Today’s video talks about a central concept in financial mathematics: pricing options using the Black Scholl formula.
Interest and profits
If you’re familiar with the topic, you’ll notice that for the sake of simplicity, I’ve ignored the impact of interest rates. But in the general case, we consider that there is a risk-free investment that provides interest \(r\), and we can also optionally add the fact that the underlying asset provides a certain profit \(q \).
Then we have a more general formula
This is for an option communicate. to get a choice whorewe get a slightly different formula but very similar in shape.
Europeans versus Americans
A point I didn’t mention in the video: I talked here about options that can only be exercised at one time, i.e. on the expiry date. This is called traditional European options. But there is also American optionswhich can be exercised at any time up to the maturity date, and for which things are more complicated.
Geometric Brownian motion
In terms of geometric Brownian motion, I pass quickly but it’s a bit more accurate than assuming percentage changes follow a Gaussian, because we’re working instead on yields. What we assume is that if we observe \(x_n\) the value of the origin at time n, then the quantity \(x_{n + 1}/x_n\) follows the law of the ordinary logarithm.
Black Scholl’s neutral delta cap
I won’t explain the Black Scholl formula here, but I will try to give some intuition regarding delta neutral hedging. First of all, it is necessary to clearly understand the purpose of this method: it is To eliminate risks, any risks. This does not mean that we are looking for a strategy that allows the bank to make money in the process, but rather a strategy that allows it to always charge the same thing, regardless of the discrepancy in the penny rate.
To do this, let’s imagine the following: the bank has just sold an option, and it doesn’t want to use the “wait and see” strategy, since this involves risk (it will cost more or less depending on the evolution of the price of the asset.) So it decides to acquire a certain amount \(\ Delta\) of the underlying (we imagine we can get partial amounts of it and without transaction costs).
At this point, the value of its position is as follows: it is Shorts of choice (since she sold it) and long of \(\delta\) implied units (which you bought). If the value of the option V and the value of the underlying S are indicated, then the value of the portfolio is thus
\(P = -V + \Delta S\).
Now imagine that the price of the underlying changes, the value of the portfolio will also change. We can calculate the derivative of P with respect to S, and simply find it
\(\frac{\partial P}{\partial S} = – \frac{\partial V}{\partial S} + \Delta \).
And what we see is that This derivative can be zero if we choose a particular value from \(\delta\)
\(\Delta = \frac{\partial V}{\partial S}\).
The fact that the derivative, for this value, is zero is interesting: it indicates that at that position, Portfolio value is not sensitive to small change in the price of the asset. This is exactly what we are looking for: eliminating risk.
It can then be shown that this means that the value of the option follows a particular differential equation. This is similar to the heat equation and reduces it by changing the variable, allowing an explicit calculation of V.
Of course from the bank’s point of view, when time passes and the value of the fundamental changes, it will be necessary to recalculate \(\delta \) and so Adjusting his position by continuously buying or selling a little bit of the underlying. The more time passes, the more the position develops toward a situation in which the bank owns either an entire unit of the asset (if the price at maturity is higher than the strike price), or zero.
It is very helpful to visualize this By tracing for a given exercise price Option value curves at different maturity periods depending on the price of the asset. Below is an example of maturities ranging from 1 to 180 days on my wheat example with a strike price of €220.
Remember that the quantity \(\Delta \) that a bank must have at any time to cover itself is equal to is derived from this curve. We see that far from the deadline (180 days), this quantity is less than 1 and does not depend very much on the share price. The closer to maturity, the closer the curve is to the blue curve at the bottom of it, which is the option’s value at maturity: that is, simply 0 below the strike price, and the spread with the strike price if one above. However, the derivative of this curve is 0 below the strike price, and 1 above.
So we can see that the closer the expiry date is, the more this strategy leads us to own an entire unit of the asset if we are above the strike price, and nothing if we are below.
The point Gilles d’Heu made to me? reka is that the trading strategy is to adjust the delta neutral hedge according to his views on the evolution of the price of the asset. If we think the stock is undervalued, we’ll hold just over delta, and vice versa.