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I'm trying to price a PIK coupon with an Embedded Option by the borrower to pay in cash. Without the Embedded Option, it is simply a zero-coupon bond paying Principal*(1 + coupon rate)^n at the end.

What I'm not sure of is how to account for the Embedded Option. What I tried to do was to imagine the borrower having a series of Bond Put options with strike price equal to the interest that could have been paid in cash.

Example: Principal = 10,000 Interest = 10% (compounding annually)

So here the annual interest is 100 (ignoring compounding). What I attempted was to think as the borrower having a put option (at each coupon payment date) to sell a bond with a principal of 100 with 10% coupon at a strike price of 100. So if the borrower exercises all the put options, it's as if they are paying with more bonds instead of cash.

But issue with this replication is that the principal and (therefore interest) changes due to the compounding effect if the borrower chooses not to pay in cash. Could anyone please suggest a better to replicate this structure?

Thank you!

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I faced exactly the same issue. I don't know a great answer, but I'll tell you what I did.

Practically, the PIK coupons are almost always fixed (but I have seen floater PIKs too). The decision by the bond issuer whether to pay cash coupon or to choose PIK (i.e. give more bond to the bond holder - effecively, borrow more money from the bondholder on the same terms as the original bond) is driven by the bond issuer's ability to borrow money elsewhere more cheaply, but sometimes by other considerations that you cannot realistically model. The borrower's cost of financing is driven more by the bond issuer's own credit spread than by the risk-free interest rates.

Sometimes bond issuers exercise options when simplistic analysis would identify the option as being out of the money. For example, I've seen lots of "make whole call" options exercised when that did not seem to make sense at the first glance.

Imagine yourself in the borower's shoes. You can choose to borrow more money at, say 5%, by not paying the 5% cash coupon on the bond, and instead exercising PIK. If you can borrow some money at 4% and pay the cash coupon, then you probably will do this. But maybe you decide that borrowing more money cheaply is too much hassle, and PIK instead. Conversely, if the cheapest rate you find now is 6%, then you probably won't pay cash. But maybe you will, so as not to signal the market that you don't have the cash. The decision is somewhat similar to a consumer with a credit card balance (revolver) deciding whether to take a loan in order to pay off the card balance.

The bond issuer may also choose to PIK for a few coupon periods, and then default on the bond. PIK bond issuers are typically very high-yield / distressed. Not paying cash coupon is a predictor that they're having problems and that default is likely. So the bond issuer has more than 2 choices: pay cash, PIK, default, maybe even others.

I did not feel that trying to model this decision as a "Black-Scholes" option would help much. Think of the following analogies from consumer finance. When a consumer exercises their right to prepay a mortgage, it is a call. When a consumer draws on a credit card (revolver loan) - pays for a purchase with the card, or takes out a cash davance - it is a put. When a consumer pays off some or all of the balance, it is a call. But trying to model these transactions as puts or calls in order to predict whether a consumer would exercise them would not help much either.

When a bank is trying to price a credit card portfolio, then they cannot just conservatively assume that every consumer will pay off their card balances right away. They have pretty complicated models, that may even take into account the consumers' propensity to act irrationally.

With PIK bonds, I chose to assume simply that the bond issuer will choose to pay cash coupon every time, instead of trying to figure out the liklihood of it happening. This is "conservative" in accounting sense for the bond holder, who earns less interest. Only if someone (such as the bond issuer themselves) indicates that part or all a particular coupon will be PIK, then add this information in the bond's static indicative data. You also need to know the current factor of the bond, ideally from knowing all the historical PIKs in the bond's static indicative data.

Please note also that if PIK is chosen, then the bond factor does not "jump" on a coupon date, unlike amortizations/prepayments. Rather, the factor grows every day when PIK accrues. Take a look at Are pure PIK bonds' payoffs known from the start? , it may help you.

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