# Tag Info

41

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 ...

31

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 ...

27

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 ...

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I give you a brief outline about some key properties of Lévy processes. Lévy processes have stationary and independent increments but do not necessarily have continuous sample paths. In fact, Brownian motion is the only Levy process with continuous sample paths. Some Lévy processes (e.g. Poisson process) have single, rare but large jumps (finite activity) ...

17

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 ...

17

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 $a$, $\sigma_0$, ...

17

Life Without a Risk-Neutral Measure How would we price assets without the measure $\mathbb Q$? Well, we would start with some version of the Euler equation $P_t=\mathbb{E}_t[M_{t+1}P_{t+1}]$, where $M$ is the stochastic discount factor (SDF). This equation holds under very weak assumptions (law of one price) and uses real-world probabilities. So, we take the ...

15

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 ...

15

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.). ...

14

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 ...

11

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: ... 11 This is a loaded question. Ross' recovery theorem has both flaws and insights. The single answer thus far did a great job of addressing the flaws from an economics perspective. No one questions that the math is wrong: it is correct. Here is a mathematical insight from Ross' work. Abstracting from the finance and economics, the purely probabilistic content of ... 11 Think of moving volatility in the other direction. As volatility approaches zero, any call strike strictly smaller than the ATM strike, K<K_{ATM}, will have zero probability of ending in the money, and the corresponding option value will be zero. An infinitesimally small change in stock price will not move K past K_{ATM}, so the option value ... 11 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) ... 11 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 ... 11 There are several interpretations for \Phi(d_1) and \Phi(d_2). As you know, \begin{align*} C(t,S_t)=S_te^{-q(T-t)}\Phi(d_1) -Ke^{-r(T-t)}\Phi(d_2). \end{align*} Exercise Probabilities We can show that \begin{align*} \mathbb{Q}_S[\{S_T\geq K\}]&=e^{-q(T-t)}\Phi(d_1), \\ \mathbb{Q}[\{S_T\geq K\}] &=e^{-r(T-t)}\Phi(d_2). \end{align*} Thus, \Phi(... 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)... 10 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)} ... 10 It is, of course, possible to price such a contract in a no-arbitrage market. Indeed, if f is a sufficiently smooth function, then you can price all contracts paying f(S_T). Note that your specific payoff has no optionality and that the payoff may be negative. Bakshi and Madan (2000) discuss the economic meaning of a derivative paying \cos(S_T) in the ... 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 ... 9dS / S = \mu dt + \sigma dW \\ \\ dS / S -r dt= \mu dt - rdt + \sigma dW \\ \\ dS / S -r dt= [\frac{(\mu - r)}{\sigma}dt + dW]\sigma \\$$Then, Girsanov tells us that, as long as the risk premium is bounded from below, we can write [\frac{(\mu - r)}{\sigma}dt + dW]\sigma as \sigma d\tilde{W} where \tilde{W} is simply another brownian motion with ... 9 Recently I came across an interesting intuitive explanation: Suppose driftless market. Market price is 105, strike price is 100. Call option costs 8, put option 3. (intrinsic value of call is 5, time value of both is 3) Now the market starts drifting upwards massively. You say, that you would probably price call higher, e.g. at 10. Would you also price put ... 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 You're missing the point of the risk-neutral framework. The idea is as follows: assume the real probability measure called \mathbb{P}. The thing is, because investors are not risk-neutral, you cannot write that v_0 = E_\mathbb{P} [ e^{-rT} V_T]. Using the Fundamental Theorem of Asset Pricing, you know that if the market is arbitrage-free, then there ... 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 Very simply, Ross' framework assumes a great deal to extract the true pricing kernel. Time homogeneity, additively separable state dependent utility, (discrete time Markovian structure - though these have been relaxed.) In particular, there are two schools of criticism, one is that time homogeneity makes little sense in the real market. In fact, the Recovery ... 9 Options have an asymmetric payoff profile: The payoffs are zero for almost all cases and positive else (as we well know). If the option is OTM, most of its payoffs are zero. A rise in volatility will hence increase the likelihood for instead positive payoffs from a change in the underlying price (i.e. delta increases). If the option is already ITM, most(... 9 A swaption in which the underlying swap starts at a date materially after the expiration date is called a midcurve swaption. The implied volatilities of these can not be obtained from the regular swaption surface. Market makers calculate implied volatilities for midcurves in s number of ways. One popular method is to compute the volatility of the forward ... 9 Your d_1 is wrong - the minus sign is wrong. It should be:$$ d_1 = (\ln(S_0/K) + \sigma^2/2)/\sigma.  See e.g. here.

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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 ...

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