Pricing Options using Black Scholes Merton
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- čas přidán 31. 05. 2024
- These classes are all based on the book Trading and Pricing Financial Derivatives, available on Amazon at this link. amzn.to/2WIoAL0
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The Black-Scholes or Black-Scholes-Merton model is a mathematical model for the dynamics of a financial market containing derivative investment instruments. From the partial differential equation in the model, known as the Black-Scholes equation, one can deduce the Black-Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price regardless of the risk of the security and its expected return. The formula led to a boom in options trading and is widely used, although often with adjustments and corrections, by options market participants.
Based on works previously developed by academics and practitioners, such as Louis Bachelier and Ed Thorp among others, Fischer Black and Myron Scholes demonstrated in the late 1960s that a dynamic revision of a portfolio removes the expected return of the security, thus inventing the risk neutral argument. After three years of efforts, the formula was published in 1973 in an article entitled "The Pricing of Options and Corporate Liabilities", in the Journal of Political Economy. Robert C. Merton was the first to publish a paper expanding the mathematical understanding of the options pricing model, and coined the term "Black-Scholes options pricing model". Merton and Scholes received the 1997 Nobel Memorial Prize in Economic Sciences for their work, the committee citing their discovery of the risk neutral dynamic revision as a breakthrough that separates the option from the risk of the underlying security. Although ineligible for the prize because of his death in 1995, Black was mentioned as a contributor by the Swedish Academy.
The key idea behind the model is to hedge the option by buying and selling the underlying asset in in line with its delta and, as a consequence, to eliminate risk. This type of hedging is called "dynamic delta hedging" and is the basis of more complicated hedging strategies such as those engaged in by investment banks and hedge funds.
The model's assumptions have been relaxed and generalized in many directions, leading to a plethora of models that are currently used in derivative pricing and risk management. It is the insights of the model, as exemplified in the Black-Scholes formula, that are frequently used by market participants, as distinguished from the actual prices. These insights include no-arbitrage bounds and risk-neutral pricing. Further, the Black-Scholes equation, a partial differential equation that governs the price of the option, enables pricing using numerical methods when an explicit formula is not possible.
The Black-Scholes formula has only one parameter that cannot be directly observed in the market: the average future volatility of the underlying asset, but this can be backed out from the price of other options.
In this video we learn about the model, the assumptions required for the model and about what goes in to it.
We also learn about Implied volatility and the VIX Index. The VIX Index is a calculation designed to produce a measure of constant, 30-day expected volatility of the U.S. stock market, derived from real-time, mid-quote prices of S&P 500® Index (SPXSM) call and put options. On a global basis, it is one of the most recognized measures of volatility -- widely reported by financial media and closely followed by a variety of market participants as a daily market indicator.
pricing options using black scholes merton
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Awesome video sir. Please what parameter is "d"?
Hey Patrick, great video! That being said, I am still a little confused about why we are using implied volatility. The whole point of pricing an option is to substantiate what the derivative is worth on an intrinsic level and then compare that to what the market is pricing these options at. If we back inputs out of current option prices, how is this not counterintuitive to the whole point of valuing the option? What if the market as a whole is wrong about the perceived volatility? We would have to assume efficient market theory is correct (which I refuse to believe lol) ... anyways, pardon the lengthy question, just trying to better understand practical application of this model
Good question!
1. We "back out" the volatility input for the contract since it can only be determined in the future. Also, the computed IV can be compared across contracts/series/underlyings, so that computation result is valuable even if it isn't a theoretical price.
2. No need to believe efficient markets! You can use the pricing model to compute the IV for a contract and decide that the market has "mispriced" the contract from your perspective.
The pricing model is not a trading decision engine -- it doesn't need to produce specific outputs, nobody must follow it, and it cannot decide whether a price is distorted. It's an optional (but widespread) tool that can build confidence while trading.
@@cole6167 ahhhh I see, so essentially we’re looking at priced volatility differentials and buying those we believe to be bargains based on comps?
@@santiagoheier5961 that's the idea! That one approach can prompt quite a variety of trading decisions like:
- "IV is higher on QQQ than on SPY, so I will sell puts there"
- "IV is strangely low 120 days to expiration, so I will buy those in a calendar instead of selling a vertical"
- "IV tends to fall over time, so my next risk-on trade will bias toward selling premium rather than buying it"
No wonder such a versatile tool is so widely used!
@@cole6167 Thanks Cole! I sincerely appreciate the response, I’ll have too look into learning more about these trading strategies so I can develop a couple of my own and apply them to large potential gain opportunities that are not heavily identified in the market (something that sounds easy but I’m sure will be very challenging to actually execute lol)
Just search practitioners black Scholes model
Huh, this made a lot of things click. Thank you
Black Shoals does not consider volatility or assumes that it is constant but it is not the case. When trading, I often see options pricing vary from what my assumptions are.
Did you consider transaction costs, dividends etc?
Correct. Part of opening the position might include comparing your expectations of _the change in implied volatility_ to the price/IV prevailing in the market at order entry.
You could argue that this "oversimplification" in the pricing model must be compensated for somehow when trading (by either ignoring possible changes in IV, taking a speculative position on IV changes, preemptively hedging for changes in IV, etc).
Like Patrick said, volatility over the life of the contract (which lay ahead in the future) is not known.
So this is less about computation with the model and more about decision-making in one's trading practice while facing unknown factors.
Is it possible to trade vix index by using quantitative approach?
why wouldn't it be? The VIX has tradable options and futures and relationships to highly liquid products (S&P500 options) -- may be a very good candidate for empirical-/model-driven trading...
Volatility a “bit” of a chicken egg problem 😂😂😂