15
votes
Accepted
Solution of Merton's Jump-Diffusion SDE
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 ...
- 14.5k
5
votes
Accepted
Question about calculating asset volatility using Black-Scholes and the Merton Model (Differentiation Question)
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/...
- 66
4
votes
Accepted
Variance of the log returns in jump diffusion with time-varying jump sizes
Note: Time-dependent parameters can be introduced quite easily into affine jump diffusion models. Even if the corresponding (time) integrals cannot be solved in closed form, option pricing and moment ...
- 6,260
4
votes
Accepted
Estimation of Default Probability using Merton's model
As you see in the third equation on that Mathworks page, the Merton model postulates that the value of equity equals the value on a residual claim on a company's assets after the creditor has been ...
- 6,260
4
votes
Accepted
Why is the value of debt modeled as a short put option in Merton's model?
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 ...
- 9,272
4
votes
Merton model riskless self-financing derivation
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^...
- 21k
4
votes
Accepted
Hamilton-Jacobi-Bellman equation in Merton Model
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 $[t,t+dt[$, ...
- 14.5k
3
votes
Accepted
rationale for maturity adjustment formula in basel IRB formula
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". ...
- 1,307
3
votes
How do we solve bellman's equation in Merton's model
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,...
- 14.5k
3
votes
Accepted
Formula for Merton jump diffusion call price
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 ...
- 5,949
3
votes
Pricing Exotics: Monte-Carlo is too slow?
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 ...
- 3,232
3
votes
Solution of Merton's Jump-Diffusion SDE
Use Ito for jumps
$$ dS_t = \frac{\partial S_t}{\partial t} dt + \frac{\partial S_t}{\partial W_t}dW_t + \frac{1}{2}\frac{\partial^2 S_t}{\partial W_t^2} dt + \frac{\partial S_t}{\partial N_t}d N_t $$...
- 624
2
votes
Accepted
Simple question on jump-diffusion
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 ...
- 14.5k
2
votes
Accepted
Price of European calls in Merton's Model
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 (...
- 430
2
votes
Understanding and simulating the jumps in Merton's Jump-Diffusion SDE?
For anyone else searching for good Merton Jump Diffusion examples, found a much better notated reference here:
https://www.codearmo.com/python-tutorial/merton-jump-diffusion-model-python
2
votes
Accepted
Merton's Jump diffusion model: Specify poisson rate
$\lambda$ 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,...
- 2,177
2
votes
Accepted
Use of PIT vs TTC PD in a Merton one-factor model
The first equation is already a PIT PD if $\displaystyle PD_{i}$
is substituted by TTC PD. The challenges of using this model are:
(1) $\displaystyle \rho _{i}$, the asset correlation, is very ...
- 256
2
votes
Merton model d1 and probability of default
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).
- 2,540
2
votes
Accepted
Merton portfolio allocation problem proportions/weights >1 or <0?
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 ...
- 1,456
1
vote
Finding Equity Volatility for the Standard Merton Model of Corporate Debt
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 ...
- 8,436
1
vote
Relationship between risk free rate and credit spread in the Merton model
By formula (14) in Merton (1974) the difference of the yield to maturity $R(\tau)$ of the firm's risky debt and the riskless rate $r$ is
$$\tag{14}
R(\tau)-r=\frac{-1}{\tau}\log\Big\{\Phi[h_2(d,\sigma^...
- 1,958
1
vote
Accepted
Feynman-Kac representation of Black-Cox model
I'm not sure if this answers your question, but what you call the 'pde solution' does come directly from your probabilistic setup.
With $t=0$, we have:
$$ E \left[ e^{- rT}p 1_{x_T\geq p, \tau_b\geq ...
- 4,968
1
vote
Accepted
Cauchy-Euler ODE with indicator function in coefficient
I solved it for the case $\mu = r_1$, the solution in $\mathbb{C}^1$ takes the guessed form $$F(V) = \begin{cases} A_0 + A_1 V + A_2 V^{-x} \; \text{ if } \; V>k \\ B_0 + B_1 V + B_2 V^{-y} \; \...
- 327
1
vote
Yearly ytm calculation on stock using binomial model
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 ...
- 6,174
1
vote
What are the best relative value frameworks for Corporate Credit?
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 ...
- 988
1
vote
Relationship between asset volatility and debt and equity value
You're right about the equity value increasing with higher volatility. You're wrong about the debt value. That decreases with higher volatility, because it is short an option. And yes, equity ...
- 16.2k
1
vote
Model the share price under the Merton Credit model
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 ...
- 31
1
vote
Merton model for Probability of Default - What liabilities?
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 ...
- 2,985
1
vote
Pricing Exotics: Monte-Carlo is too slow?
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?...
- 111
1
vote
Accepted
Numerical Methods for Merton Model
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 ...
- 116
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