# 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

• credit spread or CDS spread of yourself (or of your collateral).
• determine a volatility of your credit spread

The first bullet point seems the more inmportant one, it allows you to calculate the intrinsic value of the option. You can get an idea of your credit spread if you shop around a bit and get some offers without optional components.

The second part is more challenging, you need some history to determine a volatility. Otherwise you can not calculate a time value.

Price option 2

To value an option on a fixed bond you also need your current credit spread to determine the intrinsic value and some volatility of the bond price, which is determined by credit spread and interest movement.

Calculate Bond price volatility

Often in these cases the credit spread is assumed to be constant, which reduces option 1 to it's intrinsic value and option 2 can be calculated with the black model using available volatilities for swaptions.