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9

Feynman–Kac Theorem: Assume that $F$ is a solution to the boundary value problem \begin{align} &F_t+\mu(t,x)F_x+\frac{1}{2}\sigma^2(t,x)F_{xx}-rF=0\\ &F(T,x)=\Phi(x), \end{align} Assume furthermore that the process $e^{-r_s}\sigma(s,X_s)F_s$ is in $\mathcal L^2$ where \begin{align} dX_s=\mu(s,x)ds+\sigma(s,x)dW_s, \end{align} then $F$ has the ...


5

Your question is not clear. What you might want to say is what distribution should the futures price follow, under the risk-neutral or physical probability measure. In this sense, it will depend on your intention. For potential future exposure, you may want to use the physical measure for the price evolution, while the distribution will depend on your model ...


4

Given one satisfies margin requirements anyone can short exchange traded options as long as local regulators permit (American retail investors at present are not permitted, for example, to trade futures options . As long as there is a market and one finds a willing counterpart nothing speaks against shorting options contracts. Some brokers might require a ...


4

First we write dynamic of ${{x}_{t}}=\ln ({{S}_{t}})$ \begin{align} & d{{x}_{t}}=({{r}_{t}}-\delta -\frac{1}{2}\sigma _{t}^{2})t+{{\sigma }_{t}}d{{W}_{1}}(t) \\ & d{{\sigma }_{t}}=a({{\sigma }_{t}},t)dt+b({{\sigma }_{t}},t)d{{W}_{2}}(t) \\ & d{{r}_{t}}=\alpha ({{r}_{t}},t)dt+\beta ({{r}_{t}},t)d{{W}_{3}}(t) \\ \end{align} Let \begin{align} ...


3

For a two-factor option pricing model with underlying variables $S$ and $r$ defined as above, if we assume there is no correlation between the two Wiener processes $W_1$ and $W_2$, one finds the generalized Black-Scholes PDE \begin{align} V_t+\frac{1}{2}\sigma^2V_{SS}+r\,S\,V_S-r\,V+\frac{1}{2}\Sigma\,^2\,V_{rr}+\kappa(\theta-r)V_r=0 \end{align} This ...


3

Answering my own question as it could be useful for others. Actually package fOptions is vectorized. The only constraint (and that make sense) is that you can't compute at the same time 2 different greeks, or mix up calls and puts. So assuming that you want to compute the delta of a set of puts, the code will be the following: ...


3

I know one article (download) that explaining how to calculate local vol surface from IV surface and also chapter 18 of this book is very good In this context. However you know that Dupire’s (1994) formula for local volatility is \begin{align} \sigma_L(k,T)=\sqrt\frac{\frac{\partial C}{\partial T}}{\frac{1}{2}K^2\frac{\partial^2 C}{\partial K^2}} \end{align} ...


3

This is a bit of an old question, but I thought I'd contribute to add more weight to to what some people have been saying. A CSO (calendar spread option) is NOT a calendar spread of options. If you read it carefully, you can see the Hull quote Max Li posted is talking about a calendar spread, not a CSO. A CSO needs to be priced the same way as a spread ...


3

First Question:This derivation is a special case of a PDE for general stochastic volatility models,described in books by Lewis (2000), Musiela and Rutkowski(2011) and others. The argument is similar to the hedging argument that uses a single derivative to derive the Black-Scholes PDE. In the Black-Scholes model, a portfolio is formed with the underlying ...


3

Fact 1: if you are not good at pricing options, of course you can create a lot of arbitrage opportunities for the rest of the market. It does not matter whether the reason is in dividends or anything else. Fact 2: if you are good in pricing options, you price the dividend effect in advance. Consider the situation of the European calls, and suppose that both ...


3

When $\sigma=0$ , the boundary condition is little more complicated: \begin{align} P_t+(r-\delta)SP_S +\alpha P_r +\beta^2\frac{1}{2} P_{rr}-rP=0 \end{align} When $\sigma\rightarrow\infty$ , we have \begin{align} P(S,\infty,r,t)=0 \end{align} When $r=0$ , then \begin{align} P_t+aP_\sigma+\frac{1}{2}b^2P_{\sigma\sigma}+\sigma S b \rho_{12}P_{S\sigma}=0 ...


2

In general, $v = \frac{\partial C}{\partial \sigma} > 0$ and $\theta = \frac{\partial C}{\partial t} < 0$. If maturity $T$ increases than $C$ increases. Suppose volatility is non-constant. Then if $T$ increases, the option value is more volatile, since the stock price is more volatile. Since $v > 0$ the option price must increase. He claims that ...


2

The quick answer is: There's no such thing as a free lunch, the no-arbitrage principle. The longer answer: To push the price down, you must short a massive amount of shares relative to float (tradeable shares). Don't forget that to have a short position, you must actually sell the shares. If you short large volume, then the market price will go down, but ...


2

One does not estimate the local volatility at a given $T$ and $K$. Instead, Dupire's formula actually gives $\sigma(T,K)$ for all $T$ and $K$. $$ \sigma^2(t_0,S_0;T,K)= \frac{\frac{\partial C}{\partial T} + (r - q)K \frac{\partial C}{\partial K} + qC}{\frac{1}{2} K^2 \frac{\partial^2C}{\partial K^2}} $$ where $C(t_0,S_0;T,K)$ are the call prices for ...


2

Generally no, because 'dividends' are already 'priced into' the options. Which means, if an ATM call cost 0.50, and stock price drops by 1.00(amount of dividend), the ATM becomes OTM, but it may still cost 0.50, because the initial price of 0.50 already factored in the dividend.


2

Of course you can sell options and you can certainly sell options on most major indices. Thinkorswim (TDAmeritrade) offers and excellent platform. Moreover, one can short options without "full" account privileges provided a defined risk trade is entered (such as an iron condor or call spread)


2

It depends on the derivatives exchange but e.g. Eurex exchange can also be used by retail investors as long as they are qualified (concerning their max. risk level) and their bank offers access to it (some at least do that).


2

Assuming zero interest, the put option has the price \begin{align*} KN(-d_2)-S_0N(-d_1), \end{align*} and delta $-N(-d_1)$. When $N(-d_1)$ units of stocks are shorted and invested in bonds, the total value in bonds is $KN(-d_2)$, which is indeed greater than the option price. However, as you have shorted $N(-d_1)$ units of stocks, your portfolio value is ...


1

The market does not follow Black-Scholes assumptions, as you clearly know : there is a skew and vol levels change. Neither does it follow any other particular known model. So when you say "dynamically hedge" you have to understand this as an approximate hedge that still leaves some significant risk. Vols will move, and not always together and in the way ...


1

There does exist some volatility control indexes (e.g., see page 37 of the S & P index methodology, which can be downloaded from http://ca.spindices.com/documents/methodologies/methodology-index-math.pdf?force_download=true), and also options on them, which are usually embedded in certain structured notes (e.g., google "Risk Aligned Deposit Notes"). ...


1

This is the Black Scholes Call Price: \begin{align} C(S, t) &= N(d_1)S - N(d_2) Ke^{-r(T - t)} \\ d_1 &= \frac{1}{\sigma\sqrt{T - t}}\left[\ln\left(\frac{S}{K}\right) + \left(r + \frac{\sigma^2}{2}\right)(T - t)\right] \\ d_2 &= \frac{1}{\sigma\sqrt{T - t}}\left[\ln\left(\frac{S}{K}\right) + \left(r - \frac{\sigma^2}{2}\right)(T - ...


1

an up and out call involves an absorbing barrier. There is no pricing formula with a reflective barrier.


1

In optimiazation system, you have to weight the price for the different maturities in a way that reflect your confidence in each data point (influenced by liquidity). One way to do so is to weight, each price by its Black-Scholoes Vega (see Tankov (2003)). So when minimazing the squared differences of the sum your weighted option prices, you can use the ...



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