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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 ...

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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 ...

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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 / ...

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This is just to expand a bit on vonjd's answer. The approximate formula mentioned by vonjd is due to Brenner and Subrahmanyam ("A simple solution to compute the Implied Standard Deviation", Financial Analysts Journal (1988), pp. 80-83). I do not have a free link to the paper so let me just give a quick and dirty derivation here. For the at-the-money ...

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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 ...

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This one is the best approximation I have ever seen: If you hate computers and computer languages don't give up it's still hope! What about taking Black-Scholes in your head instead? If the option is about at-the-money-forward and it is a short time to maturity then you can use the following approximation: call = put = StockPrice * 0.4 * ...

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I have worked on this topic extensively (pricing and calculating IV in production) and believe can offer an informed opinion. First of all Mathworks - the company that creates Matlab is not a trading firm so you should probably not rely on their advice so much. There are few closed form options pricing models, and all have practical shortcomings. ...

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Actually there are more than just ideas and hints concerning this topic. There is an intuitive model and solution to your question already using machinery of option theory. But don't worry, it's not a surprise that you didn't find any useful literature in your search because the proposed solution actually comes from a very different topic. In addition to ...

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The price of a binary option, ignoring interest rates, is basically the same as the CDF $\phi(S)$ (or $1-\phi(S)$ ) of the terminal probability distribution. Generally that terminal distribution will be lognormal from the Black-Scholes model, or close to it. Option price is $$C = e^{-rT} \int_K^\infty \psi(S_T) dS_T$$ for calls and $$P = e^{-rT} ... 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 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 ... 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. 8 There is a simple solution if there is no drift, as the probability p(x,t) obeys a simple diffusion equation: \mathrm{d}(p)/\mathrm{d}t = \frac{1}{2} \sigma^2 \frac{\mathrm{d}(\mathrm{d}(p))}{\mathrm{d}x^2}, here x is the price difference \text{price}(t) - \text{price}(t=0). Of course there is a simple solution to the diffusion equation (using ... 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 The Black-Scholes 'normal-vol' formula leads quickly to a similar approximation to the one described by olaker. Click here for a paper which contains a formal derivation of the call and put prices based on a normal model (ie a brownian motion rather than a geometric brownian motion). The formula for the call price is:$$\text{Call} = (F-K)N(d_1) + ...

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I believe this is a nice paper for you to start with. Check out what references it cited and who cited it. Markov Chain Monte Carlo Analysis of Option Pricing Models "Use the Markov Chain Monte Carlo (MCMC) method to investigate a large class of continuous-time option pricing models. These include: constant-volatility, stochastic volatility, price ...

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You need to compute your greeks as finite differences, but the full procedure may be pretty tricky. I will use vega $\aleph$ as the example here. Let's begin by designating your Monte Carlo estimator as a function $V(\sigma,s,M)$ where $\sigma$ is the volatility as usual, $s$ is the seed to your random number generator, and $M$ is the sample count. To ...

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This is in fact a tricky matter. As you say one way is to calculate delta by an analytic formula, i.e. calculate the first derivative of the option pricing formula you are using with respect to the underlying's spot price. The second way is to do it numerically, i.e. change the spot price by a small value $dS$, calculate the value of the option and then ...

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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 ...

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The skew is almost always bid for puts on the stock market. When stocks go down, people tend to panic and volatility goes up as a result. Since the puts get more vega when the market goes down, they trade at higher vols. Read up on stochastic volatility for a more in-depth explanation.

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The main component of that option premium is (forward-looking) volatility $\sigma$. The very simplest formula you could use for ATM options is the Bachelier model $$\text{Call}_T = \sigma S \sqrt{\frac{T}{2\pi}}$$ where the time to expiration is $T$ and $S$ is the current underlying price. This formula is "wrong" strictly ...

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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 ...

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I do not know such a software - but we can think about the code. There are tow points which you have to define properly: which assets (correspondently, payoffs) are you allowed to replicate the complicated option? as barrycarter has already asked - what should be the form of the input? Further procedure should be quite easy. You are trying to find a ...

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Note first that this key equation is only assumed to hold true under some extra assumptions. Typically those assumptions are taken to be about absence of arbitrage, though it is possible to weaken them somewhat if you are willing to consider portfolio arguments or collectively agreeable objective function. Anyway, the argument is this: if all the risk can ...

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In addition to what vonjd already posted I would recommend you to look at the E.G. Haug's article - The Options Genius. Wilmott.com. You can find some aproximations of BS not only for vanilla european call and put but even for some exotics. For example: chooser option: call = put = $0.4F_{0} e^{-\mu T}\sigma(\sqrt{T}-\sqrt{t})$ asian option: call = put = ...

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You are typically interested in evaluating $E\left[ f(X_T)-f(\bar{X}_T^{(n)}) \right]$ (refered as the weak convergence) $X_t$ the solution of the sde : $dX_t^x=b(X_t^x)dt+\sigma(X_t^x)dW_t$ $\bar{X}_t^{(n)}=b(\underline{t},X_{\underline{t}}^{(n)})\cdot (t-\underline{t})+\sigma(\underline{t},X_{\underline{t}}^{(n)})\cdot (W_{\underline{t}}-W_t)$ is your ...

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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.

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As with most derivatives that have early exercise, you are going to want to price this using a grid scheme. I have priced callable loans with floors using the Generalized Vasicek model at my old hedge fund, and it is fairly easy to handle. As a matter of fact my students are doing that very problem as homework this week, and my reference implementation ...

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Theta decay doesn't depend on the in the moneyness. A 70 delta call and a 30 delta call have very close theta decay at any given moment. They are slightly different because of skew with 70 delta put having slightly bigger theta. Theta is the decay of extrinsic value. In practical trading, you can assume your decay distribution (using your graph is fine) ...

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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 ...

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