36

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 ...


29

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 ...


28

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. Barone-...


23

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,...


23

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 ...


22

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) + \frac{\...


16

We assume that the short interest rate $r_t$ follows the Hull-White model, that is, the short rate $r$ and the stock price $S$ satisfies a system of SDEs of the form \begin{align*} dr_t &= (\theta_t -a\, r_t)dt + \sigma_0 dW_t^1,\\ dS_t &= S_t\Big[r_t dt + \sigma \Big(\rho dW_t^1 + \sqrt{1-\rho^2} dW_t^2\Big)\Big], \end{align*} where $a$, $\sigma_0$, ...


14

Assume the price follows a lognormal process. We can convert it into a problem of finding the probability of a standard Brownian motion particle starting from $0$ and hitting $x$ before time $t$, or its first passage time $\tau_x$ being less than $t$. This can be derived through the reflection principle. The paths crossing $x$ are exactly paired up by the ...


14

Agree with all of vonjd's points though I like to add the following: First of all, market practitioners do not read options prices or set options prices in the market, they price the option through models primarily on the basis of implied volatility. Im plied volatility is actually traded, options prices is what comes out on the other side. I know there was ...


13

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 ...


13

This is an interesting and not so easy question. Here's my 2 cents: First, you should distinguish between mathematical models for the dynamics of an underlying asset (Black-Scholes, Merton, Heston etc.) and numerical methods designed to calculate financial instruments' prices under given modelling assumptions (lattices, Fourier inversion techniques etc.). ...


12

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 ...


12

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 ...


12

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 '...


11

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} \...


11

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) ...


10

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 ...


10

Q: What does the risk-neutral price represent if the option is not replicable? In an incomplete market, there is no unique martingale measure but instead a set $Q$ of equivalent martingale measures. Consequently, there is an interval of arbitrage-free prices: $ \Big( inf_{\mathbf{Q} \in Q} E_{\mathbf{Q}}[DX], sup_{\mathbf{Q} \in Q} E_{\mathbf{Q}}[DX] \Big)...


9

Scott Mixon argues in What Does Implied Volatility Skew Measure that among all measures of implied volatility skew, the (25 delta put volatility - 25 delta call volatility)/50 delta volatility is the most descriptive and least redundant (volatility is Black-Scholes implied volatility). His paper, recently published in the Journal of Derivatives, gives a ...


9

There are many different ways a pricing model can be better : It can allow to reproduce the observed market price (Fit criterion) It takes into account a specific recognized behaviour of the underlying S, say the forward smile dynamic. If you write a product whose value is mostly derived from said behaviour, you dont want to miss that aspect. (Don't fill ...


9

Regarding conventions One thing to keep in mind in all questions about "what's right and what's not?" is that conventions don't always matter as much as one would think. When a trader marks his vols by looking up option prices on the market, he is going to mark them using the pricing model which his quants implemented. So whether he uses one convention or ...


9

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 ...


9

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: $$ ...


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.


9

A stochastic volatility model for a single risky asset can't be complete because you have two sources of randomness. But you can easily make it complete by adding a derivative whose value depends on the volatility. For example, if you add a variance swap in the Heston model then it becomes complete. This allows you to calibrate the model. But your ...


9

We assume that, under the risk-neutral measure, the stock process $\{S_t, t \ge 0\}$ satisfies an SDE of the form \begin{align*} dS_t = r S_t dt + \sigma dW_t, \end{align*} where $r$ is the constant interest rate, $\sigma$ is the constant volatility, and $\{W_t, t \ge 0\}$ is standard Brownian motion. For $0 \le t \le T$, \begin{align*} S_T = S_t e^{r(T-t)} ...


9

1. What does it mean by the vol surface is the current view of vol? The local volatility model is calibrated to vanillas prices (and equivalently their implied volatilities), which reflect the market's view of the volatility, in order to use it to use it to price other options that one will hedge with the vanillas. Where a Black-Scholes model (no smile) ...


9

From an equities perspective, there are two concepts that should not be confused in my opinion and context should make the distinction self-explicit: Forward variance swap volatility (A) Forward implied volatility smile (B) I really recommend reading Bergomi's "Stochastic Volatility Modeling" which is an excellent book for equity practitioners. The topics ...


9

Of course making money is always the key issue. That (not completely facetious) comment aside: On the practical side, in many firms IT is struggling with being clear, transparent, and intuitive in their handling of multiple curves and their associated risks. Stumbling over your own systems is an annoying way to lose money. These risks can be surprisingly ...


8

VIX is calculated from a basket of SPX options, and VIX futures expire into following expiration, e.g. September VIX futures that will expire next Wednesday will use SPX October options chain to calculate settlement value. If $B$ is the value of the basket then VIX value at expiration is $\sqrt{ B }$. Then VIX futures price is the expectation of the basket $...


Only top voted, non community-wiki answers of a minimum length are eligible