# How to price this Bond

I have below Bond -

Issue date : 1/1/2020
Principal 1,000
Coupon : 8% pa
Frequency : Semi-annual
Tenor: 2 years

This Bond has 2 specific characteristics -

1. At the maturity NO principal will be paid
2. After 1st year, Bond can be exited on 6/1/2021 (i.e. Put option attached)

Without the 2nd condition, this Bond can easily be priced using discounted CF method. However given that the 2nd option, how can I price this Bond?

Is there any software implementation to price this type of Putable Bond?

Modified based on StackG's comment

StackG pointed a lack of clarity on the payment upon premature exit, So I added the payoff profile of this bond as below -

1. At the maturity principal will be paid based on the prevailing one month average of S&P index + last coupon
2. After 1st year, Bond can be exited on 6/1/2021 (i.e. Put option attached). In that case, the prevailing one month average of S&P index as on 6/1/2021 will be paid + accrued coupon

How can I price this Putable Bond?

Your pointer will be highly appreciated.

• Putable bonds are pretty common in Russian and Indian markets, but I don't understand this one, you may want to clarify. If the bondholder exercises the put in 1 year, then then the bond issuer repays the principal. But if the bondholder doesn't exercise the put, then the bond issuer never repays the principal?? – Dimitri Vulis Aug 14 '20 at 19:42
• There would be no Principal payment even on the Maturity i.e only Coupon payment (4 times as for semi-annual). This bond probably more looks like Annuity with additional optionality – Bogaso Aug 14 '20 at 19:45
• So if 2nd condition isn't exercised, this bond is simply 4 semi-annual cashflows. If it IS issued, what happens? Unless I'm missing something, it never makes sense for the holder to exercise the put if it results in cancelling the future cashflows for no payment now? Or is there also a put price attached to the contract? – StackG Aug 15 '20 at 6:18
• I see it. But for Principal paying bond also this would be the case, isnt it? – Bogaso Aug 15 '20 at 7:46
• I don't think so. If the put is at par, then it would make sense to exercise the put if rates go above the bond yield, since you could then reinvest the cash at the higher rate in the market. – StackG Aug 15 '20 at 11:45

This bond pays four coupon cashflows of \\$40 each at 6 month intervals. By themselves, that would be very easy to price from a discount curve, if there is no credit risk. The additional complication is the payoff of the principal, which pays the last month average of the SPX index. For simplicity, I'm going to assume throughout that it simply pays off the SPX value at expiry date, but it won't affect the result much.

Option pricing theory tells us that the price of a security is the discounted expectation of its payoff in the risk-neutral measure \begin{align} C(0) = \delta(0,t) \cdot {\mathbb E} \Bigl[ C(t)\Bigr] \end{align} where $$\delta(0,t)$$ is the discount factor at time $$t$$ and $$C$$ is the option price at the time (ie. $$C(t)$$ is the payoff at expiry)

Now this equation works for any tradable security, at any time (it says assets are forced to be martingales in the pricing measure, because otherwise arbitrageurs will come in and trade them until it is true). So assuming we can trade the SPX index (via ETFs or futures perhaps), it holds for the SPX index too, and reversing the order of the equation we have \begin{align} {\mathbb E} \Bigl[ SPX(t_2)\Bigr] = SPX(t_1) \cdot {\frac 1 {\delta(t_1,t_2)}} \end{align} which just says that our expected value of SPX at $$t_2$$, viewed from $$t_1$$, is just the spot value divided by the dcf

Now we need to think about our decision at $$t_1$$. We're faced with a choice of either receiving $$SPX(t_1)$$ straight-away, or else receiving $$SPX(t_2)$$ at $$t_2$$ and the two remaining coupon payments. But the value of the payment at $$t_2$$ is just $$\delta(t_1, t_2) \cdot {\mathbb E} \Bigl[ SPX(t_2)\Bigr]$$ which is just equal to $$SPX(t_1)$$, so it is NEVER a sensible decision to exercise the option at $$t_1$$

So you can simply price the option assuming it runs to expiry, the price is: \begin{align} \delta(0,0.5) c_{0.5} + \delta(0,1) c_1 + \delta(0, 1.5) c_{1.5} + \delta(0,2) c_2 + SPX(0) \end{align}

• nb. if dividends were 8% it might make sense to exercise... but SPX typically pays <2% so I'm ignoring this – StackG Aug 17 '20 at 15:10

i find the modification very confusing and so different from the original question. how did the S&P 500 come into this? it's hard to confuse "no principal paid" in BOLD with principal is paid based at least partially on S&P 500.. i think OP needs to rethink the question/idea.

one minor question for clarification: is the put option a one-time privilege? or can it be exercise anytime on or after key date?

one thing i notice is if you put early, you are giving up a bond with a variable value (like a convertible bond)....... i believe you might look into numiere (correct term?)

and it is hard to get away from explicitly modelling the 2 year to 1 year interest rate.

• Hi Stephen - let me try to address your queries. 1) Yes I agree that originally my question was incomplete, but later modified. However I did not delete my original writings just because some discussions already had happened to bring further clarifications (which I did) and wanted to keep the context. However if viewers advise to delete the earlier parts, I am happy to do that 2) Technically nothing can stop a non-standard bond's payoff to be linked with a 3rd asset, here it is linked to S&P. 3) the Put option privilege is one time and can only be exercised on that particular date only. – Bogaso Aug 17 '20 at 7:42
• ... 4) modelling an Instrument is hard doesnt necessarily mean that such instrument should not be traded or existed. Let me know if any further clarification is required. – Bogaso Aug 17 '20 at 7:42
• Expanded my argument below – StackG Aug 17 '20 at 14:01