Bond Forward Prices and Changes of Numeraire

# Hull Martingales and measures problem 27.16 7e?


# Hull Martingales and measures problem 27.16 7e?












Here's a question from Hull's Options Futures and Other derivatives which I'd appreciate if someone helped me to clarify. The question is from the chapter "Martingales and Measures"

Suppose that the price of a zero-coupon bond maturing at time T follows the process:

\begin{align} \frac{dP(t,T)}{P(t,T)} = \mu_P dt + \sigma_P dW_t^{\mathbb{P}} \\ \end{align}

and the price of a derivative dependent on the bond follows the process

\begin{align} \frac{df}{f} = \mu_f dt + \sigma_f dW_t^{\mathbb{P}} \\ \end{align}

Assume only one source of uncertainty and that f provides no income.

(a) What is the forward price $F$ of $f$ for a contract maturing at time $T$?

(b) What is the process followed by $F$ in a world that is forward risk neutral with respect to $P(t,T)$?

(c) What is the process followed by $F$ in the traditional risk-neutral world?

Now the answers are:

(a) $F=\frac{f}{P(t,T)}$, and from here we can derive $F$ dynamics as:

\begin{align} \frac{dF}{F} = (\mu_f-\mu_p + \sigma_P^2 - \sigma_f \sigma_P )dt + (\sigma_f -\sigma_P) dW_t^{\mathbb{P}} \\ \end{align}

(b) Since $F=\frac{f}{P(t,T)}$ has the numeraire $P(t,T)$ we can expect it to be a martingale under this measure so that the dynamics are:

\begin{align} \frac{dF}{F} = (\sigma_f -\sigma_P) dW_t^{\mathbb{P}} \\ \end{align}

(c) This is where I'm confused, the solution apparently is

\begin{align} \frac{dF}{F} = (\mu_f-\mu_p + \sigma_P^2 - \sigma_f \sigma_P )dt + (\sigma_f -\sigma_P) dW_t^{\mathbb{P}} \\ \frac{dF}{F} = ((r + \lambda \sigma_f)-(r + \lambda \sigma_P) + \sigma_P^2 - \sigma_f \sigma_P )dt + (\sigma_f -\sigma_P) dW_t^{\mathbb{P}} \end{align}

Now since we are talking about the risk-neutral world we must choose $B_t$ as the numeraire. With

\begin{align} \frac{dB_t}{B_t} = (r)dt \end{align}

In this chapter Hull says that if we choose $\lambda = \sigma_g$ the process of $f/g$ will become a Martingale. Assuming $f$ and $g$ follow also the same dynamics with the same source of uncertainty.

So in this case we might choose $\lambda = 0$ since there is no brownian motion in the $B_t$ dynamics.

This would lead us to believe that the solution is:

\begin{align} \frac{dF}{F} = (\mu_f-\mu_p + \sigma_P^2 - \sigma_f \sigma_P )dt + (\sigma_f -\sigma_P) dW_t^{\mathbb{P}} \\ \frac{dF}{F} = (\sigma_P^2 - \sigma_f \sigma_P )dt + (\sigma_f -\sigma_P) dW_t^{\mathbb{P}} \quad (1) \end{align}

This last equation is the solution proposed in the Solutions Manual of the book.

My problem here is that to my understanding, if (1) are the actual dynamics of $F$ under the traditional risk-neutral measure then $F/B$ should be a martingale, but when doing the calculation it doesn't.

Is my interpretation of the question incorrect? or could the answer be wrong?

Much help appreciated