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29

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 ...

22

One starts with the Black-Scholes equation $$\frac{\partial C}{\partial t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2 C}{\partial S^2}+ rS\frac{\partial C}{\partial S}-rC=0,\qquad\qquad\qquad\qquad\qquad(1)$$ supplemented with the terminal and boundary conditions (in the case of a European call) $$C(S,T)=\max(S-K,0),\qquad\qquad\qquad\qquad\qquad\qquad\qquad\... 22 In general there are two basic ways to make money out of your option pricing models: Sell side (market maker, risk neutral): You use these models to calculate your greeks to hedge your portfolio, so that you live on the spread. Buy side (market/risk taker): You use your model to find mispriced options in the market and buy/sell accordingly. (A third ... 21 In the Interest Rates field there is one paradox in nowadays market conditions (i.e. since the crisis) that is quite tricky to properly understand. This is the fact that one need several curves to have a correct pricing of simple interest derivatives such as Swap with floating index set to some Libor reference. Simply and crudely speaking, you have to ... 20 You may want to look at Chapter 5 - "The Quest for the Option Formula" from the Derivatives book. The book is available online for free and it has a very decent review of approaches that were used 20-30 years before the Black-Scholes-Merton equation. 19 Black-Scholes itself didn't change a lot but we can now adjust it to deal with a lot more complicated factors to price more complicated contracts: stochastic volatility (Heston, Gatheral) stochastic rates (Hull) credit risk dividends Other methods (computing intensive) have also evolved to deal with various types of contracts where BS is not very ... 16 There are a wide variety of models (by which I mean the theoretical / mathematical formulation of how the underlying financial variable(s) of interest behave). The most popular ones differ depending on the asset class under consideration (though some are mathematically the same and named differently). Some examples are: Black-Scholes / Black / Garman-... 16 The reason for put and call volatilities to appear different is that the implied vol has been calculated using different drift parameters than those implied by the market. Let's take everything in the model as given except the interest rate r and the volatility \sigma. For European options we have the Black-Scholes formula for put and call values V_{P,... 13 The man who grasps principles can successfully select his own methods. The man who tries methods, ignoring principles, is sure to have trouble. ~ Ralph Waldo Emerson ~ Black-Scholes made it possible for an idiot with a calculator to imagine that he was smart enough to judge the value of options ... it has always been possible to determine option value -- ... 13 Consider a more financially plausible model than Black-Scholes: one where the stock can suddenly go bankrupt due to fraud, and the volatility varies over time. Neither model is perfect, but the new one (call it SVJ) will be "less wrong". Mathematically, we no longer have the Black-Scholes SDE based on a single stochastic generator W$$ \frac{dS}{S} = \...

12

I think this slightly misses the point. Before Black-Scholes options prices were set entirely by human judgement, just like prices in many other markets are set, which is why this model was so important. Peter Bernstein has a good recollection of this kind of behavior in "Capital Ideas".

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Short Version : Two main uses I'm doing an arbitrage/statarb strategy (volatility for instance) which should not be dependant on the Delta (I'm an arbitragist). I HAVE to keep a product in my portfolio, but I don't want to be EXPOSED to it (I'm a market maker). Long Version : The goal of Dynamic Hedging is not down the line to earn risk free rate of ...

11

There is also the so-called Hakansson’s paradox that can be found in Derman's article on dynamic replication. Hakansson’s so-called paradox (Hakansson 1979, Merton 1992) encapsulates the skepticism about dynamic replication: if options can only be priced because they can be replicated, then, since they can be replicated, why are they needed at ...

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A very good book addressing such "puzzles of finance" — highly recommended! Puzzles of Finance: Six Practical Problems and Their Remarkable Solutions by Mark P. Kritzman The paradoxes that are treated here are: Siegel's Paradox. Likelihood of Loss. Time Diversification. Why the Expected Return Is Not To Be Expected. Half Stocks All the Time or All ...

10

There is a missing link to early options pricing literature which had been overlooked. Put-call parity along with static delta hedging were understood in actionable detail well before BSM and trading and risk management were accomplished through heuristic methods which indeed continued to be used after BSM. Would point to "Why we Have Never Used the Black-...

10

Options and futures were common instruments in France at the end of the 19th century. Louis Bachelier, in his 1900 thesis, derives the price of a European option when the underlying asset is normally distributed. Interestingly, he seems to have some strong opinions about mathematical finance in his introduction to his thesis: The calculus of ...

10

I feel that the best way to answer your question is to first quote your problematic idea and then carefully explain the subtle alternative. :) The derivation of the Black-Scholes PDE is based on the assumption that the price of the option should change in time in such a way that ... And my question is: Why do we assume that the price of the option has ...

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I think you are interpreting too much into the matter. The $-\frac12\sigma^2$ is just a correction term that comes from Jensen's inequality. You need this when switching from supposedly symmetric returns (normal distribution) to the skewed price process (log-normal distribution). I think there are no deeper truths to be found here.

10

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 ...

9

Except in highly unusual cases, financial PDEs lack analytic solutions. The mathematical tools used are Monte Carlo, plus the usual ones for solving PDEs on grids, almost always one of the following: Trees, for very simple cases Explicit finite differencing, for throwaway projects or very specific cases Implicit or Crank-Nicolson finite differencing for ...

9

With $15\%$ annual volatility we have $15\%/\sqrt{252}\approx0.94\%$ daily volatility. To go from $27$ to $28$ is a $1/27\approx 3.7\%$ move which is $3.7/0.94\approx 3.9$ standard deviations. For a normal distribution this is about $0.005\%$ probability which is in line with your result.

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Parrondo's paradox is a paradox in game theory that describes a losing strategy that wins in the long term. It seems the paradox is only used in textbook examples of finance and has little applications in practice, though.

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An equity represents ownership of a company and may be thought of as a derivative on the cash flow. For this reason, equities are valued through discounted cash-flow (DCF) analysis. An option is a right, though not an obligation, to buy or sell an asset at a fixed price at some point in the future. As per Black-Scholes, the value of an at-the-money option ...

8

You can look at equity as a call option on the firm. In theory this illustrates the differences between holding equity or debt. The quick and dirty is that equity holders own the firm, but only after the debt holders are repaid. If you have a simple levered firm with one outstanding debt issue, it as though the equity holders have a call option on the firm ...

8

You should look at Paul Willmott's Frequently Asked Questions In Quantitative Finance. He offers 12 (I think) ways of deriving BS and I think you'll find what you look for there. The cool thing is that you really have many different approaches; one is the classic PDE, one is done using change of measure, one is done using binary trees, and so on.... Really ...

8

Black Scholes in java? This guy, Christian Fries http://www.christian-fries.de/, has some impressive codes and a book on these topics. You can find http://www.christian-fries.de/finmath/book/index.html the contents as well as the library itself http://www.finmath.net/java/. As well as your request, you'll see LIBOR model, HJM model, binomial model, etc. ...

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Sure. The formula for vega (you probably recall) is $$v(\sigma) = S n( d_1(\sigma) )\sqrt{T-t}$$ The gaussian PDF, $n(\cdot)$, is strictly non-convex, having a local maximum at zero. There is therefore a corresponding maximum of vega occurring where the strike $K_\text{max}$ solves $$d_1(\sigma)=0$$ which works out to $$K_\text{max} = S \exp((... 8 In this scenario, the "joint dynamics" are trivially computed since the option value is a known deterministic function of stock price. For example, the mean of the option value for time \tau is$$ \mu_O = \int_0^\infty BSM( S_\tau ) p(S_\tau) dS_\tau $$which is best computed using quadrature as available in standard numerical libraries like scipy. The ... 8 Your question is not really well formulated since you do not specify at which time the delta is equal to 0.5. What you claim is in fact only true for an ATM call option at the time of maturity. In the Black-Scholes model the price of a call option on the asset S with with strike price K and time of maturity T equals$$c(t,S(t),K,T) = S(t)\Phi\left(\...

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To recover the Black-Scholes pricing equation, you should first express the standard normal cdf in terms of its characteristic function analogous to the Heston solution: $$N(x) = \frac{1}{2} - \frac{1}{\pi} \int_0^{\infty} Re [\frac{e^{-i\phi x} f(\phi)}{i\phi}] d\phi$$ where $f(\phi)$ is the characteristic function of the standard normal distribution:  ...

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