# Why Drifts are not in the Black Scholes Formula

This question has puzzled me for a while.

We all know geometric brownian motions have drifts $\mu$:

$dS / S = \mu dt + \sigma dW$

and different stocks have different drifts of $\mu$. Why would the drifts go away in Black Scholes? Intuitively, everything else being equal, if a stock has higher drift, shouldn't it have higher probability of finishing in-the-money (and higher probability of having higher payoff), and the call option should be worth more?

Is there an intuitive and easy-to-understand answer? thanks.

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amazon.com/Heard-Street-Quantitative-Questions-Interviews/dp/… look at question 2.49. You will get an explanation without any mathematics. (Girsanov theorem might be overkill here) –  Were_cat Jun 18 '13 at 23:02

Here couple pointers that may make it clearer:

• Drift can be replaced by the risk-free rate through a mathematical construct called risk-neutral probability pricing.

• Why can we get away with that without introducing errors? The reason lies in the ability to setup a hedge portfolio, thus the market will not compensate us for the drift above and beyond the risk free rate under risk-neutral probability pricing.

• As long as such hedge exists and couple other conditions are met (please look up Girsanov's Theorem) we can introduce a risk-neutral measure so that when applying it to the differential equation and through application of Ito Calculus the drift term vanishes, which greatly simplifies the underlying math.

• It is not easy to come up with a way to reliably measure drift as it is not well known and hard to estimate in reality. Similarly, real probabilities are different for every underlying asset and are difficult to estimate because investors require different risk premiums for each and every asset. Hence, being able to take the detour through risk neutral pricing greatly simplifies the derivation not only mathematically but also intuitively. Please note that the volatility of the risk neutral paths and paths under real-world probabilities are identical, what differs is the drift term vs risk-free rate.

• Keep in mind that not all derivative functions can be derived through risk-neutral pricing, in fact I venture to estimate that less than 20% of all traded derivatives by investment banks can be priced through risk-neutral pricing (in terms of number of different instruments not trading volume).

• In order to understand the technical reason of the elimination of drift through risk-neutral pricing you need to walk through the mathematics and one of the cleanest and most intuitive approaches I have seen was in Steven Shreve's book, Stochastic Calculus for Finance II, page 218-220 (2004 edition)

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Being on the sell side and selling options you can intuitively think of it like this:

An option is like any other product that is being produced out of ingredients and because of the competitive situation of the producer this is done by the cheapest possible production process.

The ingredients are in a simple (Black Scholes) setting a stock and and a risk free bond. So this is where the name "derivative" comes from: It is derived from other, simpler products (the underlying).

Now to make it even simpler think of a call with a zero strike on the underlying. How would you price such a call? Well, you would price it like the stock (up to a correction for the risk free interest rate)! Here the same question could be asked: Shouldn't it be priced higher with the underlying having higher drift? The answer is no, because this is already included in the price of the underlying! You would kind of double count this effect.

With "real" options the reason is the same, the only thing that changes is that you have to adapt the proportion of the stock when the price of the underlying changes. But the original reasoning stays the same: The drift does not enter into the price of the derivative because it is already included in the price of the underlying.

A very readable intuitive paper on this issue is by a giant of the field, Emanuel Derman. It can be found here: The Boy's Guide to Pricing & Hedging

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I love this short read you referenced, +1 alone for this one. Shows me time and again that the top researchers in this and related fields all strive for simplicity because reputation is earned by explaining stochastic calculus to an ape rather than how to peel a banana to an astronaut. –  Matt Wolf Jun 19 '13 at 1:39
I take a different view from yours: For me there is a fundamental difference between the price of a derivative and the value of the same. What risk neutral pricing does is assisting in finding the price of a derivative. The value may vastly differ depending on the inputs to the pricing model. That differential for me explains why some people buy, others sell. Of course each participant has a slightly different pricing model though most are surprisingly similar. What differs is the estimation of value. That for me is where buy and sell side guys differ in their view –  Matt Wolf Jun 20 '13 at 5:09
...and which ultimately causes each individual to buy when he thinks price < value and sell if price > value. The pricing algorithm itself is equally applied to buy and sell side demand and supply. Given I traded today on the sell side, and tomorrow I was trading on the buy side, I would apply the exact same pricing models and derivations, nothing changes there. What changes is my risk transfer needs and thus where I see value. As risk seller I see value in the spread I can provide, as risk buyer I see value in the alpha generated from underlying assets vs hedge risk protection. –  Matt Wolf Jun 20 '13 at 5:10
In a sense yes. Though Q != P, the pricing in both, Q and P, must agree because of the no-arbitrage requirement. But assigning a specific probability measure to either pricing or valuation could lead down a dangerous path. Purely intuitively speaking though I like your comparison and agree. –  Matt Wolf Jun 20 '13 at 7:03
This is argument is fine, but it presupposes the existence of continuous self-financing portfolios. Since those do not and will probably never exist in the real world, as an intuitive argument I don't think this works. –  Andrew Dabrowski Nov 5 '13 at 14:45

Recently I came across an interesting intuitive explanation:

Suppose driftless market. Market price is 105, strike price is 100. Call option costs 8, put option 3. (intrinsic value of call is 5, time value of both is 3)

Now the market starts drifting upwards massively. You say, that you would probably price call higher, e.g. at 10. Would you also price put higher? Probably not, since it's less probable that it will end up in the money. You would probably price put even lower, e.g. at 2.

But in that case, you clearly violate put/call parity, and it's possible for me to buy the call, and completely hedge it with the put, some cash and some stocks and earn riskless profit from you. Due to put/call parity, the price of put must rise together with price of call (they must have the same time value), which is inconsistent with dependence on drift.

I disregarded interest rates for clarity.

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This is basically Derman-Taleb's argument that put-call parity implies the BS equations. I like it! But Merton apparently did not, in fact he sicced some attack dogs on DT. Anyone more knowledgeable care to step in and arbitrate? –  Andrew Dabrowski Nov 5 '13 at 14:09

$$dS / S = \mu dt + \sigma dW \\ \\ dS / S -r dt= \mu dt - rdt + \sigma dW \\ \\ dS / S -r dt= [\frac{(\mu - r)}{\sigma}dt + dW]\sigma \\$$ Then, Girsanov tells us that, as long as the risk premium is bounded from below, we can write $[\frac{(\mu - r)}{\sigma}dt + dW]\sigma$ as $\sigma d\tilde{W}$ where $\tilde{W}$ is simply another brownian motion with mean 0 and variance equal to its time step. Hence, $$dS / S = rdt + \sigma d\tilde{W} \\$$

Another way to think about this is that, in the Black-Scholes world of option pricing, markets are assumed to be friction-less and hence, all assets can be perfectly hedged. If we can perfectly hedge our underlying asset, then in theory, it should have no volatility and should grow at the risk free rate, r.

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+1 Great answer. –  SRKX Jun 18 '13 at 15:49
Did you mean to say that in a world where we can perfectly hedge the risk premium has to be equal to the risk free rate? I am asking because the asset clearly has volatility even after changing measure to a risk neutral probability measure. In fact volatility is identical before and after. –  Matt Wolf Jun 19 '13 at 0:45
I'm puzzled by something: unless $\mu=r$ the mean of ${(μ−r)\over σ}dt+dW$ is not 0, so how can it be equivalent to anything with mean 0? –  Andrew Dabrowski Nov 5 '13 at 14:12
@AndrewDabrowski: Check out the Girsanov Theorem. It lets you transform one brownian motion into another. –  fbt Apr 22 '14 at 22:56

If someone wants simple intuition, here is what happened to the drift. It did go into the formula, believe it or not, but it came into the B-S math sort of in two ways so it cancelled out in the end.

It disappears because Black-Scholes assumes people may have different preferences for risk, but at least everyone is consistent on their own preference.

There is a drift in Black-Scholes. There needs to be some way to say how much return (or drift) you personally must get to take a certain amount of risk. More accurately, we must know how much return above the risk-free rate (or risk premium) you must get to accept one $\sigma$ of return risk.

So, imagine we had a formula that required us to somehow calibrate it for your personal tastes. How do we calibrate you, then? If you want to buy an option on stock A, we need to know how much excess return you need to suffer the risk of a stock just like stock A. If we could find another stock, B, that had identical return and risk prospects, then somehow we could "plug in" that stock's numbers to the formula for you.

But wait - why look for an identical stock? Stock A is out there and we could calibrate you with it. Stock A is the very stock to use then!

Sounds a bit fishy? Not really. Presumably, you are neither buying nor selling stock A today because you don't believe it is either overvalued or undervalued. Thus, whatever you believe in your head about the risk and return to price an option on A can use stock A as the calibration. Imagine doing just the drift term: if you think stock A is going up 15%, well, then, 15% is the right drift to use for stock A options for you. As long as you are consistent it all falls out and we don't need the drift.

This makes more sense (and this is the one of the hedges that keeps getting talked about) if you imagine that you think you have secret information that a company's stock will jump 10% later today. If you really though that you would think the stock is really, really undervalued right now, true? But, you could also buy a call and sell a put that expires tomorrow with identical strikes. That would also go up 10% later today if you are right!

If you try the example I just gave in your head, you can see that, if you are right, the call options are woefully undervalued and puts overvalued) by 10%. So, the accumulation of everyone else's preferences and beliefs - which go into both the stock and the options prices - are very different from yours. You, in fact, think that stock A has a lot more return at very low risk to occur compared to everyone else...

It doesn't spit out by giving you your own price for the option (we all share one price), but it gives a price based on a collective estimate of drift - though, oddly, we don't know what that estimate is. However, option prices would seem too low if you estimate a higher drift or lower volatility, so you would load up on options.

It definitely can fall out of the math a couple of ways, via risk-neutral pricing and Girasnov's theorem, or via a Taylor series expansion that can't ignore one of the second order terms due to quadratic variation, etc. It can also be thought of via a hedge, but that is also hard to understand. None of these is intuitive.

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Let me provide an intuitive answer that I just thought of (correct me if I am wrong). So starting with two Stochastic Differential Equations (SDE)

$\frac{dS_t}{S_t}=μdt+σdW_t$

$\frac{dD_t}{D_t}=-rdt$

(I am assuming our risk-free rate to be constant as is done in most introductory financial math courses) Notice: $D_t = e^{-rt}$ is the solution to the ODE above which is our continuous discounting factor. No need for Ito's lemma since r is not a stochastic process! $$-$$$$-$$ Now a question to ask is: Is our current measure under $\mathbb{P}$ risk neutral? Well basically in a risk-neutral world we want our discounted price of our asset to not change or to be said in another way to be a martingale. An intuitive explanation for this is that we believe that an asset price's drift should at least rise by as much as the risk-free rate. Otherwise, we could just put our money in the bank risk-free, so maybe we think part of the drift upwards for the asset price is due to the risk-free rate. So we have to discount the asset. You can of course also use the portfolio replication argument as well to be concrete. So we want:

$D_tS_t$ to be a martingale. Let's try to make it so.

$d(D_tS_t) = D_tdS_t + S_tdD_t$

$= D_tS_t(μdt+σdW_t) + S_t(-rD_t)dt = D_tS_t[(μ-r)dt + σdW_t]$

Well let's choose a $dW_t = d\tilde{W}_t - \theta dt$ such that the drift term is eliminated in $D_tS_t$ so that it becomes a martingale (all Ito processes w/ no drift are martingales).

$\theta = \frac{μ-r}{σ}$ leads to $d(D_tS_t) = D_tS_t[(μ-r)dt + σ(d\tilde{W}_t-\frac{μ-r}{σ}dt)] = σD_tS_tdW_t$ Now it's a martingale ! $$-$$$$-$$

$\frac{dS_t}{S_t} = μdt + σdW_t = rdt + σd\tilde{B}_t$

Well for those Girsanov fans out there, why do we not need to worry about calculating $\frac{d\mathbb{\tilde{P}}}{\mathbb{dP}} = e^{-\int^t_0\theta dB_s - \frac{1}{2}\int^t_0\theta^2dt}$? Well because I didn't even state a $\mathbb{P}$-measure and the question was to find the dynamics of the asset price movement under the risk neutral measure $d\mathbb{\tilde{P}}$. And under $d\mathbb{\tilde{P}}$, $d\tilde{B}_t$ is a brownian motion.

$\therefore \frac{dS_t}{S_t}=rdt+σdW_t$ under risk neutral measure.

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