Hot answers tagged option-pricing
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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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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 ...
5
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 ...
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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 ...
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FDMs represent PDEs over a simple grid shape; the different implementations are just different recurrence relations to approximate the solutions to the PDE between boundary values (e.g., for options pricing, $T=[t_\mathrm{now},t_\mathrm{maturity}]$ and $S=[\mathrm{deep\_itm},\mathrm{deep\_otm}])$.
FEM is a general name for a lot of different ...
4
First we must define what we mean by implied volatility. Let $c_{BS}(t,S(t),K,T;\sigma)$ denote the price of the call option with strike price $K$ and maturity $T$ in the Black-Scholes model with the volatility $\sigma$ (emphasized in the argument). Furthermore, let $c_{MA}(t,S(t),K,T;\sigma)$ denote the corresponding price on the market.
The volatility ...
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The use of risk-neutral measure is based on the ability to arbitrage away the instantaneous risk of contingent claims. Although for forward contracts the hedge quantity is 1.0, in the general contingent claims case we must assume it varies instantaneously with the market state.
The Girsanov Theorem tells us what the difference is, instantaneously, between ...
4
He is approximating $C(S_{t+1})$ around $t$:
$$C(S_{t+1})=C(S_{t}) + \frac{\partial C(S_{t})}{\partial S_{t}}(S_{t+1}-S_{t})+\frac{(S_{t+1}-S_t)²}{2}\frac{\partial^{2}C(S_{t})}{\partial S_{t}^{2}} + ...$$
In addition, he takes the time value of $C(S_t)$ into account
(and I look only at the time contribution here):
$$C(S_{t+1})-C(S_t)=\Delta ...
4
The put call parity is given as follows:
$$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 ...
3
I don't have the original reference here but what about Merton's Jump-Diffusion model?
Stochastic Calculus for Finance II: Continuous-Time Models by Steven Shreve has some chapters about modelling with jump-processes. I think it is a slightly easier introduction to the topic than Cont/Tankov.
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a. is correct, but you should derive it using appropriate logic, not just guessing the answer. Ie the drift of discounted stock should be 0. Define a bond dB = rBdt. d(S/B) should have no drift. This can help you find the correct mu. You can find the sde for S/B using two dimensional ito
b. don't really know about market price of risk.
c. In this case the ...
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I think that you are missing one key condition on the call prices that I would say is standard, namely that the call prices should be bounded below by an "intrinsic" value. Specifically, we would expect $C(K) \ge (S-e^{-rT}K)_+$, and this can easily be seen to yield a static arbitrage if violated. This condition (in a slightly different form) can be found ...
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There are two excellent choices for implementing prediction markets:
(1) Use book orders that stand until filled, just as intrade.com does.
(2) Use an automated market maker (like Robin Hanson's) that stands ready to make trades.
The book orders model is very simple to implement, but can suffer from very wide Bid/Ask spreads. And, it can be tough to bet ...
3
I think you should not just ask what the implied vol is of a basket of equity derivatives but you should aim to generate a volatility surface. A spot implied vol gives you nothing to work with. What you need is an implied vol surface in order to understand the smile and skew effects when you quote basket options in the market and/or as price taker. Take a ...
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Implied volatility skew is simply collection of implied volatilities on the same underlying instrument for a given expiration. Term "implied volatility skew" is only loosely connected to statistical definition of skewness.
Implied volatility surface is the collection of implied volatilities on the same underlying for several expirations.
If BS formula were ...
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.
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As far as PDEs (deterministic) are concerned we have the notion of a "strong solution" (directly solving the differential operator in the strong formulation of the problem) and the "weak solution" that deals with a weak formulation of the problem.
For the strong formulation, finite differences are the way to go since they are the natural discretization of ...
2
Fourier Transform seems a good method for option pricing by take advantage of Fast Fourier Transform technique, such as the following paper written by Peter Carr and Dilip B. Madan:
http://portal.tugraz.at/portal/page/portal/Files/i5060/files/staff/mueller/FinanzSeminar2012/CarrMadan_OptionValuationUsingtheFastFourierTransform_1999.pdf
2
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 ...
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Jim Gatherals Book deals with the models you mention and gives an intuitive understanding about calibration and issues that arise. Mostly basic stuff, but very useful if you're just starting out. Also very understandable without an extensive math background.
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Let $\beta$ denote the relative machine precision, usually $\beta = 1E-16$. Assume the you can evaluate the value V up to precision $\alpha$. The best you can get is $\alpha = \beta \cdot V$ if $V$ is not underflow or overflow.
Then you can calculate the finite difference up to precision $4 \alpha / \epsilon$ (the 4 might be a rough estimate, but it comes ...
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Once you have slogged through all the relatively useless theoretical literature, this paper is a rediscovery (and pretty good write-up) of how basket option pricing is really done in serious quant packages at the big banks.
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Have you looked at using Laplace in a Monte Carlo simulation? Here is how you price American style options within a MC framework:
http://www2.math.uu.se/research/pub/Jia1.pdf
and the Longstaff, Schwartz paper: http://escholarship.org/uc/item/43n1k4jb#page-1
Regarding the discretization of a process that draws its random variables from a Laplace ...
2
I found that sometimes going back to the source gets me the farthest. Here is what you are probably looking for:
http://www.math.ku.dk/kurser/2005-1/finmathtowork/ODD.pdf
Discrete dividends that have not been declared yet need to be estimated. Estimating and updating dividend expectations is part of the job of every single stock vol trader. You are asking ...
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If you are able to derive quantities in the LMM with unconditional expectation as functions of $L(0,T)$ and $\lambda(s,T)$ for $0 \leq s \leq T$ then expression of the time $t$ conditional expectation is exactly the same function in $L(t,T)$ and $\lambda(s,T)$ for $t \leq s \leq T$ - just replace $0$ by $t$. This is due to the fact that the model is ...
1
There is a market accepted standard to translate vanilla option prices to implied vols and backward which is the Black Scholes (BS) options pricing formula. There is no ambiguity here, everyone knows of the deficiencies of BS yet its what people use to translate between iVols <-> prices.
The numerical difficulty I see is to make more realistic ...
1
Vomma, or Volga or DvegaDvol is the second derivative of the option w.r.t volatility. In other words, it is the sensitivity of vega to changes in implied volatility.
A simple way to remember how Vomma is computed in the Black-Scholes framework is as follows:
$$\frac{\partial^2 C}{\partial \sigma^2} = Vega \left(\frac{d_1d_2}{\sigma}\right) $$
1
That looks about right
Volga: S*Sqrt(T)*d1*d2*N'(d1)/σ
Edit: I provided a link to a pdf of the following book:
http://books.google.co.jp/books/about/The_complete_guide_to_option_pricing_for.html?id=tuoJAQAAMAAJ&redir_esc=y
but took it off because it was a scanned version and I was not sure it infringes on copyrights.
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For part (d), instead of using Girsanov's theorem as phubaba suggested, I believe that we can state directly that the price is
$$V_t = e^{-r(T-t)} \mathbb{E}^Q \left[ u(S_T-K) \middle \vert \mathcal{F}_t \right],$$
where $u$ is the step function, $Q$ is the risk-neutral probability measure, and $\mathcal{F}_t$ is the filtration at time $t$, since the value ...
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The strike price provided by OptionMetrics is simply strike x 1000, so in order to calculate moneyness of the option you have to divide the strike by 1000 and then proceed in a standard manner.
In terms of filtering the moneyness of the option, there are few options. The easiest is using VOLATILITY_SURFACE table in the OptionMetrics database.
Amount of the ...
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