This post will help you understand the Black-Scholes equation in a conceptual way. No calculus or mathematical derivations.
It assumes you are somewhat familiar with it as a model to price options.
Why did I Write This?
I watched a video that combined with my prior understanding of the model to yield a more satisfying grasp of the intuition than the one I used to carry with me. Maybe my current grasp will resonate with readers in ways that extend their own intuition.
Here’s the video that prompted the post:
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Conceptual Overview
Nobody takes the model seriously as way to compute the absolute value of an option. It is used more as a thermometer to measure what the market might be saying about implied volatility. In that sense, it’s useful for comparison.
But the intuition is a great demonstration of the replication approach that characterizes arbitrage pricing techniques. The gist of the approach rests on a simple idea:
If you can replicate the cash flow of an asset with a strategy then the price of the asset should equal the cost of executing the strategy.
If the strategy and the asset have the same cash flows, then a portfolio that is short the asset and long the strategy has no risk.
If the asset trades for a higher price than the cost to execute the strategy, then:
short the asset and execute the strategy to capture the excess cash flow
This would be a riskless profit. Since the competition for riskless profits is ruthless we can infer that the price of asset would trade in line with replication costs.
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Decomposing Black Scholes
The underlying stock process
To model the price distribution of a stock, Black Scholes assumes a random walk. For our purpose, the equation detailing the process is not important. It’s sufficient to note that the process has a deterministic component and a random component.
Drift (deterministic)
This is the expected continuously compounded return of the stock. You might wonder where we get the “expected return”. Like CAPM says the expected return is a function of how volatile the stock is. The more risky, the higher expected return.
The key insight in the arbitrage pricing framework is that in the context of replication, the drift of the stock is assumed to be the risk-free rate. This might sound crazy but the key qualifier is “in the context of replication”.
When you are trying to price an option, you don’t actually care what the expected return is of the stock because you are basing the price on an offsetting strategy that sterilizes the effect of the stock price. If the offsetting strategy matches the profit of the call option, then have constructed a long/short portfolio without risk. The payoff of that portfolio only needs to be discounted by the risk-free rate.
In this replication context, the drift is simply the annualized risk-free rate.
Distribution (random)
While the drift tells us the expected return of the stock, we know that the actual return will be a random distribution centered around the expected return.
We assume a lognormal distribution because:
the returns are compounded
stock prices are bounded by zero
Volatility (aka standard deviation) tells us how wide the dispersion of returns are around the expected return.
Roughly speaking, if a stock is $100 and has a volatility of 30% and the risk-free rate is 10% then the distribution of stock prices in a year is a mean of $110 give or take 30%.
[This is not mathematically true because of compounding, but the details are a hindrance for this post]
Intuition for the lognormal distribution
The lognormal distribution looks like a lop-sided bell curve. The higher volatility (or time to expiration) the wider the spread of possible returns.
Since:
the expected return is only a function of the risk-free rate x time
the distribution is bounded by zero
the larger the volatility, the more lop-sided or positively skewed the distribution becomes.
The mass or total probability under the curve must still equal 1, but the spread of outcomes is wider (ie the right tail expands)
If the right-tail extends further, but the expected value or stock price is unchanged then the probability of the stock falling must be increasing to counterbalance the bigger upside.
This is a well-understood property of compounded returns. Google “ergodicity” or see The Volatility Drain
This idea will reappear later in an important way.
The basics of the equation
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Additional Info
Assumptions [pertinent but not exhaustive]
The inputs of spot price, risk-free rate, strike price, and time to maturity are known with high confidence
The large assumption: our estimate of volatility is correct and does not vary over the life of the contract
Definitions
N(d1) = delta or hedge ratio of the option
If you look closely at d1 you will see that is the equivalent of a z-score:
The numerator is the distance of the stock price from the strike
The denominator normalizes that distance by vol time
N(d2) = probability of the option expiring in-the-money
d2 is also a z-score but look closely —it is always less than d1
This means:
Delta is always greater than the probability of expiring in the money because delta is a hedge ratio. As a hedge ratio, it must account not just for probability but payoff.
If you have high volatility or lots of time to expiry, the right tail of a lognormal distribution is longer. So the number of shares you need to hedge the potential unbounded price of the call is larger
This means the divergence between d1 and d2 depends strictly on volatility and time to maturity.
The right-side of the equation is the strategy that replicates a long call option. (To offset the actual call we are short)
Let’s break this down step-by-step:
That strategy is a portfolio
The value of that portfolio at expiration discounted to present value must equal the value of the call option today
The portfolio has 2 components:
Shares: We need long shares of the underlying stock
Cash: We will need a loan to finance those shares
(An important idea in derivates pricing via arbitrage pricing is that we assume the strategy is “self-financing”. That means you don’t need money to start. If you respond with “But I do have some money to start”, the self-financing paradigm is already taking care of the opportunity cost using the RFR. The computation remains valid.)
The formula tells us how much of each component we need.
Drake University shows us that those quantities are actually expected values! The computations tell us, on average, the present value of how many shares you need to replicate the call option for a given time to maturity, moneyness, volatility, and cost of money.
I’ll be more explicit.
Share quantity in the portfolio
The amount of stock you need in this replicating portfolio is weighted by the expected value of the strike being in the money. Notice we say “expected value” which is not just probability but probability x payoff.
The phrase expected value of the shares going in the money is what determines the delta or hedge ratio of the option.
Delta = N(d1)
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Share quantity
S*N(d1)
Cash quantity in the portfolio
We need cash to finance the purchase of those shares. If we are short the call and it goes in the money we know we will receive the strike price at expiration because the long option holder will exercise the call and we will sell shares to them.
If the shares were 100% to be in-the-money then we know we would receive the strike price at expiration. For example, if you sold a call option struck at $125 and it was 100% to be in-the-money, you are certain to sell the stock at $125 and receive that much cash at that future date.
Of course, the option is not 100% to be in the money. So we discount the strike in 2 ways:
By the probability that it will be in the money
By the risk-free rate, to get it in present value terms
We can now say, on average, you will receive the present value of the strike weighted by its probability of being in the money.
Recall:
Probability of strike being in-the-money = N(d2)
Again, this is just expected value logic. We weight the present value of the strike by its probability of being in the money.
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Cash quantity
PV(strike) * N(d2)
To bring this to life, let’s set up an example to refer to.
Pricing a European-style call option with the following terms:
1 year to maturity
Spot price = $100
Strike price = 125
Risk-free rate = 10%
Volatility = 30%
You can solve the formula with an online calculator, programming it into Excel or the language of your choice. Back in 2000, I programmed it into one of these:
Ok, here’s the output:
The call option is worth $7.20
It has about a .40 delta
It has about a 29% chance of expiring in-the-money
We’ll refer back to this.
Animating the equation
My recap of the replicating portfolio:
Where does the idea that “you need the cash to buy the shares” show up?
That’s the source of the minus sign in Drake's representation:
The Motion Animating The Equation
Portfolio component: cash loan
In the model, how do you borrow money to buy shares?
You sell a T-bill (zero coupon bond) with a face value of the probability-weighted strike.
The probability-weighted strike is the amount of cash we expect to receive at maturity from the shares we sell.
Strike * N(d2)
$125 * 28.8% = $36
If we sell a 1 year T-bill with a face of $36, then today we receive the present value of $36:
$36e^(-.10%) = $32.57
Portfolio component: shares
The delta-weighted share quantity tells us how much stock we need to own today to hedge the value of the stock conditional upon the strike being in-the-money:
S* N(d1)
$100 * .397 = $39.77
We need to own $39.77 worth of stock to be hedged against the possibility of the stock going in the money.
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The value of the call option emerges
We borrow $32.57 today
We invest it in the stock.
We need more stock to cover the contingency that the call gets assigned. On average we need:
$39.77 - $32.57 = $7.20
The value of the call option is therefore the price that reflects the full cost to replicate its payoff!
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Decompose the p/l:
The loan cost the interest on the T-Bill:
$32.57 - $36= ($3.43)
In expectancy terms, I will be selling you $39.77 worth of shares for only $36:
$36 - $39.77 = ($3.77)
Net P/L = ($7.20)
The replicating portfolio will cost you $7.20 in expectancy, therefore that must be the value of the call option!
The recap table:
Final Observations
The impact of the risk-free rate
Notice that a higher risk-free rate:
increases N(d1) otherwise known as the hedge ratio or delta.
This is because the “drift” or forward price is mechanically higher.
increases N(d2) which is a positive function of N(d1)
Simple intuition agrees. If you increase the forward price of the asset, then any fixed strike price is mechanically more in-the-money or less out-of-the-money.
However…
The value of the call option will increase overall as you raise the risk-free rate.
Why?
The higher the risk-free rate, the less proceeds you earn on the sale of the t-bill. It trades at a larger discount to face value.
The net effect:
The first term: S*N(d1) grows larger as N(d1) grows larger
The second term: PV(strike) * N(d2) does not grow as quickly because while N(d2) grows PV(strike) shrinks
Therefore the call option which is the number of shares you need minus the amount of shares you can actually afford to finance expands in value.
It costs more to finance the replicating portfolio!
This is why call options have a positive rho, the greek for sensitivity to interest rates.
Put options have a negative rho and increase in value as the risk-free rate falls.
The impact of volatility on delta and probability of expiring in-the-money
Let’s look at our option again. We’ll keep all the inputs the same except the volatility to see how it affects a few key parameters.
We’ll work from obvious to less obvious.
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Pinned for reference
How Volatility Impacts The Call Price
Higher volatility increases the value of all options.
The maximum a call can be worth is the stock price itself. Here we can see that an annual volatility of 400% makes the 125-strike call nearly worth the spot price of $100!
How Volatility Impacts N(d1) aka delta
Higher volatility pushes the deltas of out-of-the-money options towards 1.00
Intuitively, if you increase volatility to the maximum, the option price asymptotically approaches the spot price. Once the volatility is so high the option is simply worth the spot price. Which means it will move dollar for dollar with the spot price. If spot goes to $105, the option value will also go to $105.
The hedge ratio or delta is clearly 1.00 at that point.
How Volatility Impacts N(d2) aka probability of expiring in-the-money
Higher volatility has an ambiguous effect on N(d2)
For “reasonable” increases in the level of volatility the 125 call is more likely to finish in the money. At a 75% volatility, the call has about a 25% chance of expiring in-the-money.
If the volatility was merely 5% the 125-strike would be several standard deviations away and highly unlikely to expire in-the-money.
But at higher levels of volatility, N(d2) shrinks again. The 125-call has about a 5% chance of expiring in-the-money with volatility at 15% or 300%!
What’s happening?
Increasing volatility starts to raise the probability that an out-of-the-money call will be in-the-money at expiration…but at some point, the increased volatility squeezes the mass of the lognormal distribution so far to the left that the distributional effect overpowers the higher volatility effect.
Remember as volatility increases, the Gaussian shape gets smushed with a median outcome closer to zero while being counter-balanced by a long, skinny right tail.
Conflicting Forces
Divergence between N(d1) and N(d2) on the call price
At sufficiently higher levels of volatility the delta, which keeps increasing, and the probability of expiring in-the-money diverge:
Let’s walk through the impact of the divergence by re-visiting the
The quantity of shares you need to own to hedge the short call keeps increasing with the delta. At extremely high levels of volatile you need to own shares in a 1 to 1 ratio against the short call.
At extremely high level of volatility, N(d2) approaches zero as the stock, despite its long right tail, has a most likely scenario of losing most of its value. Since the present value of the strike is weighted by a near zero probability, you don't expect to sell any stock at expiration to the counterparty who owns the call. There's no future expected cash flow from a sale to borrow cash against. In other words, you cannot finance the 100% delta worth of shares you need to buy to hedge.
The upshot: the replicating portfolio simply requires you to buy the shares and there's no way to finance them. So the call option is just worth the share price itself!
Delta of in-the-money options as you vary vol
Let’s switch the strike from 125 to 75. The rest of the inputs are unchanged, we are just looking at the call that’s in-the-money by $25 instead of out-of-the-money by $25
We’ll cut right to the chart.
Notice that the call starts at 100% delta and nearly 100% chance of expiring in-the-money. But as we start raising volatility:
N(d2) just continues to fall throughout the vol increase. Unlike the OTM call which saw an initial increase in N(d2) before the lognormal effect overpowered it, the ITM call just becomes less likely to expire ITM.
This is intuitive. If volatility is low a call so deep ITM is very likely to find itself ITM at expiration. But if you raise volatility, well, that probability has nowhere to go but down since increasing the spread of possible returns introduces the possibility that it will not be ITM by expiration.
The lognormal distribution getting “smushed” to the left as vol increases reinforces this effect on N(d2) for the ITM option.
It’s the delta that now sees an inflecting curve. Initially, the delta of the call drops from 100% to about 80% as the increased spread of possible returns effectively makes the call feel less in-the-money. As we raise vol, that $25 of being in-the-money creeps more towards feeling at-the-money.
This is the idea behind the intuition you’ve probably heard — raising vol pushes ITM option deltas towards .50
But…as vol increases to “very high” levels, the slim but long right tail of possible stock returns starts to push the hedge ratio (the amount of shares you’d need to replicate the option) back up towards 100%!
The call price still increases towards the spot price of $100 as vol gets ridiculously high.
Dynamic hedging
If you run many Monte Carlo simulations according to the assumptions of the B-S model including the underlying distribution, on average, the call value would match the predicted model value.
If you worked through my game example The Snake Eyes Options you’ll recognize that this is the same thinking. It’s just expected value and conditionals. Black-Scholes just requires calculus to handle the continuous rather than discrete underlying distribution.
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Simpler versions: first round interview question I faced in 1999
You flip a single die and will paid $1 times the number that comes up. How much would you pay to play?
Suppose I let you take a mulligan on the roll. Now how much would you pay?
Congratulations, you just priced an option.
Of course, any one simulation will have massive noise in the option value compared to the Black Scholes value.
If you sell an option worth $7.00 at $7.20 you expect to make 20 cents on average. However, the standard deviation of the p/l will be massive. (We are even making the unrealistic assumption that the vagaries of the stock are drawn from the underlying lognormal distribution, mean, and volatility that we used to generate the model price).
If you hedge the delta of the option periodically, you are attempting to replicate it for the cost of $7.00 and pocket the difference. The more you hedge, the closer that replication value will approach $7.00.
Hedging itself is a cost that the model assumes to be free (no slippage or transaction costs).
Another thing you could do in this made-up world we are using for demonstrating principles — you could sell overpriced options across many names and not hedge at all. In theory, you should still earn that $.20 and diversification handles the variance instead of hedging.
Of course, delta risks will swamp option edge. If you sell that option contract at 31% vol when the underlying go on to realize 30% that will be invisible compared to the wild swings in p/l that come from where the stock eventually ends up.
In other words, isolating a trade where you think the asset is going to move 2% per day instead of 2.2% a day is not best expressed by just letting your deltas run. Your trade thesis hinged on mispriced volatility and that is not what you are betting on when you don’t hedge at all.
In the real world, hedging is discrete. It can be automated or discretionary. It is a rabbit hole of obsession amongst vol traders. The main point is that hedging reduces variance in exchange for a cost.
[Buy Financial Hacking for chapters worth of discussion on this topic. An especially useful bit from the author’s sims was how the reduction of risk is proportional to the square root of how often you hedge. For example, if you want to halve the variance of your trades from whatever it is now, hedge 4x as often].
Because replication (ie hedging) is costly, in practice it is a bit of an art depending on how diversified a portfolio is and how positions are sized.
For my extensive overview of hedging, in general, see:
