Valuing Credit-Risky Bonds with Survival and Recovery

# Observed market price for the August-Greece-paid bonds were the NPV of the bond or of an option?


# Observed market price for the August-Greece-paid bonds were the NPV of the bond or of an option?












The bonds which Greece has paid had been valued by market as junk once, just before their payment. Given that the observed market value is the net present value of the instrument, why were they so low?

Because the value to be received has been discounted by the market via a huge credit spread related to the issuer=Greece?

Or because they have been reflecting the optionality of Greece between paying this debt and defaulting for everything(=cost of default)? When you consider the probability (of survival) to be/get the amount $K_\text{up}$, and the probability of default to be/get $K_\text{down}$, it is valuable as an option (with its associated NPV/price). The cash flows to come to be considered as a barrier basket option (with multiple cash settlements).

Shouldn't be bonds better priced with options'theory, than with fix income's one?

## Answer by BlueTrin (score 1)

https://quant.stackexchange.com/a/4095

Bonds are traditionally valued using the discount curve, the credit spread to determine the probability of default and usually you still have to adjust the price using a spread because bonds can become illiquid.

If you ignore the last spread the price is sum of discounted cashflows except that at every point of time you need to compute the implied probability of default. If the bond defaults, the market assumes that you receive a recovery. By computing the value of the coupon leg conditionally on survival and the default leg for the default case, you should obtain a NPV which is much closer to the traded price of this security.

Usually you still need to adjust this price by a spread as the market can trade the security at a different price than your theoretical price and this is especially true for illiquid bonds. You could generalise this by saying that the market tries to anticipate what is the distribution of the expected cashflows and asks for a premium for uncertainty.

Assuming a flat rate, the basic NPV of the cashflows is: $$ B = P_T \frac{1}{(1 + r)^T} + \sum C_t \frac{1}{(1 + r)^t} $$

Some would try to assume that you need to add a credit spread over the discounting rate: $$ B = P_T \frac{1}{(1 + r + c)^T} + \sum C_t \frac{1}{(1 + r +c)^t} $$

This is still incorrect as the implied default probability from the CDS spread. You would want to compute: $$ B = P_T \frac{1}{(1 + r)^T)}surv(T) + \sum C_t \frac{1}{(1 + r +c)^t}surv(t) $$

where $surv(t)$ is a function the cumulative probability of survival at time t.

But you are still incorrect, the bonds actually pay you a recovery in the form of cash or new bonds in the case of default. Assuming that this recovery is represented by an amount $R$ and the instantaneous default probability at time t is represented by a function $q(t)$: $$ B = P_T \frac{1}{(1 + r)^T)}surv(T) + \sum C_t \frac{1}{(1 + r)^t}surv(t) + R \int_{t=0}^T \frac{1}{(1 + r)^t} \cdot q(t) $$

Even with all these adjustments, you will still not be matching the market price of a security, you can consider the difference is likely to be due to some liquidity issues or error in your estimation of your recovery or other parameters.

There are various basis methods which will yield different results, the best is to find which one the people who trade this market seems to be using and try to match the market in addition to calculate a fair theoretical value.

HTH