# Tag Info

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 ...

8

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 ...

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 $$\frac{\partial f}{\partial t} = \frac12 \sigma^2 x^2 ... 7 You may need to differentiate between the use of options and the pricing of options. How options are used has no bearing on the price of such options. Options can be used as leveraged investments or as insurance or as hedges. Any such use does not change the fair value derived for the option. By the way you are in fact compensated the risk premium but it is ... 6 A simple intuitive answer why the OTM Call is more expensive than the OTM Put is because of the skewness of the log-normal distribution. Think about it, what is the probability that the stock price is above 110 at expiration and what is the probability it is below 90? This should answer your question. Written in probability terms: The median of the ... 6 To recover the Black-Scholes pricing equation, you should first express the standard normal cdf in terms of its characteristic function analogous to the Heston solution:$$ N(x) = \frac{1}{2} - \frac{1}{\pi} \int_0^{\infty} Re [\frac{e^{-i\phi x} f(\phi)}{i\phi}] d\phi $$where f(\phi) is the characteristic function of the standard normal distribution:$$ ...

5

While the translation between implied volatility (iVol) and options prices is of a strictly mathematical nature (when you feed 10 market makers with the same iVol you most likely get 10 identical or close to identical option prices in vanilla structures). What is on the other side more of an art than science is how to assess whether iVols/prices trade ...

4

The put call parity is given as follows: $$c_t-p_t = S_t - \frac{X}{e^{r(T-t)}}$$ If you assume $r=0$, you get $$c_t-p_t = S_t - X$$ So, $c_t \neq p_t$. The rationale behind it is much more financial than mathematical. You have to look at the payoff on both side of the equation, and you see that both portfolio will give the same payoff at time $T$ (the ...

4

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 ...

4

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 ...

3

On a single-option basis, there is this paper comparing methods by Mark Joshi. It doesn't specifically examine portfolios, but there's a reason for that. Portfolio and scenario computations are embarrassingly parallel, so once you have achieved your most efficient available option pricer, the rest is simply about wise distribution of your computational ...

3

The first Google result seems clear enough: A seagull option is structured through the purchase of a call spread and the sale of a put option (or vice versa)....This structure is appropriate when volatility is high but expected to fall, and the price is expected to trade with a lack of certainty on direction. So, for example, you might buy the 105% ...

3

As you correctly pointed out volume has no place in the pricing models of most any option(Unless of course you create an option whose underlying or is volume in some way or if volume is used as some sort of barrier). The reason is simple: The contingent payoff and hence the probability of ending up in the money is not a function of volume. Why the market ...

3

Yes. The risk neutral and the real path share the same volatility, so the difference is in the drift rate, where the risk-neutral path drifts with the risk-free rate r. You may want to check out Paul Willmots book, esp. ch. 26, for applications.

3

$$dS / S = \mu dt + \sigma dW \\ \\ dS / S -r dt= \mu dt - rdt + \sigma dW \\ \\ dS / S -r dt= [\frac{(\mu - r)}{\sigma}dt + dW]\sigma \\$$ 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 ...

3

The following is a standard exercise that will help you answer your own question. Consider a one-period binomial lattice for a stock with a constant risk-free rate. Determine the initial cost of a portfolio that perfectly hedges a contingent claim with payoff xu in the upstate and xd in the downstate (you can do this so long as the up and down price are ...

3

Maybe this could also be a comment but I think an it is not possible to answer this question with a 'yes and here is how you do it'. It has been tried, e.g. by me for a university research project. In this research we focused primarily on aggregation of returns and the main problem was the tractability of the resulting distributions and expressions, also ...

3

In binomial tree models, there is no such a thing as a path. The binomial tree represents information about the distribution of the zero-curve at a given time and preserve enough information between different times to let you compute conditional expectations. Generally, you can not price path-dependant instruments in a model based on trees—because there is ...

3

First find minimum value of j that assures the option is in the money. This would be function of u,d,n,S,K,p,q Any particular value of j has a probability associated with it. You gave formula above. Then you need sum probabilities from j=min j needed to n. See wikipedia binomial distribution to find formula for sum. Sum is based on cdf of binomial ...

3

Two parts Real world vs risk neutral: Can we even estimate risk neutral volatility using historical data? There is a difference in distribution of the underlying stock price under the real world and risk neutral measures. Luckily, changing to the risk neutral measure does not affect volatility, only the drift. Thus, a real world measure of volatility will ...

3

No, you cannot decompose a barrier option as a linear combination of European options. You can find the derivation of the formula in Musiela & Rutkowsi pg.235, for example. But I can tell you that your formula is wrong because if $S_t<S_0e^b$ the price should be zero but in your equation this does not happen. Also note that your equation is nothing ...

3

Assuming the filtration is generated by Brownian Motion, you know that the price of a contingent claim is just the expectation under the risk neutral measure $Q$. Hence for the first one $$E_Q[\int_0^TS_udu]$$ where $S$ has the dynamic: $S_t=S_0e^{\sigma W^Q_t-(\frac{1}{2}\sigma^2-r)t}$, where $W^Q_t=W_t+\frac{\mu-r}{\sigma}$ is the Girsanov chagned ...

3

So we have the identity $$g(S,\sigma, t, C,C_t,C_S,...)=g(S, t,\sigma, V,V_t,V_S,...)$$ where $S$, $\sigma$, and $t$ are independent variables and $V=V(S,\sigma,t)$, $C=C(S,\sigma,t)$ are some unknown functions. But we can also treat the above identity formally and assume that the functions $C,C_t,C_S,...,V,V_t,V_S,...$ are themselves independent ...

3

I think you need to go even one step further than vonjd went in his reply. If liquid trading of the underlying is not possible, not only the arbitrage argument underlying risk neutral pricing breaks down. In that case there is simply no reason why the prices of those two assets (the option and its underlying) should be related in any way at all. So in my ...

3

Behavioral Finance is a wide topic, which I believe is still today underestimated by many financial professionals. How can it be used by quants? Well, in portfolio optimization it can be used "as an overlay" in the form of constraints where the optimal portfolio can not be too different from the current portfolio, because clients have behavioral biases ...

2

Here is a paper by the infamous Mark Rubinstein that should get you started. http://www.haas.berkeley.edu/groups/finance/WP/rpf232.pdf And here the trinomial tree version: http://www.ederman.com/new/docs/gs-implied_trinomial_trees.pdf by no lesser than Derman and Kani. This may also help with the actual computations: ...

2

It may be the case with certain exotics that greeks are derived analytically through approximations. In that case at certain boundaries you may get different results from such approximation over the numerical approach. Why do you not approach the numerical case similarly than most banks and hedge funds when they "shock" their options books: Simply shift your ...

2

The standard realized volatility calculation assumes an underlying model: geometric Brownian motion with constant drift and volatility. Then realized vol squared is an unbiased estimator of the process volatility squared. If you want to move beyond Black Scholes then you have two possibilities: look at a different formal model and the estimators for its ...

2

I checked out a paper which deals with out-of-sample option pricing (http://repec.kse.org.ua/pdf/KSE_dp38.pdf, especially following pp. 40-) and I believe it is a sound approach to test whether the addition of structural parameters ads value in pricing capability to more parsimonious models. Their approach is to derive additional parameters (I use the ...

2

Let $C$ be the price of the option, $S_t=S_0e^{X_t}$ be the stock price, $r$ be the risk-free rate, $K$ be the strike price, $T$ be the maturity time, $m=S_0/K$, $f$ be the density of $X_T$ and $\phi$ be the characteristic function $E(e^{i\xi X_T})$ which we assume is known.  C = e^{-rT}E((S_T-K)^+) = e^{-rT}S_0\int_{-\infty}^\infty ...

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