Questions tagged [merton-model]
The merton-model tag has no usage guidance.
50
questions
0
votes
0
answers
33
views
Understanding the application of Asset-Correlation to credit risk models
Suppose we have a portfolio of $n$ credits. In order the estimate the Portfolio Value at Risk (99,9) we use a standard vasicek model with the Ability to pay variable $A_i=\sqrt{\rho}x+\sqrt{1-\rho}z_i$...
- 1
0
votes
1
answer
66
views
Can someone provide an example of how arbitrage would be used when an american call option can be bought for less than max(final stock - strike,0)? [closed]
"Final stock" means the stock price at expiration, and "strike" means strike price. If a call option had to be purchased for more than the max(final stock - strike,0)then you would ...
0
votes
1
answer
94
views
Solving Equation for estimation risk averse parameter
Let the portfolio value follow the SDE:
$$V_t=(\mu w+r(1-w))\cdot V_t\cdot dt +\sigma \cdot w\cdot V_t \cdot dB_t $$
where $\mu$ = drift of the portfolio,
$\sigma$=standard deviation of the portfolio, ...
- 37
0
votes
0
answers
254
views
Maximum Likelihood Estimation for Merton's Jump Diffusion Model
I'm wondering if there is a consensus about the
a) most accurate and
b) most computationally efficient
way to estimate parameters for Merton's (1976) jump diffusion model. In this model, the ...
- 121
1
vote
1
answer
124
views
Does Gordy Formula measure default risk & downgrade risk?
The Gordy Formula used for measuring Credit Risk as proposed in Basel Rules is based on the asymptotic single risk factor model. It is derived from a Merton Model. The Merton Model only knows to stati,...
- 289
5
votes
1
answer
443
views
Variance of the log returns in jump diffusion with time-varying jump sizes
I'm trying to calculate the variance $\mathrm{var}\left(\log\frac{S\left(t\right)}{S\left(0\right)}\right)$, where the dynamics of the stock $S$ follows a jump-diffusion process given by $$\frac{dS\...
- 103
1
vote
0
answers
93
views
Simulating the same stock price with different methods/distributions
I would like to ask if we could simulate stock price paths with different methods/techniques.
What I mean is : say we have a specific stock price hence we can extract historical mean and standard ...
- 129
1
vote
1
answer
631
views
Relationship between risk free rate and credit spread in the Merton model
Based on Merton model of credit risk, I understand that investing in a risky debt is the same as buying a treasury bond and writing a put option on the firm's assets with a strike price equal to the ...
- 11
2
votes
1
answer
140
views
Feynman-Kac representation of Black-Cox model
Consider the standard setup from Black and Cox (1976, Journal of Finance).
A firm issues a defaultable coupon bond to finance a productive asset that follows a geometric brownian motion:
$$dx_t = \mu ...
- 327
4
votes
1
answer
106
views
Cauchy-Euler ODE with indicator function in coefficient
Consider the following Cauchy-Euler ODE, which is in particular the asset pricing equation for a (perpetual coupon defaultable) bond:
$$\frac12 \sigma^2 V^2 F_{vv}(V,t) + \mu V F_{v}(V,t) - r F(V,t) + ...
- 327
1
vote
0
answers
75
views
Derivation defaultable bond price in Leland 1994 (Merton)
Consider the model in Leland (Journal of Finance, 1994).
The partial differential equation that describes the price of the (perpetual coupon defaultable) bond is:
$$\frac12 \sigma^2 V^2 F_{vv}(V,t) + \...
- 327
2
votes
0
answers
117
views
The distribution of the jump diffusion process
In the Merton jump diffusion model the process of the share price can be expressed as $$S_{t}=S_{0}\cdot\exp\left\{ X_{t}\right\} ,$$ where $$X_{t}=\mu t+\sigma W_{t}+\sum_{i=1}^{N_{t}}Y_{i}.$$
Here $...
- 195
1
vote
1
answer
341
views
Estimation of Default Probability using Merton's model
There is an explanation of Risk Neutral Default Probability using a Firm's Equity price here - https://www.mathworks.com/help/risk/default-probability-using-the-...
- 373
0
votes
0
answers
98
views
Can Merton's continuous-time portfolio model be reformulated without a utility function?
Under the standard Merton optimization problem the agent maximizes expected utility
$$J(\pi,c) =\mathbb{E}\Big[\int_0^TU(c_tX_t) dt + U(X_T)\Big],$$
where the dynamics of wealth of the agent satisfy
$...
- 2,980
2
votes
0
answers
79
views
How can you interpret one of the parameters of optimal consumption at the Merton portfolio problem?
Statement: Let the dynamics of wealth of the agent satisfy
$$dX_{t} =
\pi_tX_t\Big(\mu dt+\sigma dB_{t}\Big)- c_t X_t dt, \qquad \textrm{with}\quad X_0=x_0 \in \mathbb{R},$$
where $(\pi,c)$ is an ...
- 21
4
votes
1
answer
2k
views
Use of PIT vs TTC PD in a Merton one-factor model
Under one-factor Merton framework, like Basel, you use unconditional PDs as input of the portfolio model and this "unconditional" means it is a TTC-PD.
Given a i-th borrower, the default ...
- 77
3
votes
0
answers
356
views
Black-Cox yield spreads
From Lando (2004)* I am trying to replicate the following figure (Section 2.6 Default Barriers: The Black-Cox Setup):
The spreads are computed as follows:
$$s(T) = \frac{1}{T}\ln\frac{D}{B_0}-r$$
...
- 116
1
vote
1
answer
200
views
Merton model d1 and probability of default
What is the value of $d_1$ when the probability of default is 50%
I know that:
$$
\begin{aligned}
d_2 &= 0 \\
\mathcal{N}(d_2) &= 50\%\\
1- \mathcal{N}(d_2) &= \mathcal{N}(-d_2) = 50\%
\...
1
vote
1
answer
191
views
Yearly ytm calculation on stock using binomial model
So I have been given this problem in class, and although I have no issues doing the binomial model on options, I cannot seem to get my head around the problem when its calculating ytm on just a stock.
...
- 11
2
votes
1
answer
222
views
What are the best relative value frameworks for Corporate Credit?
Fixed Income (Credit) fair value models in the literature tend to be variations on cross-sectional regressions. For a recent example in a factor-model setting, see here.
My understanding is that this ...
- 73
2
votes
1
answer
2k
views
rationale for maturity adjustment formula in basel IRB formula
For capital requirement, rwa is computed as a product of terms including a K (unexpected losses).
(As shown is the summary from wikipedia :
https://en.m.wikipedia.org/wiki/Advanced_IRB
)
K is ...
- 365
6
votes
0
answers
83
views
Formal proof market incompleteness under jump diffusion
Does anyone have formal proof of markets incompleteness under jump diffusion ?
I am familiar with the intuitive approach as mentioned in Tankov (delta), yet I am looking for a formal approach and ...
- 61
8
votes
1
answer
4k
views
Understanding and simulating the jumps in Merton's Jump-Diffusion SDE?
I found this great post deriving the solution to the Merton Jump-Diffusion SDE
$$S_t = S_0\exp\left(\left(\mu - \frac{\sigma^2}{2}\right)t + \sigma W_t\right)\prod_{j=0}^{N_t}V_j$$
The first part of ...
- 405
2
votes
1
answer
380
views
Merton's Jump diffusion model: Specify poisson rate
Currently applying the Merton's jump diffusion to test how Option price change as parameters change. However, I am struggling to specify the poisson rate $\lambda$. We know that:
$P(\text{There is a ...
- 1,396
1
vote
0
answers
119
views
Optimal allocation problem by finite differences
I am attempting to apply implicit finite difference to solve Merton's problem of optimal portfolio allocation for constant parameters.
The equation to solve is the Hamilton-Jacobi-Bellman equation:
$$...
- 113
2
votes
1
answer
614
views
Relationship between asset volatility and debt and equity value
So how I understand it, higher asset volatility implies a higher call option price. The Merton Model holds that the value of equity is a call option. This therefore implies that the equity value must ...
- 21
2
votes
0
answers
126
views
Hedging jump models with a infinite number of derivatives
First of all, I inform you that I am not a financial mathematician and have vague knowledge about an incomplete market.
Stochastic volatility models are incomplete so derivatives cannot be ...
- 305
1
vote
1
answer
321
views
Question about calculating asset volatility using Black-Scholes and the Merton Model (Differentiation Question)
I have a problem where I need to relever equity volatility and take into account debt. I'm trying to solve a system of nonlinear equations for $\sigma_v,V$ using $f(\sigma_v,V) = VN(d_1)-De^{-R_ft}N(...
- 75
0
votes
1
answer
282
views
Finding Equity Volatility for the Standard Merton Model of Corporate Debt
I am working on a project studying historical accuracy of the standard Merton Model, but am struggling to follow the required inputs. I seem to read conflicting definitions of the information I need.
...
- 1
2
votes
1
answer
372
views
Price of European calls in Merton's Model
The stock price is modeled by $$S_t = S_0 e^{bt +\sigma B_t + \sum_{k=1}^{N_t} Y_k}$$
with $B_t$ Brownian motion, $Y_k$ iid $N(\mu,\delta^2)$, $N_t$ a Poisson process independent of $(B_t)$ and $Y_k$ ...
- 183
1
vote
1
answer
147
views
Model the share price under the Merton Credit model
The project I'm working on requires me to model the share price of a firm through time using the Merton and Black-Cox credit models. The model is used here to induce the leverage effect in the share ...
- 51
0
votes
2
answers
1k
views
Merton model for Probability of Default - What liabilities?
In Merton structural model for credit risk (74), the company's Assets and Liabilities are used to imply the default probability of the firm.
At the end, we don't need to know the assets value, and ...
0
votes
1
answer
523
views
Default Probability calculation. How to solve system of 2 non linear equations?
I am trying to repeat calculations from Hull(options futures and other derivatives) chapter "Using Equity Prices to Estimate Default Probabilities". I want to solve system of 2 equations:
\begin{...
- 11
1
vote
1
answer
456
views
How do we solve bellman's equation in Merton's model
Studying the expected utility maximization problem in Merton's model, I'm having some difficulties.
Let $t$ be a starting time, $T$ the final finite Time.
We define,
\begin{equation}
V(t,x)=\underset{\...
- 287
2
votes
1
answer
1k
views
Formula for Merton jump diffusion call price
What is the formula for a call price in Merton's jump diffusion model?
I am asking because I was taught: $$BS \left[ S = S_0e^{ n(m + \frac{v}{2}) - C \cdot T} , vol = \sqrt{\sigma^2 + nv/T} \ \right]...
- 125
3
votes
1
answer
1k
views
Hamilton-Jacobi-Bellman equation in Merton Model
I'm trying to study the Merton Model for portfolio optimization and the document doesn't explain a quite important step :
if $$V(t,x)=\sup\{E[U(X_T(\phi))~|~X_t=x]~~ |~~\phi~~\text{an admissible ...
- 287
1
vote
2
answers
552
views
Pricing Exotics: Monte-Carlo is too slow?
I want to price exotic options under the exponential VG model and Merton's model to compare both models.
To price exotics under Merton's model, I have written the code below. The output is the price ...
- 431
2
votes
1
answer
203
views
Merton portfolio allocation problem proportions/weights >1 or <0?
In the classical Merton portfolio problem, lets assume:
$$ dX_t \, = \, \frac{\pi_t X_t}{S_t} S_t(\mu dt +\sigma dW_t)
= \pi_t X_t (\mu dt +\sigma dW_t) $$
ie: zero interest rates for simplicity.
...
- 123
3
votes
3
answers
468
views
Merton model riskless self-financing derivation
Suppose $dA_t = A_t[\mu dt+\sigma dW_t]$ (assets' value) under the physical measure, plus the other assumptions of the Merton model.
Suppose further that debt and equity are tradeable assets that ...
- 103
5
votes
0
answers
2k
views
Calibration of Merton's jump diffusion model
Setting
In my financial engineering project I'm working on a new calibration formalism for jump-diffusion models and in particular Merton's jump diffusion model. A jump diffusion process $\{X(t), t \...
- 160
3
votes
1
answer
496
views
Numerical Methods for Merton Model
The stochastic differential equation for an underlying with jumps in Merton model is:
$$d{{S}_{t}}=\mu \,{{S}_{t}}dt+\sigma \,{{S}_{t}}\,d{{W}_{t}}^{P}+(J-1){{S}_{t}}d{{q}_{t}}$$
where
$t \quad\,\,\, ...
- 323
2
votes
1
answer
5k
views
How to find volatility of Asset given volatility of Stock in Merton model?
I encounter a problem in one of my project to find the 1 year, 2 year and 3 year Asset volatility. We are given 2015 Bell Canada's financial report and a software to do this. The financial report can ...
- 23
5
votes
1
answer
570
views
Simple question on jump-diffusion
In the textbook by Shreve in sec. 11.7.2 a jump-diffusion process is introduced. More precisely
$$
dS_t = \alpha\,S_t\,dt+\sigma\,S_t\,dW_t+S_{t-}\,d\left(Q_t-\beta\,\lambda\,t\right)\quad (1)
$$
...
- 465
16
votes
2
answers
5k
views
Solution of Merton's Jump-Diffusion SDE
In many textbooks and also in the original Merton's paper the solution of the SDE
$$
dS_t = S_t\,\mu\,dt+S_t\,\sigma\,dW_t+S_{t^-}\,d\left(\sum_{j=1}^{N_t}V_j-1\right)
$$
is written as
$$
S_t = ...
- 465
2
votes
0
answers
157
views
Future value of the debt under Merton model
The author Malz states the future value of the firm's debt under the Merton model can be found from:
$$
D_{t} = D - \max(D - A_{t} , 0)
$$
(where $D$ is the par value of the debt, $A_{t}$ is the ...
- 1,549
3
votes
4
answers
9k
views
Why is the value of debt modeled as a short put option in Merton's model?
Can someone give me an intuitive understanding of why the Merton model models the value of the debt from the lender's point of view as a short put with a risk free bond?
I'm not well versed in this ...
- 1,549
2
votes
1
answer
229
views
Question about the stochastic differential equation in the Merton model
in the following stochastic differential equation merton model we have $$\frac{ds}{s}=(\alpha-\lambda k)dt+\sigma dW+dq$$
where $\alpha$ is the instantaneous expected return on the stock; $\sigma^2$...
- 123
4
votes
2
answers
481
views
How to interpret negative asset volatility numerical results in Merton model?
I am currently working on my thesis where I discuss the Merton default probability model. I have a huge sample of US firms for the period 1990-2010. I use both numerical and complex iterative approach ...
- 63
1
vote
0
answers
489
views
How to calculate modeled asset volatility by industry factor?
Currently I am working with huge data frame which consists of a lot firms. For each firm in my sample I calculated asset volatility ( I am using Merton default probability model, so I have used 2 ...
- 11
8
votes
2
answers
7k
views
How to use Merton model to calculate default probability with monthly stock prices?
I want to calculate the estimated default probability with only given data the monthly returns for the last 20 years, the risk-free rate ($R_f$), equity value (EV) and the face value of debt ($D$). My ...
- 81