Bachelier Call Pricing with Interest Rates and Model Conventions

# Bachelier model call option pricing formula


# Bachelier model call option pricing formula












Does anybody have the Bachelier model call option pricing formula for $r > 0$?

All the references I've read assume $r = 0$. I don't speak French, so I can't read Bachelier's original paper.

## Answer by Gordon (score 22)

https://quant.stackexchange.com/a/32882

We assume that, under the risk-neutral measure, the stock process $\{S_t, t \ge 0\}$ satisfies an SDE of the form \begin{align*} dS_t = r S_t dt + \sigma dW_t, \end{align*} where $r$ is the constant interest rate, $\sigma$ is the constant volatility, and $\{W_t, t \ge 0\}$ is standard Brownian motion. For $0 \le t \le T$, \begin{align*} S_T = S_t e^{r(T-t)} + \sigma\int_t^T e^{r(T-s)}dW_s. \end{align*} That is, \begin{align*} S_T \mid S_t &\sim N\left(S_t e^{r(T-t)},\, \frac{\sigma^2}{2r}\left(e^{2r(T-t)}-1 \right) \right)\\ &\sim S_t e^{r(T-t)} + \sqrt{\frac{\sigma^2}{2r}\left(e^{2r(T-t)}-1 \right)}\,\xi, \end{align*} where $\xi$ is standard normal random variable. Then \begin{align*} C_t &= e^{-r(T-t)}E\left(\left(S_T-K\right)^+ \mid \mathcal{F}_t \right)\\ &=e^{-r(T-t)}E\left(\left(S_t e^{r(T-t)} + \sqrt{\frac{\sigma^2}{2r}\left(e^{2r(T-t)}-1 \right)}\,\xi-K\right)^+ \mid \mathcal{F}_t \right)\\ &=e^{-r(T-t)}\sqrt{\frac{\sigma^2}{2r}\left(e^{2r(T-t)}-1 \right)}E\left(\left(\xi -\frac{K-S_t e^{r(T-t)}}{\sqrt{\frac{\sigma^2}{2r}\left(e^{2r(T-t)}-1 \right)}}\right)^+ \mid \mathcal{F}_t \right)\\ &=e^{-r(T-t)}\left(S_t e^{r(T-t)}-K\right)\Phi\left(\frac{S_t e^{r(T-t)}-K}{\sqrt{\frac{\sigma^2}{2r}\left(e^{2r(T-t)}-1 \right)}}\right) \\ &\qquad + e^{-r(T-t)}\sqrt{\frac{\sigma^2}{2r}\left(e^{2r(T-t)}-1 \right)}\,\phi\left(\frac{S_t e^{r(T-t)}-K}{\sqrt{\frac{\sigma^2}{2r}\left(e^{2r(T-t)}-1 \right)}}\right), \end{align*} where $\Phi$ is the cumulative distribution function of a standard normal random variable, and $\phi$ is the corresponding density function.

> Comments

Let $K^*=e^{-r(T-t)}K,$ and $$v^2(t, T) = \frac{\sigma^2}{2r}\left(1-e^{-2r(T-t)}\right).$$ Then, we can re-express the price as \begin{align*} C_t &= \left(S_t-K^*\right)\Phi\left(\frac{S_t-K^*}{v(t, T)}\right) +v(t, T)\,\phi\left(\frac{S_t-K^*}{v(t, T)}\right). \end{align*} See also Section 3.3 of the book Martingale Methods in Financial Modeling; however, note that there are a few typos in this book.

One other possibility is to assume that \begin{align*} S_t = e^{rt}(S_0 + \sigma W_t). \end{align*} Then the corresponding option price can be similarly obtained. See also the book mentioned above.

## Answer by NSZ (score 9)

https://quant.stackexchange.com/a/32868

It's pretty simple to derive with basic knowledge of stochastic calculus. But since you are looking for the easy answer here it is:

$$C_t=e^{-r(T-t)}\sigma\sqrt{T-t} (D \Phi(D)+\phi(D))$$ where $D=\frac{F_{t,T}-K}{\sigma \sqrt{T-t}}$ and $\Phi(\cdot)$ and $\phi(\cdot)$ are respectively the normal cdf and pdf. $F_{t,T}=S_te^{r(T-t)}$ is the forward price.

## Answer by jherek (score 6)

https://quant.stackexchange.com/a/37487

You might want to differentiate between the growth rate $\mu$ and the discount rate $r$.

@Gordon's solution is the most logical thing to do, given the question. However, in practice, it is not uncommon to model the forward process $F$ instead of asset spot process $S$. Interestingly, unlike in the Black-Scholes case, the forward process and the spot process do not have the same volatility in the Bachelier model.

@NSZ's solution amounts to assuming a lognormal forward process $$dF = \sigma dW$$ with a growth rate $\mu$ and $F(t,T) = S(t) e^{\mu(T-t)}$.

We apply Ito's Lemma to $f(t,F) = F e^{\mu(t-T)}$ to obtain in terms of $S$: $$dS = \mu S dt + \sigma e^{\mu(t-T)} dW\,.$$

Under the forward model, the call option price with a drift is obtained from the standard Bachelier option price: $$ C(t,T) = e^{-r (T-t)} \left[ (F-K) \Phi\left(\frac{F-K}{\sigma\sqrt{T-t}}\right) + \sigma\sqrt{T-t} \phi\left(\frac{F-K}{\sigma\sqrt{T-t}}\right)\right]\,,$$ where $\Phi$ is the cumulative normal distribution function and $\phi$ is the normal probability density function, and $F=F(t,T)=S(t)e^{\mu (T-t)}$.

## Answer by jaehyukchoi49 (score 4)

https://quant.stackexchange.com/a/71148

Here's an extensive reference on the Bachelier model including the option price formula:

Choi J, Kwak M, Tee CW, Wang Y (2022) A Black–Scholes user’s guide to the Bachelier model. Journal of Futures Markets 42:959–980. https://doi.org/10.1002/fut.22315. [Arxiv Download]

Abstract:

> To cope with the negative oil futures price caused by the COVID–19 recession, global commodity futures exchanges temporarily switched the option model from Black–Scholes to Bachelier in 2020. This study reviews the literature on Bachelier's pioneering option pricing model and summarizes the practical results on volatility conversion, risk management, stochastic volatility, and barrier options pricing to facilitate the model transition. In particular, using the displaced Black–Scholes model as a model family with the Black–Scholes and Bachelier models as special cases, we not only connect the two models but also present a continuous spectrum of model choices.

## Answer by Con Fluentsy (score 2)

https://quant.stackexchange.com/a/66344

The Complete Book of Option Pricing Models 2007 McGraw Hill By Espen Gardner Haug has all option pricing models, including an excel VBA and c++ implementation of Bachelier's original option pricing model. This was and probably is the most definitive collection of option pricing models out there, I have not seen one since which surpasses it. Most option models since, are adaptations or tinkering with existing models at the time rather than true innovations.