As a bank manager you are approached by a company seeking funds to start a busi$-$

ness. They need funds to purchase fixed assets, the total amount needed is $V(0)=2000,$

the equity holders are going to provide the necessary funds after the bank proposes the

conditions of the loan, in particular the loan size $D(0).$

The assets are illiquid so we have zero recovery in case of the default. The only liquid

asset is cash generated by the sales which are determined by the process satisfying a

stochastic equation. Suppose that, tentatively $-$ this will be modified, that

$dx(t)={}_{x}x(t)dt+{}_{x}x(t)d{W}_x(t)$

for some parameters ${}_x,{}_x$ and a Wiener process $W_x$ living on some probability space

$,$tin$[0,T].$

The process $x(t)$ represents daily sales which can be executed just once a day $($like$,$

if $S(t)$ is the stock price, it can be sold at any time but just once$).$ Note that such a

process can be calibrated on the basis of past sales or predicted in a business plans.

We assume that the sales are converted to cash immediately by means of factoring,

but in this process some value is lost $($which reflects the possible delay or even the default

of our customer$).$ The money received from the customers is $F(Y(t))x(t),$ where

$F$:$(-,+)[0,1]$

is a function of some form, assumed, for example

$F(x)=\frac{1}{2}+\frac{1}{}arctan(x)$

and $Y(t)$ is an auxiliary stochastic process satisfying

$dY(t)={}_Y(m-Y(t))dt+{}_{Y}d{W}_Y(t)$

for some ${}_Y,{}_Y,m$ and Wiener process $W_Y$ living on the same space as $W_x,$ correlated to

$W_x$ with constant $$ correlation coefficient. A stochastic discount factor allows modelling

of the random length of the payment delay period, which is a realistic feature. The

process $Y$ is mean$-$reverting and represents the financial condition of the customers with

parameters chosen to reflect a typical delay in invoice payments.

To find the cash generated by the sales we have to subtract the costs: variable $cx(t)$

for some cin$[0,1)$ and fixed costs, at the level $C_f$ for one day.

We assume that time is discrete, $t=\frac{n}{360},n=0,$dots,$360($the year is assumed to have

$360$ days$).$

The debt is paid either as a series of monthly instalments $($annuity$)$ or at the end of

the year $($zero$-$coupon bond$).$ We denote by $D(t)$ the amount of debt paid at $t.$

So the daily cash flow $CF(t)$ for $t>0$ is given by

$CF(t)=F(Y(t))x(t)-cx(t)-C_f-D(t)$

The cumulated cash is denoted by$.$

$C(t)={\displaystyle \underset{st}{\overset{?}{}}}CF(s),C(0)=0.$

We assume that the daily cash flow has impact on the sales, a coefficient $-$ force of this

impact and we accordingly modify the equation for $x$ :

$dx(t)={}_{x}x(t)dt+dC(t)+{}_{x}x(t)d{W}_x(t).$

The default takes place once the cumulative cash hits zero.

Tasks:

Design a simulation of the stream of cash flow and find the payoffs of debt and equity

at time $T.(50$ marks$)$