The short rate under the Vasicek one factor model under the real-world measure $\mathbb{P} $ follows : $$ dr(t)=(a\theta - (a+\lambda \sigma)r(t))dt + \sigma dW(t),$$ $$ r(0)=r_0 $$
where $ \lambda $ is the market price of risk .ie. risk premium
Now, under the Vasicek model, $\lambda$ is usually chosen to be constant .ie. $ \lambda(t)=\lambda$
But what if this was not the case, and instead, the risk premium was time-dependent .ie. $ \lambda(t) $? In that case:
- What would be my mathematical starting point if I wanted to determine the value of $ \lambda(t) $?
- Would the traditional formula of $ \lambda = (\mu - r_f)/\sigma $ work in this case?
Is this even a valid point to think about? Apologies if this doesn't make sense and thank you in advance.