.1^2}\sum_{s=1}^{4}{V_{s,2}Q_{s,2}} At 15.... 1 Not a complete answer, but some thoughts below - First you need to bifurcate the names into two categories - (1) Traded Credit, (2) Illiquid credit. For Traded credit underliers, fairly reliable market quotes are available for CDS and bonds. These can be used to back out a credit curve, and then you could go with the approach 2 ("Structural Model based on ... 1 The maturity adjustment is there to take into account the risk of changing default probabilities in future years. Parameters are according to Basel calibrated from "observed... capital market data". It is covered in some detail in section 4.6 devoted to the subject in Basel's explainer on IRB riskweights. 1 If the company is publicly traded you can use the current market capitalization and the implied volatility of that stock (a choice must be made here, it seems sensible to use the ATM implied volatility). For non-publicly traded companies it doesn't seem possible to use the Merton model as these figures can't be obtained. 1 Your latter statement is correct. Under the Merton model, Firm Value (FV) = Value of Equity + Value of Debt. The percentage changes in FV are then assumed to be GBM. So, the value of equity will be the Black-Scholes call price. And the value of debt will be the face value of a zero coupon bond minus the Black-Scholes put price. Black-Cox is an extension of ... 1 The Kealhoffer-Merton-Vasicek (KMV) model is derivative of Merton. Essentially, it codifies the calibration process and extends the framework to empirical distributions. The following entry, Modeling Default Risk, contains one such passage regarding KMV’s parameterization of liabilities: Oldrich Vasicek and Stephen Kealhofer have extended the Black-... 1 I'm definitely not an expert on this topic, but it seems to me that: \hat{\pi}_t is here defined as \begin{align} \hat{\pi}_t &= \underset{\pi_t \in \Bbb{R}}{\text{argsup}} \left( \frac{dV(t,x)}{dx} x (r+ \pi_t(\mu-r)) + \frac{1}{2} \frac{d^2V(t,x)}{dx^2} x^2 \pi_t^2 \sigma^2\right) \\ &= \underset{\pi_t \in \Bbb{R}}{\text{argsup}}\, I(\pi_t) \... 1 I assume you are using MatLab. You may consider pre-generating all 1,000 random numbers once before for-loop by exploiting array coding. Another approach, have you ever tried using Quasi Monte Carlo? Generating Quasi-Random Numbers QMC ensures faster convergence and MatLab has functions that can generate quasi-random sequence very fast (a billion under a ... 1 You should take a look at the BENCHOP project. There we benchmarked around 15 different numerical methods against 6 option pricing problems. One of the problems was the Merton model. The methods were split into 4 families: Monte Carlo, Fourier, Finite Difference, and Radial Basis Function methods. This is the paper containing the results: http://dx.doi.... 1 The delta of an option is the amount the option value will change according to the change in the underlying. The Book value of a company is typically it's assets minus liabilities. This can differ from market value (which is the share price * number of shares outstanding). The picture you provided looks like an option calculator, with inputs: Stock price, ... 1 Could it be that your problem is only due to the t^- notation convention? Think of it that way, it is only worth distinguishing S_{t^-} from S_t at a jump time. Elsewhere, knowing that Brownian motion paths are continuous, you'll always have S_t = S_{t^-}. Thus you could also write the SDE:\frac {dS_t}{S_{t^-}} = \alpha dt+\sigma dW_t+ d\...

1

Although I, admittedly, did not go hunting through your code for an error, I have seen this phenomenon before using this model. This model (like all other models) isn't perfect. This is especially true when you can only observe those parameters that come from the balance sheet quarterly. There are scenarios where no asset vol can imply the current market ...

1

your steps are a bit too complicated to me. in Step 1 and 2 you do two things: you caculate monthly log-returns and then their standard decviation. Given prices $P_t$ indexed by time the log return is given by $$r_t = \ln(P_t/P_{t-1}) = \ln(P_t) - \ln(P_{t-1}).$$ The formula for standard-deviation (the sample estimator of it) should be clear:  \sigma =...

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