Questions tagged [merton-model]

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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 ...
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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 ...
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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) + ...
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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) + \...
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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 $...
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1 answer
219 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-...
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87 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 $...
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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 ...
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4 votes
1 answer
748 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 ...
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189 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: $$s(T) = \frac{1}{T}\ln\frac{D}{B_0}-r$$ ...
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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\% \...
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1 answer
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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. ...
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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 ...
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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 ...
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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 ...
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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 ...
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1 answer
278 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 ...
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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: $$...
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1 answer
514 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 ...
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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 ...
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1 vote
1 answer
271 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(...
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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. ...
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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 $Y_k$ ...
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1 vote
1 answer
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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 ...
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2 answers
875 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 ...
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0 votes
1 answer
448 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{...
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1 answer
397 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{\...
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3 votes
1 answer
869 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]...
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954 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 ...
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1 vote
2 answers
514 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 ...
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2 votes
1 answer
170 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. ...
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3 votes
3 answers
409 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 ...
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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 \...
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3 votes
1 answer
432 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\,\,\, ...
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2 votes
1 answer
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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 ...
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5 votes
1 answer
507 views

Simple question on jump-diffusion

In the textbook by Shreve in sec. 11.7.2 a jump-diffusion process is introduced. More precisely $$ dS_t = \alpha\,S_t\,dt+\sigma\,S_t\,dW_t+S_{t-}\,d\left(Q_t-\beta\,\lambda\,t\right)\quad (1) $$ ...
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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 = ...
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2 votes
0 answers
147 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 ...
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3 votes
4 answers
8k 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 ...
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2 votes
1 answer
212 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; $\sigma^2$...
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4 votes
2 answers
456 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 ...
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1 vote
0 answers
474 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 ...
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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 ...
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