# Option on Loan rate

I have been trying to get my head around pricing an option none of the traditional option types fit the structure.

I want to get a loan of $100 , 5Y maturity . The lender gives me the following terms. 1. On day 1 there is a fixed rate offer on the table . Lets say 2.5% . 2. Once locked in I have 6 months to decide if I want the loan. 3. End of this period I can either accept the terms or reject it. No penalty. 4. If I accept it , I have a 30 day option(European for simplicity sake) where I can either take this loan at 2.5% , take the prevailing USD Libor + 50 bps or pay a penalty and cancel the facility. Essentially it is 30 day option on Loan rate , of notional$100 and tenor 5Y and the penalty is the embedded option premium and option can be represented as min(2.5%,3M Libor+50) . Whats the best way of pricing this penalty i.e the embedded option premium.

I will update if I find a solution , in the meantime any thoughts will great.

Combining the 2 Options

1. The option on the floating rate loan is comparable to an option to sell credit protection on yourself (or your collateral).
2. To value the option on the fixed loan you have to take interest and credit spread into consideration. It's comparable to a call option on a bond.

These two options are not independent since you can exercise maximal one of them and never both. As an approximation you can simply use the higher value of the two options as joint option value. Note that this will always underestimate the real value.

Payouts:

In the first 6 month your payout is $$\max(NPV(250bp), NPV(Libor + 50bp), 0)$$ where $NPV(250bp)$ is meant to be the present value of the loan with 2.5% fixed rate and $NPV(Libor + 50bp)$ the present value of the floating rate loan. Note that I understand the $NPV$ from your perspective. Hence I used maximum and not minimum to describe the payout.

After you agreed to the loan in the 30 days period the payout becomes $$\max(NPV(250bp), NPV(Libor + 50bp), -p) = \max(NPV(250bp+s_{p}), NPV(Libor + 50bp+s_{p}), 0) - p$$ with $s_{p}$ being the penalty $p$ expressed as a running spread. So I think you are right, in that you can see $p$ as an option premium, but it is an premium that is payed at maturity and you have to add $s_{p}$ on the coupon of the underlying loan.

Price option 1

Pricing of CDS Options

To price a CDS Option you need to know