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18

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) ...


18

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 ...


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 / ...


15

You may want to look at Chapter 5 - "The Quest for the Option Formula" from the Derivatives. The book is available online for free and it has a very descent review of approaches that were used 20-30 years before the Black-Scholes-Merton equation.


14

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 ...


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

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 ...


12

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

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".


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 ...


10

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} = ...


10

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.


9

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 ...


9

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 ...


9

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 ...


8

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

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 ...


8

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 ...


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

It is not possible for what most people think of as options, but there are classes of options for which an ODE is used. For a nontrivial example, think of perpetual American-exercise options. Because of perpetual exercise, the option value is independent of time. In place of the Black-Scholes PDE $$ \frac{\partial f}{\partial t} = \frac12 \sigma^2 x^2 ...


7

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 ...


7

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 ...


7

Two quick points: Recall that the derivation involves continuous time and $(t, t+\Delta t)$ arguments---so the granularity is (at the margin) infinite. And hence time zero does not really get reached until we actually are at expiry. Generally speaking want the number of business days, not calendar days, and holidays do matter. So one generally uses the ...


6

Fundamentally this is no different from other simulation-based estimation---see this little experiment in R: R> set.seed(42) R> rowMeans(replicate(200,sapply(1:6, +> FUN=function(x) mean(rnorm(10^x)), simplify=TRUE))) [1] -2.47827e-02 -9.46800e-03 2.38226e-03 -1.08650e-03 9.41395e-05 1.06759e-05 R> We are calculating the mean of ...


6

You may want to look into these two open source projects: QuantLib which is aimed at providing a comprehensive software framework for quantitative finance. This is written in C++. JQuantLib the 100% Java implementation based on the first project.


6

A discrete-time model only works in no-arbitrage land with discrete asset values. Furthermore, the number of allowable asset values per timestep is limited by the number of available securities. The tree is the classic example of this. Binomial trees "work", but if you make a one-step trinomial tree, you will find that you can no longer form a risk-free ...


6

Well as far as I know it is a really hard but interesting question. Asymptotics of smile in the strike direction is not known in a model free way as far as I know. I think I can remember that nevertheless you have upper and lower bounds if you know something about the underlying dynamics and especially the first moment of explosion. I can't remember the ...


6

All this assumes the absence of arbitrage: As you probably know without dividends it's is never optimal to early exercise a call option on a non dividend paying stock because then the time value is lost, if $r$ is non-negative. Early exercise of an American put option can be optimal if the option is sufficiently far in the money and $r > 0$. Then you ...


6

This claim is false. A deep in-the-money option with very high volatility can have both large time value and high delta. As a counterexample, consider a call option with: K = 100 (strike price) S = 300 (spot price) r = 0 T = 1 vol = 150% This gives a Black-Scholes value of approximately \$230, so the time value is \$30, but the delta is 93.1%.


6

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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