How to price this Bond

Question

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.

Thanks for your time.

Answer 1

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}

Answer 2

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.