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Look the first answer of this thread: How to derive the implied probability distribution from B-S volatilities? Also many papers in Dupire volatility have your formula derivation. For example, look at (10) in http://www.javaquant.net/papers/DupireLocalVolatility.pdf

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Calendar spreads have a number of disadvantages for trading Vega: Vega in different months are generally not additive, some traders use root-time-Vega but it does not remove the additional risk. You are trading time spread not just volatility, so be careful Calendar spreads are affected by dividends and rate changes - another source of risk. A ...

1

You can construct delta and gamma neutral option portfolio, but: It won't generally stay neutral forever, so you would still have to constantly rebalance it by trading additional options (thus paying more transaction costs and creating mess in the portofolio). Anything will break the neutrality - underlying move, time passage, implied volatility change ...

1

Apart from the usual risks measured by Greeks there's risk associated with volatility dynamics. Volatility surface moves with stock movement and is usually dependant on stock price level. This risk is usually modelled by extensions to volatility models that take underlying price into account or stochastic volatility models (e.g. SABR). The way to do ...

0

The convention in the world of finance is that notionals are always nonnegative. In mathematical finance, notional could be real-valued, so let's call it MFNotional to distinguish. If MFnotional is negative, then it means short an option. Here is an example to indicate how I arrived at N1 = N/ΔK. Suppose K0 = 0.8, ΔK= 0.2, and N = 1m SGD. Then K2 = 0.8 ...

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You can find an exact algorithm with a step-by-step explanation here: https://www.dropbox.com/s/t4fq067kzx26mhw/project_paper.pdf As you can see from the URL it is an archived document because the original site is unfortunately long gone and the tool referenced in the paper with it :-( But it should be helpful anyway to understand what is going on. Notice ...

2

the answer for calculating the prices can be found here - see chapter: Black–Scholes valuation ;) The put-call parity in that case is pretty straight forward: $P=Se^{-qT}-C$. Using the results presented on the Wikipedia page in the aforementioned section this can be proved as follows $P=Se^{-qT}-C$ $=Se^{-qT}-Se^{-qT}\Phi(d_1)$ ...

2

By derivating the Black-Scholes formula in function of r (ρ=∂C/∂r), you get ρ_call=0.01TKe^(-rT) N(d_2 )=ρ_put+0.01TKe^(-rT) You can see that call prices increase (and put prices decrease) if interest rates (risk-free) increase.

3

Assume $p_i(x)$ is a payoff of one particular option. You can try to reproduce the diagram using a bunch of options with strikes on the breakpoints (underlying is useless, because its payoff can always be modelled by buy&sell of a certain call and put). Then you can create a system of k equations with n unknowns (number of each kind of option). All other ...

2

Personally I think there is no easy answer to this question. Economically a rise of interest rates often means an increased demand for capital. Banks need more money to lend to the industry thus they increase rates to entice consumers. On the other hand a demand for capital on the side of the economy often means increased market activity - companies want ...

2

"However, this way I have a -P cash flow at time 0." - yes, and this is one of the ways to hedge a forward. There is no free lunch - you are cutting risks and paying the price of a put for it. Hedging is a process of limiting your risks, and you certainly can't guarantee a positive overall cashflow, but you do guarantee you won't loose more than P. By ...

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For dividends, you could pull the info from dividend futures, or use the implied dividends backed out via put-call parity. Risk-free rates you can get from whatever yield curve you are using for discounting, or more generally (as Raphael mentioned) using whatever your specific funding curve is.

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You get the repo rate from asking your broker. Most of the time the underlying is not on "special" so you get whatever standard rate appears in your contract. You obtain dividends from BDVD or a similar service.

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In what follows, I assume no default and no frictions. A perfect hedge for your short position in a binary call would be a long position in a digital call with the same notional, strike price, and maturity date. Your goal is to super-replicate the payoff from a long digital call by a static position with a pair of co-terminal vanilla calls, one with strike ...

0

This is perhaps not a concrete solution to your problem but the space in the comments is limited :) In your setupt you are not actually pricing an option on a basket but on a dynamically allocated portfolio. Thus conventional pricing and hedging approaches won't apply. Also you are underestimating porfolio optimization algarithms. To find an optimal ...

1

I would define the weights $w_1,\ldots,w_n$ as whatever number you want and the basket given by $$B_t = \sum_{i=1}^n \frac{w_i}{W}S_t^{(i)}\ , \qquad W = \sum_{i=1}^nw_i$$ so the weights always sum to one. This doesn't make much sense, however, because you are changing the product, not a market variable. This meaning that when the weights change, the ...

2

No the discounting factor that you use for backward induction won't change. (confer here Chapter IV) This is only seems confusiong due to the mathematical formulation. Introducing continuous dividends basically adjusts your stock price (down) by discoutning the divididend (for it is paid out and thus dicreases the stock value). Your "risk-free" stock value ...

0

What you suggest is mainly true in times of stress. The shorter maturity deals are priced with larger implied volatility to incorporate the short term volatility in the market.

3

If I understand well, you have a market with 3 states: up, flat or down. You have 3 instruments: The stock The risk-free rate (50%) The option If you can create a portfolio today with these 3 instruments that can replicate de payoff of the option you have to price, then the law of one price tells you that the price of the option should be the price of ...

1

you get a volatility skew by imposing a neumann-like barrier if market makers think a stock won't surpass a certain threshold, a skew is inevitable if one were to match the pricing under a barrier with the BS formula https://en.wikipedia.org/wiki/User:Barrieroption/sandbox

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 ...

1

I think the main point of your question lies in the assumption that you cannot (delta) hedge your option. When you cannot hedge the argument for risk-neutrality breaks down and you have to use real world probabilities.

0

Arnold and Crack (2000) is an extension of the binomial option pricing model that uses real world rather than risk neutral probabilities. Our model, in both its one-period and multi-period forms, is a direct generalization of the Cox, Ross, and Rubinstein (CRR) binomial option pricing model (Cox et al., 1979). CRR do not give enough information to ...

0

Yes, Black Scholes breaks down absolutely for real market conditions at deep in the monies. Please remember that this isn't my specialty, so I can only give my observations. The reason to me seems that only variance can be specified because BS assumes lognormality. Prices certainly aren't lognormally distributed. The best fit I've found is logVariance ...

1

Most likely you are looking at bid prices which are lower that fair (theoretical) price. It is very common that bid price of an ITM option is below the lower bound as bid-ask spreads are wide. The IV of ITM call at theoretical price should match IV of OTM put at corresponding strike. If this does not happen then check your forward price, rates and dividends. ...

1

The lower bound is not just a BS-specific bound. It is a no-arbitrage bound and so if the price is lower than this, you have an arbitrage opportunity (some good explanation here). It doesn't mean it is present in the market necessarily, because mid price is not necessarily the price you can trade and when you take spread into account this is likely to go ...

0

My back testing has shown that in the case of really big market moves, dynamic delta hedging of short positions can increase risk in comparison to no hedging, and in fact cause large losses. For confirmation, see http://sisla06.samsi.info/fmse/lp/Presentations/JumpHedging.pdf which concludes just that.

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I can speak from experience that options with next to no volume and ridiculously large spreads have market makers that accept nothing short of 5% effective spreads, right below liquidation value for deep in the money, and quickly nothing for out of the money. Also, the parameters should be expected to move against your fund flows very quickly. I've found ...

0

If I understand correctly you have calculated our investors expected payoff using his probabilities to 11.177USD. He wants a three percent return so the value he assigns is 11.177/1.03 = 10.85USD. Simple as that. You can then have another argument a la Black and Scholes to show that you can replicate the payoff to another cost. If that cost is lower, your ...

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With respect to what you need, you have to consider different aspects of optimal trading: the Almgren-Chriss framework (cited by Anna, since Jim and Alex -amongst others- extended it) focus on obtaining an optimal trading rate, it is nice but not really what you need. You can nevertheless use it to plan / schedule your trading during the day. but what you ...

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