Convertible bonds. Consider an underlying asset price process (left(S_{t} ight)_{t in mathbb{R}_{+}})given by [d S_{t}=r S_{t} d

Convertible bonds. Consider an underlying asset price process \(\left(S_{t}\right)_{t \in \mathbb{R}_{+}}\)given by

\[d S_{t}=r S_{t} d t+\sigma S_{t} d B_{t}^{(1)}\]

and a short-term interest rate process \(\left(r_{t}\right)_{t \in \mathbb{R}_{+}}\)given by

\[d r_{t}=\gamma\left(t, r_{t}\right) d t+\eta\left(t, r_{t}\right) d B_{t}^{(2)}\]

where \(\left(B_{t}^{(1)}\right)_{t \in \mathbb{R}_{+}}\)and \(\left(B_{t}^{(2)}\right)_{t \in \mathbb{R}_{+}}\)are two correlated Brownian motions under the riskneutral probability measure \(\mathbb{P}^{*}\), with \(d B_{t}^{(1)} \cdot d B_{t}^{(2)}=ho d t\). A convertible bond is made of a corporate bond priced \(P(t, T)\) at time \(t \in[0, T]\), that can be exchanged into a quantity \(\alpha>0\) of the underlying company's stock priced \(S_{\tau}\) at a future time \(\tau\), whichever has a higher value, where \(\alpha\) is a conversion rate.

a) Find the payoff of the convertible bond at time \(\tau\).

b) Rewrite the convertible bond payoff at time \(\tau\) as the linear combination of \(P(\tau, T)\) and a call option payoff on \(S_{\tau}\), whose strike price is to be determined.

c) Write down the convertible bond price at time \(t \in[0, \tau]\) as a function \(C\left(t, S_{t}, r_{t}\right)\) of the underlying asset price and interest rate, using a discounted conditional expectation, and show that the discounted corporate bond price \[\mathrm{e}^{-\int_{0}^{t} r_{s} d s} C\left(t, S_{t}, r_{t}\right), \quad t \in[0, \tau]\]
is a martingale.

d) Write down \(d\left(\mathrm{e}^{-\int_{0}^{t} r_{s} d s} C\left(t, S_{t}, r_{t}\right)\right)\) using the Itô formula and derive the pricing PDE satisfied by the function \(C(t, x, y)\) together with its terminal condition.

e) Taking the bond price \(P(t, T)\) as a numéraire, price the convertible bond as a European option with strike price \(K=1\) on an underlying asset priced \(Z_{t}:=S_{t} / P(t, T), t \in[0, \tau]\) under the forward measure \(\widehat{\mathbb{P}}\) with maturity \(T\).

f) Assuming the bond price dynamics \[d P(t, T)=r_{t} P(t, T) d t+\sigma_{B}(t) P(t, T) d B_{t}\]
determine the dynamics of the process \(\left(Z_{t}\right)_{t \in \mathbb{R}_{+}}\)under the forward measure \(\widehat{\mathbb{P}}\).
g) Assuming that \(\left(Z_{t}\right)_{t \in \mathbb{R}_{+}}\)can be modeled as a geometric Brownian motion, price the convertible bond using the Black-Scholes formula.