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.
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4$\begingroup$ The dissertation has been translated into English. $\endgroup$– Dave HarrisCommented Mar 7, 2017 at 4:41
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1$\begingroup$ (1) IIRC Bachelier did not include non-zero interest rates in his model. (2) Bachelier's paper was translated into English starting on Page 17 of the book 'The Random Character of Stock Market Prices' (3) It should not be difficult to discount the payoff at a chosen interest rate $\endgroup$– nbbo2Commented Mar 7, 2017 at 4:53
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5 Answers
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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.
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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.
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$\begingroup$ Can U explain the last equality before the comment. I can't see why u can go from $E(....)^+$ to write it in terms of the $\Phi$ and $\phi$ $\endgroup$– SanjayCommented Mar 20, 2018 at 21:42
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1$\begingroup$ @Sanjay: Can you compute the expectation $E\big((\xi-\alpha)^+\big)$, where $\alpha>0$ and $\xi$ is standard normal? $\endgroup$– GordonCommented Mar 21, 2018 at 13:51
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$\begingroup$ That is exactly the computation i do wrong (I think?) $E[ (\xi-a)1_{Z>a}] = E[\xi1_{Z>a}]-E[a1_{Z>a}] =\int_a^ \infty x\phi(x)dx-a\Phi(-a)$. I can't rewrite this term into the desired result. If I can show that: $\int_a^ \infty x\phi(x)dx=\phi(-a)$, then I think I will be able to understand your proof. $\endgroup$– SanjayCommented Mar 21, 2018 at 14:39
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3$\begingroup$ Note that $$ \int_a^{\infty} x \phi(x) dx = \frac{1}{\sqrt{2\pi}}\int_a^{\infty} x e^{-x^2/2} dx = \frac{1}{\sqrt{2\pi}} (- e^{-x^2/2})|_a^{\infty} = \phi(a). $$ $\endgroup$– GordonCommented Mar 21, 2018 at 15:40
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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.
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$\begingroup$ Shouldn't the interest rate appear in your expression for $D$? I.e. I would have expected $D = \left( S_0 e^{r (T - t)} - K \right) / \sigma \sqrt{T - t}$. $\endgroup$ Commented Mar 7, 2017 at 9:31
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$\begingroup$ If $S=r dt + \sigma dW$ under risk neutral, then I would expect the forward to satisfy $dF = \sigma dW$, so $F_{t,T} = S_{t} + r(T-t)$, as opposed to $F_{t,T}=e^{r(T-t)}S_t$. Am I missing something? $\endgroup$– OlafCommented Mar 7, 2017 at 10:10
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2$\begingroup$ No because the drift would be $rS_tdt$ so that the discounted process is a martingale under the risk neutral measure. $\endgroup$– NSZCommented Mar 7, 2017 at 10:12
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1$\begingroup$ What is the dynamics of your stock process? I have difficulty to re-produce your results with both stock process assumptions in my answer above. $\endgroup$– GordonCommented Mar 8, 2017 at 21:44
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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 $F=F(t,T)=S(t)e^{\mu (T-t)}$.
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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.
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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.
