merton-model 0.0.8
Merton model distance to default



## Introduction
The Merton model was first developed by economist Robert Merton in 1974. The model makes a claim about default probabilities based on the capital structure of the firm. It models the value of equity as a call option on the total assets of the underlying firm. If assets are below the call option’s strike price at time $T$, the firm defaults, the value of equity goes to zero, and the remaining value of the firm is divided equally among debt holders. 

The model has some limitations. The version of the model we considered makes the Black-Scholes assumptions, which are generally accepted to be false. Furthermore, it does not address the possibility of a bankruptcy before time $T$.

We used several resources. Of particular utility was a Society of Actuaries article entitled "Structural Credit Risk Modeling: Merton and Beyond" by Wang, "Options, Futures, and Other Derivatives" 9th ed. by Hull, and Journal of Banking & Finance article "Credit spreads with dynamic debt" by Das and Kim.

## Functions

`get_d(V, T, K, r, sigma_V)`

* V: Value of the firm.
* T: Time until maturity.
* K: Strike price; related to the firm's debt level.
* r: Risk-free rate.
* sigma_V: Firm volatility.

`distance(V, T, K, r, sigma_V)` 

The Merton model distance to default. Corresponds to $d_2$ in the Black-Scholes framework.

* V: Value of the firm.
* T: Time until maturity.
* K: Strike price; related to the firm's debt level.
* r: Risk-free rate.
* sigma_V: Firm volatility.

`prob_default(V, T, K, r, sigma_V)`

The probability of default using the Merton model. Corresponds to $N(-d_2)$ in the Black-Scholes framework.

* V: Value of the firm.
* T: Time until maturity.
* K: Strike price; related to the firm's debt level.
* r: Risk-free rate.
* sigma_V: Firm volatility.

`call(V, T, K, r, sigma_V)`

Value of a call option under the Black-Scholes framework. Corresponds to the value of equity under the Merton model. Formula on page 604 of "Options, Futures, and Other Derivatives" 9th ed. by Hull.

* V: Value of the firm.
* T: Time until maturity.
* K: Strike price; related to the firm's debt level.
* r: Risk-free rate.
* sigma_V: Firm volatility.

`put(V, T, K, r, sigma_V, phi = 1)`

Value of a put option under the Black-Scholes framework. Corresponds to value of insurance required to make debt risk free. Formula on page 604 of "Options, Futures, and Other Derivatives" 9th ed. by Hull.
    
* V: Value of the firm.
* T: Time until maturity.
* K: Strike price; related to the firm's debt level.
* r: Risk-free rate.
* sigma_V: Firm volatility.
* phi: Fraction of firm's value retained in the case of default.

`put_ui(V, T, K, H, r, sigma_V, phi = 1)`

Value of an up-and-in put option under the Black-Scholes framework. Formula on page 605 of "Options, Futures, and Other Derivatives" 9th ed. by Hull.
    
* V: Value of the firm.
* T: Time until maturity.
* K: Strike price; related to the firm's debt level.
* r: Risk-free rate.
* sigma_V: Firm volatility.
* phi: Fraction of firm's value retained in the case of default.

`put_uo(V, T, K, H, r, sigma_V, phi = 1)`

Value of an up-and-out put option under the Black-Scholes framework. Formula on page 605 of "Options, Futures, and Other Derivatives" 9th ed. by Hull.
    
* V: Value of the firm.
* T: Time until maturity.
* K: Strike price; related to the firm's debt level.
* H: Value of the firm which deactivates the up-and-out option.
* r: Risk-free rate.
* sigma_V: Firm volatility. 
* phi: Fraction of firm's value retained in the case of default.

`put_di(V, T, K, H, r, sigma_V, phi = 1)`

Value of a down-and-in put option under the Black-Scholes framework. Formula on page 606 of "Options, Futures, and Other Derivatives" 9th ed. by Hull.
    
* V: Value of the firm.
* T: Time until maturity.
* K: Strike price; related to the firm's debt level.
* H: Value of the firm which deactivates the up-and-out option.
* r: Risk-free rate.
* sigma_V: Firm volatility. 
* phi: Fraction of firm's value retained in the case of default.

`put_do(V, T, K, H, r, sigma_V, phi = 1)`

Value of a down-and-out put option under the Black-Scholes framework. Formula on page 606 of "Options, Futures, and Other Derivatives" 9th ed. by Hull.
    
* V: Value of the firm.
* T: Time until maturity.
* K: Strike price; related to the firm's debt level.
* H: Value of the firm which deactivates the up-and-out option.
* r: Risk-free rate.
* sigma_V: Firm volatility. 
* phi: Fraction of firm's value retained in the case of default.

`prob_default_mc(V, T, K, r, sigma_V, trails = 10000, steps = None)`

Probability of default if possible for some values of t prior to maturity. Default occurs if the value of the firm is below K at time Δt, 2Δt, ..., or T = steps · Δt.
If T is measured in years, steps = floor(12 * T + 1) corresponds to possibility of default at each monthly payment. Probability obtained using Monte Carlo simulation.
    
* V: Value of the firm.
* T: Time until maturity.
* K: Strike price; related to the firm's debt level.
* r: Risk-free rate.
* sigma_V: Firm volatility. 
* trails: Monte Carlo simulations used to obtain approximation; antithetic values also considered.
* steps: Number of times the firm can default within each simulation.

`spread(V, T, K, r, sigma_V, phi = 1)`

CDS spread given default only possible at T. Uses the Black-Scholes framework. Value tends to be below market CDS spreads.
    
* V: Value of the firm.
* T: Time until maturity.
* K: Strike price; related to the firm's debt level.
* r: Risk-free rate.
* sigma_V: Firm volatility. 
* phi: Fraction of firm's value retained in the case of default.

`spread_mc(V, T, K, r, sigma_V, phi = 1, trails = 10000, steps = None)`

CDS spread if default possible for some values of t prior to maturity. Default occurs if the value of the firm is below K at time Δt, 2Δt, ..., or T = steps · Δt. If T is measured in years, steps floor(12 * T + 1) corresponds to monthly payments. In the case of default, partial payment assumed to be made at time T. Probability obtained using Monte Carlo simulation.
    
* V: Value of the firm.
* T: Time until maturity.
* K: Strike price; related to the firm's debt level.
* r: Risk-free rate.
* sigma_V: Firm volatility. 
* phi: Fraction of firm's value retained in the case of default.
* trails: Monte Carlo simulations used to obtain approximation; antithetic values also considered.
* steps: Number of times the firm can default within each simulation.

`spread_das(V, T, K_1, K_2, H, r, sigma_V, phi = 1)`

CDS spread if default possible only at time T, but firm increases/decreases debt if its value goes above/below barrior. Inspired by the Journal of Banking & Finance article "Credit spreads with dynamic debt" by Das and Kim.

* V: Value of the firm.
* T: Time until maturity.
* K_1: Strike price if firm does not increase/decrease its debt level.
* K_2: Strike price if firm increase/decrease its debt level.
* H: Value of the firm which triggers it to increase/decrease its debt level from K_1 to K_2.
* r: Risk-free rate.
* sigma_V: Firm volatility. 
* phi: Fraction of firm's value retained in the case of default.

`obj(sigma_V, sigma_E, V, T, K, r)`

Objective function to be minimized. Used to obtain the volatility of the firm's value.
    
* sigma_V: Firm volatility.
* sigma_E: Equity volatility.
* V: Value of the firm.
* T: Time until maturity.
* K: Strike price; related to the firm's debt level.
* r: Risk-free rate.

`get_sigma_V(sigma_E, V, T, K, r)`

Obtain the volatility of the firm's value. Uses the Black-Scholes framework and fsolve.
    
* sigma_E: Equity volatility.
* V: Value of the firm.
* T: Time until maturity.
* K: Strike price; related to the firm's debt level.
* r: Risk-free rate.

`get_K(CL, NCL, T = 1, r = 0)`

If the value of the firm drops below K at maturity, then the firm is considered to be in default under the Merton model. Formula overweights current liabilities because they must be paid within the year or one business cycle.
    
* CL: Current liabilities.
* NCL: Noncurrent liabilities
* T: Time until maturity.
* r: Risk-free rate.

## References
Wang, Yu. “Structural Credit Risk Modeling: Merton and Beyond.” Society of Actuaries. 2009.

Sundaresan, Suresh. “A Review of Merton’s Model of the Firm’s Capital Structure with its Wide Applications.” *Annual Review of Financial Economics.* 2013. 

Hull, John. *Options, Futures, and Other Derivatives* 9th ed. Pearson. 2015.

Das, S. and Kim. "Credit spreads with dynamic debt." *Journal of Banking & Finance.* 2015.