Risk Latte - Callable Convertible Bond Asset Swap Pricing:

Callable Convertible Bond Asset Swap Pricing:

Team Latte

The two important risk factors that one should take into account in pricing an asset swap on a callable convertible bond are stock volatility and risk free interest rate volatility. Some banks assume a zero correlation between these two factors on the grounds that to do otherwise leads to spurious accuracy.

In callable CB asset swaps there is frequently value in the "credit call" i.e. calling when the credit spread tightens. Many people use a credit-as-a-function-of-stock-price model along the lines of www.creditgrades.com to give a kind of two-and-a-half factor model. There seems to be little advantage in going for a full blown three factor model as this is computationally nightmarish and shows no particular improvement in results over the 2.5 factor case.

For CBs that are conditionally callable subject to a stock trigger, just assume that the CB will be called if it can be. This is a sensible and reasonably accurate approach that will slightly undervalue your callable asset swap. For CBs that are unconditionally callable, you might want to assume that the issuer will only call if parity goes say 5% over the call level. So for a CB callable unconditionally at par, model it as if it has a 105% or so call protection. This is obviously fudgy, but is more realistic. It depends on what your risk management and financial control people are prepared to agree with you.

Note 1.

If you're pricing CBs without asset swapping them, your priority is first with stochastic equity prices and second with credit. Interest rate risk comes a long way behind in third place.

If you are asset swapping (or doing something else to hedge the credit risk, such as CDS), then your priorities are equity vol and interest rate vol.

There are more sophisticated models around. The ones developed by investment banks are proprietary. Probably the best commercial vendor is www.ito33.com. It's worth spending some time reading through their web site.

Note 2.

The really interesting component in the CB asset swap is the option to unwind the swap, and call back the CB. We price this American option as an overlying of the CB price process itself, with intrinsic value: Max(CB - K(t), 0) , where K(t) is the strike price of the option. K(t) is roughly the "investment value" or the fixed-income component of the CB at the inception of the trade. Technically, K(t) = par + value of swapping the fixed coupons of the CB against LIBOR + asset swap spread.

As the callable asset swap is underlain by the CB, and the CB is underlain by the underlying equity, the asset swap is ultimately underlain by the equity, and verifies the same PDE as all equity derivatives:

dV/dt + 0.5*v^2*S^2*d2V/dS2 + (r + p)*S*dV/dS = r*V + p*(V - Recovery)

where p is the hazard rate or instantaneous credit spread and "Recovery" the value recovered in the event of default.

with the terminal condition:

V(S,T) = Max(CB(S,T) - K(T), 0)

and boundary condition, reflecting the American exercise feature:

V(S,t) > Max(CB(S,t) - K(t), 0)

Notice that solving the PDE for the callable asset swap requires solving for CB values as preliminary.

We have assumed deterministic interest rates, and the hazard rate can be a full function p(S,t) of equity price and time. As we mentioned earlier, stochastic equity is the main risk factor. But with the mapping p(S,t), the PDE above can be rewritten with time and credit spread as state variables, and equity volatility becomes equivalent to credit volatility. The question remains, How do you model p(S,t)? This has been extensively discussed in previous threads. Structural models like CreditGrades are one possible answer.

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