Change of Numeraire and the T-Forward Measure

For a European bond option with payoff $X = (P(T_1, T_2) - K)^+$, this becomes a double integral: we must compute the joint distribution of the stochastic discount factor $1/B(T_1)$ and the random bond price $P(T_1, T_2)$ under $\mathbb{Q}$. In all but the simplest Gaussian models, this is intractable.

• define the $T$-forward measure $\mathbb{Q}^T$ via its Radon-Nikodym density with respect to $\mathbb{Q}$;
• state and prove Proposition 7.1 (the forward-measure pricing formula);
• prove that $P(t,S)/P(t,T)$ is a $\mathbb{Q}^T$-martingale;
• state the expectation hypothesis under $\mathbb{Q}^T$ (forward rate = $\mathbb{Q}^T$-expectation of future short rate);
• relate the $T$-forward measure to the $S$-forward measure by computing $d\mathbb{Q}^S/d\mathbb{Q}^T$;
• apply the forward measure to decompose a European bond option price into two forward-measure probabilities.

1. 1. The change of numeraire — motivation
2. 1. Definition of the T-forward measure
3. 1. The forward-measure pricing formula
4. 1. Martingale property of relative bond prices
5. 1. The $\mathbb{Q}^T$-Brownian motion and forward rate
6. 1. Changing between forward measures
7. 1. Bond option pricing via the forward measure