SMM281

Wider Applications

Bonds

Monte Carlo Merton

The Merton Model is a simple extension of the Black Scholes model in which instead of modelling the value of an option with a payoff which is the amount by which the share price exceeds the strike price, we model the value of an equity and a bond where the bondholder receives all of the enterprise value up to the redemption payment amount and the equity holder receives the rest

How does this equate to the Black Scholes model

Treat the redemption payment on the bond as the strike price
The value of the equity is then calculated as if it were the call option
The value of the bond is the enterprise value of the firm MINUS the Black Scholes value of the equity

So we have the following equivalence

Black Scholes	Merton Model
Share price	Enterprise value
Strike price	Par value of Bond
Value of call option	Value of share
Value of share - call option	Value of bond

Interest Rates Models

There are many different models for interest rates which use simple stochastic techniques to to model different paths for interest rates in the future.

We can use these to project forward thousands of scenarios and from this calculate the the price of a bond would be if each path were the known future path of interest rates.

We then take the average of these prices and compare this with the actual bond price to calculate the 'shift' of $dB_t$ (the real world to risk neutral measure shift)

From this we can then re-engineer a full yield curve given our model and the knowledge of the current price of the zero coupon bond

This program: interestproj.exe shows this concept graphically

Mortgage Back Securities

Mortgage back securities achieved notoriety in the credit crunch.

They are not as complicated as often thought and can easily be modelled using Monte Carlo techniques

What is a MBS

A mortgage is a set of payments that a house buyer makes to pay off the loan he took out to buy the house

If the bank receiving those payments collects them and then agrees to sell on those payments IF they are made, then different investors can buy different tranches of payments. The tranche of payments you can buy is called a mortgage back security.

If the bank receives 100 mortgage payments each month and you buy the first ten received then you have a very secure investment as it is unlikely that over 90 people will default. If you buy the last 10 then you have a very risk asset as there is a high chance that several (if not all ten) of your payments will not be received.

The e-lecture below explains the working in detail

The spreadsheet used in this e-lecture can be found here

Ruin Theory

Ruin theory is the mathematics of predicting how likely an insurance company is to become insolvent given a certain set of starting conditions and a certain claims experience

It is difficult to analyse in a closed form mathematical way and tends to be considered by Monte Carlo simulations

We would typically consider the following factors when performing a simple ruin theory calculation:

Initial capital
Rate of premium income
Claim frequency
Claim distribution

The probability of ruin can then be calculated over any given time horizon by running thousands of random simulations as in the spreadsheet below:

Ruin Theory Spreadsheet

Multiple Decrement Models - Pensions

In a simple pension valuation you assume people work until they retire at say age 65 and then they receive a pension until they die

It is relatively easy to calculate the value of this as a financial liability

However in practice things will often be more complicated due to other states each pension scheme member might be in

For example a multiple state model can include sickness as well as death and retirement

The 'state space' may then appear as follows:

Notes

The normal state is healthy. However the life can also be Dead or Sick or Retired

A simple mortality model would have a transition rate to death

But a more sophisticated model would have transitions to sick and back (recovery)

and also transitions from sick to dead at a different rate from healthy to dead

We will ignore retirement for the time being and just analyse the multiple state model

The simple two state model: healthy and dead is simple to analyse analytically for a constant $\mu$ as it is simple exponential decay

However there is no closed form solution to the multiple state model and so we can use a Monte Carlo solution

Problem:

The sponsoring company wishes to know the cost of providing a sickness benefit to employees of £50 per day while they are sick. Develop a Monte Carlo model to price the value of this benefit given parameters of the transition rates $\mu, \sigma, \rho$ and $\lambda$ as defined in the diagram

Hint: each transition time will be distributed exponentially

Solution is Healthy Sick Dead Model.xls