Risk-Neutral Pricing and Replication of Defaultable Zero-Coupon Bonds

# Martingale measure and replicating portfolio in Risk Neutral Pricing of Defaultable Zero-Coupon Bonds


# Martingale measure and replicating portfolio in Risk Neutral Pricing of Defaultable Zero-Coupon Bonds












When pricing a defaultable zero-coupon bond the risk-neutral price is given as the expected value of the discounted payoff of the bond under a risk-neutral measure.

- My first question is how do we know that there is a measure with respect to which the discounted price process of the bond is a martingale?

- In pricing stock derivatives, the risk-neutral measure refers to the underlying stock, not the derivative. The discounted stock price process is a martingale with respect to this measure, and this measure can be derived from Girsanov's theorem and the market price of risk. Then a self-financing portfolio is derived for replicating the derivative's payoff. In the case of risk-neutral pricing of a risky bond, what is the underlying tradable asset that can be used to replicate the bond's payoff at all times?

## Answer by mmencke (score 7)

https://quant.stackexchange.com/a/63249

The risk-neutral probability measure is defined in terms of its numeraire. For the usual risk-neutral probability measure the numeraire is the bank account, $\beta(t)$. If we have a tradeable asset $X(t)$ then $\tilde{X}(t)=\frac{X(t)}{\beta(t)}$ is a martingale under $\mathbb{Q}$ meaning that $$\tilde{X}(t)=\mathbb{E}_{t}^{\mathbb{Q}}(\tilde{X}(T))$$ So we can write $$\frac{X(t)}{\beta(t)}=\mathbb{E}_{t}^{\mathbb{Q}}\left(\frac{X(T)}{\beta(T)}\right)$$ For a deterministic short rate we can write $$X(t)=\frac{\beta(t)}{\beta(T)}\mathbb{E}_{t}^{\mathbb{Q}}\left(X(T)\right)=e^{-r(T-t)}\mathbb{E}_{t}^{\mathbb{Q}}\left(X(T)\right)$$

Consider now two stocks with dynamics under $\mathbb{P}$ $$\frac{dS_{1}(t)}{S_{1}(t)}=\mu_{1}dt+\sigma_{1}dW_{1}^P(t)$$ $$\frac{dS_{2}(t)}{S_{2}(t)}=\mu_{2}dt+\sigma_{2}dW_{2}^P(t)$$ Girsanov's Theorem tells us how a $\mathbb{Q}$ Brownian Motion is related to a $\mathbb{P}$ Brownian Motion, namely $$dW_i^P(t)=dW_{i}^Q(t)+\phi_i(t)dt, \quad i\in\{1,2\}$$ Meaning that dynamics under $\mathbb{Q}$ becomes \begin{align*} \frac{dS_{i}(t)}{S_{i}(t)}&=\mu_{i}dt+\sigma_{i}\left(dW_{i}^Q(t)+\phi_{i}(t)dt\right)\\ &=\left(\mu_{i}+\phi_{i}(t)\sigma_{i}\right)dt+\sigma_{i}dW_{i}^Q(t) \end{align*} For the discounted price process to be a numeraire under $\mathbb{Q}$ we must have1 $$\mathbb{E}_{t}^{\mathbb{Q}}\left(\frac{dS_{i}(t)}{S_{i}(t)}\right)=\mathbb{E}_{t}^{\mathbb{Q}}\left(\frac{d\beta(t)}{\beta(t)}\right)$$ Which gives us the equation to solve $$\left(\mu_{i}+\phi_{i}(t)\sigma_{i}\right)dt=rdt$$ $$\iff$$ $$\phi_{i}(t)=\frac{r-\mu_{i}}{\sigma_{i}}$$ Clearly the Girsanov kernel must be different for the two stocks, so the measure is not defined in terms of the stock, but rather the numeraire. All traded assets in the economy deflated by the bank account is a martingale. Obviously you need Girsanov's theorem to change measure for a Brownian Motion, but this is specific to the Brownian Motion in question and not related to the measure in general.

Now let us consider a ZCB: $X(T)=1$. Now allowing the short rate to be stochastic. $$X(t)=\mathbb{E}_t^\mathbb{Q}\left(\frac{\beta(t)}{\beta(T)} \cdot 1\right)=\mathbb{E}_t^\mathbb{Q}\left(e^{-\int_t^Tr(s)ds}\right)$$ So we can use the same reasoning as for the stocks for this asset. For a deterministic short rate there is no Brownian Motion involved in above expression, so Girsanov's theorem does not apply here. And for a stochastic short rate it is in general not useful to define the dynamics under $\mathbb{P}$ and then use Girsanov's theorem to get the dynamics under $\mathbb{Q}$. This is due to the short rate being an unobservable quantity, so the dynamics under the physical measure $\mathbb{P}$ is not of any use2. The dynamics under $\mathbb{Q}$ can be used to calibrate the short rate model to observed market prices on ZCB's.

Now let us denote the default risk-free ZCB: $P^{rf}(t,T)$ and let us denote a default risky ZCB on counterparty A: $P^{A}(t,T)$. In case of default let us say that we recover only a fraction, $Rec$, of the market value at the time of default with a hypothetical risk-free counterparty: $Rec \cdot P^{rf}(\tau,T)$. We can thus write the cash flows of the default risky ZCB $$ \mathbb{1}_{\tau>T}+\mathbb{1}_{t<\tau<T}Rec \cdot P^{rf}(\tau,T) $$ where $\mathbb{1}_{x}$ is the indicator returning 1 if $x$ is true and 0 otherwise. We denote the default time $\tau$.

Consider a Credit Default Swap (CDS) with upfront payment on counterparty A and unit notional. In case of default this pays $(1-Rec)\cdot P^{rf}(\tau,T)$ - exactly that we do not recover. Consider a portfolio of a CDS on A plus $P^{A}(t,T)$. In case of no-default we get \$1 at time T in case of default before maturity, $t<\tau<T$, we get $$Rec \cdot P^{rf}(\tau,T)+(1-Rec)\cdot P^{rf}(\tau,T)=P^{rf}(\tau,T)$$ which delivers \$1 at time $T$. Cash only changes hands on time $t$ and $T$ and we receive \$1 for certain at time $T$. We can write the value of the porfolio $$ CDS^{A}(t,T)+P^{A}(t,T)=\mathbb{E}_t^\mathbb{Q}\left(\frac{\beta(t)}{\beta(T)}\cdot 1\right)=P^{rf}(t,T) $$ The upfront premium on $CDS^{A}$ and the price of a ZCB on A must equal the a ZCB on default risk-free counterparty otherwise it would constitute an arbitrage opportunity. If $CDS^{A}(t,T)+P^{A}(t,T)>P^{rf}(t,T)$ you can sell the portfolio and buy the default risk-free bond, pocket the difference now and have no future liabilities.

Footnotes:

- The proper way is to do Itô on $S(t)/\beta(t)$, but this should do for intuition

- See Chapter 23: Short Rate Models in Thomas Björk "Arbitrage Theory in Continuous Time" (3rd edition) for a discussion on this.