TITLE: Bachelier model option pricing
QUESTION [6 upvotes]: Consider a Brownian motion $W_t$ and Bachelier model $S_t = 1 + \mu t + \sigma W_t$ for the stock price $S_t$. Find the value of an option that pays $1(S_1 > 1)$.
Attempt: As I understand it, the answer is $P(S_1 > 1)$, which yields a big integral with parameter $\mu$. I suspect there should be a simple answer. Any help is greatly appreciated!
Update:
Kurt and Maximilian, thank you for your help. However, I don't believe the answers given are correct. As I understand it now, you are supposed to use risk neutral valuation argument by constructing delta-hedging portfolio, not evaluate the probability directly.
Interestingly, in Black-Scholes or Bachelier models, when evaluating options with risk-neutral valuation, the result usually doesn't depend on drift term $\mu$.

REPLY [5 votes]: The other two answers assume your model is already risk-neutral; however, your notation and your edit seem to indicate that in fact you have provided us with a model under the empirical measure. So to price this option we should do a change of measure to the risk-neutral measure. To do so, I assume a constant risk-free rate $r > 0$.
Let $X_t = D_t S_t$ be the discounted asset price, where $D_t$ is the discount factor ($dD_t = -rD_tdt$). Observe that the physical dynamics in the Bachelier model are given by $dS_t = \mu dt + dW_t$. We wish to argue that under the risk-neutral measure, $X_t$ is a martingale. We compute the dynamics of $X_t$ using the Itô product rule: $$dX_t = D_t dS_t + S_t dD_t = D_t(\mu dt+ dW_t) + -rD_tS_tdt = D_t\left[ (\mu - rS_t) dt + dW_t \right] = D_t d\widetilde{W}_t$$
Where we have performed the change of measure $d\widetilde{W}_t = (\mu - rS_t) dt + dW_t$. $\widetilde{W}_t$ is a Brownian motion under the risk-neutral measure. You may check this is a valid change of measure using the Novikov criterion. Replacing this in the dynamics for $S_t$, we get: $$dS_t = \mu dt + dW_t = \mu dt + d\widetilde{W}_t - (\mu - rS_t) dt  = rS_t dt + d\widetilde{W}_t$$
By using a stochastic integration factor (e.g. see here), you may check that this SDE is satisfied by the following process: $$S_T = S_te^{r(T-t)} + e^{rT}\int_t^T e^{-rs}d\widetilde{W}_s$$
By identifying this as a Gaussian process and using the Itô isometry, this gives us the following conditional distribution under the risk-neutral measure:
$$S_T\, | \, S_t \sim N \left( S_t e^{r(T-t)}, \frac{1}{2r}\left(e^{2r(T-t) - 1} \right)\right)$$
Using the above formula, the fact that the price of the option is the discounted risk-neutral expectation, and the fact that $\mathbb{Q}(S_1 > 1, S_2<1) = \mathbb{Q}(S_2<1 | S_1 > 1) \mathbb{Q}(S_1>1)$, can you compute the required option price?