# Option-Pricing Theory and Credit Risk

CFAI text, Vol 5, P.250~252 (Reading 39) Can anyone who has a good understanding of what are said in “5.6.1 Option-Pricing Theory and Credit Risk” share your learning outcome ?

Yeah, I didn’t see the value either in that chapter, but here it is: It uses put-call parity to explain the credit risks of equity and bond holders in a company. Equity (or stock) is equal to a call bought on the assets (A) with face value of liabilities or bonds (F) as strike price. MV Bond is equal to default-free bond plus a put written on the assets (A) and with the face value of liabilities or bonds (F) as strike price. Work out the payouts and you will see that’s correct.

mil82, Can you give intuitive explanations ?

If bondholders long a a default-free zero-coupon bond + Short a put on the assets, then is it that Stockholders LONG A PUT on the assets ? On the other hand, since Ao = So + F/[(1+r)^T] - Po, shall it be for Stockholders : long the stocks + long a default-free zero-coupon bond + Short a put on the assets ? I am very much confused !

The point they’re making is that the liability of the shareholders is limited to the assets of the company. The current value of the bond purchased by the bondholders ( who wrote the put ) is the difference between present value of a zero-credit-risk bond paying no coupons ( i.e. an instrument that has only market risk ) and the put option value represented in collateral form by assets of the company which are used to secure the bond. If the assets of the company are high , the put option expires worthless and bondholders get the entire face value of their loans back at maturity. If the company gets into trouble ( credit related ) then the value of the assets will decline and the bond holders will get less at maturity i.e. they will get the face value LESS some amount. The amount could be zero meaning their loan’s entire face value could be wiped out , depending on the size of the assets left after bankruptcy relative to the loans undertaken . In any case shareholders do not have to put up anything more. They already pledged the assets of the company in return for the loan . They bought a put option at a cost reflecting the size of the assets and the size of the loan

This model is called the Merton Model. You can find a lot of information about it on the web.

janakisri, Thank you for your response ! But why the payoffs to the stockholders resemble those of a call option ?

when value of underlying goes up - call option value also goes up. person who has bought a call option does not spend anything more than premium spent to buy the call. similarly shareholders value goes up when the stock value goes up. if value goes down - they are limited to paying off the value of the liabilities (premium paid in this case). They have the right to exercise the option - which since the call is out of the money - they do not exercise. with my limited understanding I am stating this.

It does say “resembles a call option” . The bond holders are not the ones to “pay” on the call option. The market (short-sellers in stock market) do that . The bondholders provide financing in the form of loans , which sets up the hurdle or “strike” . If the stock rises above the hurdle , the increased market value ( above the strike ) belongs to the shareholders , hence the call option for shareholders. On the downside , the shareholders are not the ones to pay the short-sellers when the stock falls. The bondholders pay the short-sellers through losses on their loan . This is why it resembles a put option written by bond holders

janakisri Wrote: ------------------------------------------------------- > It does say “resembles a call option” . The bond holders are not the ones to “pay” on the call > option. The market (short-sellers in stock market) do that . > Please read the statements on P.251. It is stated that “the payoffs to the stockholders resemble those of a call option”