# Tag Info

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Short Version : Two main uses I'm doing an arbitrage/statarb strategy (volatility for instance) which should not be dependant on the Delta (I'm an arbitragist). I HAVE to keep a product in my portfolio, but I don't want to be EXPOSED to it (I'm a market maker). Long Version : The goal of Dynamic Hedging is not down the line to earn risk free rate of ...

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Actually there are more than just ideas and hints concerning this topic. There is an intuitive model and solution to your question already using machinery of option theory. But don't worry, it's not a surprise that you didn't find any useful literature in your search because the proposed solution actually comes from a very different topic. In addition to ...

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I do not know such a software - but we can think about the code. There are tow points which you have to define properly: which assets (correspondently, payoffs) are you allowed to replicate the complicated option? as barrycarter has already asked - what should be the form of the input? Further procedure should be quite easy. You are trying to find a ...

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I think there is an error implicit in your question. Dynamic delta hedging, even assuming the underlying process is a continuous martingale and trading entails zero transaction costs, only eliminates the directional risk. A number of residual risks remain, most notably volatility risk, embodied in both the gamma and vega. A dynamically hedged portfolio of ...

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You have to differentiate here between the risk-taking and the market-making side. As a risk-taker, like e.g. a hedge-fund, you are right, you could just buy the bond! But as a market-maker you sell these options but don't want to bear the risk, so you have to counterbalance it. You could of course counterbalance it with another option which would be the ...

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Specifically, we have a generic conditional claim, $C$, that is a function of the diffusion process for the underlying, $S(t)$, and time $t$ so $C = C(S(t), t)$. As you pointed out, $C$ is an Ito process becuase it is a function of a stochastic process so we use Ito's Lemma to determine how the contingent claim varies as a function of the diffusion process $... 5 While another user touched on the hedging argument in order to reconcile your intuition with the correct value of the option he went off track (imho). I like to focus entirely on the hedging issue because it is key in understanding the differences in intuition and the fair price of such option. Unfortunately I have hardly ever found a simple 1-2 paragraph ... 4 You are treading controversial waters. It's hard to summarize, but at the risk of oversimplifying, there are three broad schools of thought: "Linear Models": Classic Examples are a string of papers from Jasmina Hasanhodzic and Andy Lo at MIT (scholar.google.com should give you plenty). For similar work related to Mutual Funds that you may be able to ... 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 ... 3 While not really an answer, here are my thoughts on the problem. For starters, I would approach the problem as one of whether a portfolio which is constantly rebalanced over some horizon can be replicated using vanilla options. Obviously the portfolio will have to be rolled over when the options reach expiration. Then the rest, such as payoff diagram and ... 3 Perhaps I don't understand your question correctly but one Synthetic Long Futures Construction equals "Buy one ATM Call" and "Sell one ATM Put" (see e.g. here: http://www.theoptionsguide.com/synthetic-long-futures.aspx) 3 use a vertical spread and delta hedge it. http://www.wilmott.com/messageview.cfm?catid=3&threadid=65988 3 I don't have much experience in the matter, but I've been doing some related literature research recently and I think these links can be helpful: A rather recent study from CME A (possible a bit biased) report by BlackRock A report by Lyxor (asset manager affialiated to Societe Generale) 2 There are already quite a lot of softwares that do that. Quite expensive however for most of them. Then it depends whether you're interested into a trading software (trade capture and stuff) or a pricing engine. Trading softwares : murex, misys summit, calypso ... provide tools to structure deals and value them. Then they are processed front to back. ... 2 When performing a tracking error optimization, you will obtain the same result by using the tracking error squared, which is just the variance of the relative portfolio weights. This would be just finding the minimum variance portfolio, but with conditions on the weights. For instance, it would be equivalent to instead set up the variance minimization ... 2 Let$0 \leq T < U$. Consider a European call on a U-Bond (Zero-coupon bond maturing at time U) with time of maturity$T$. What you do is that you hedge the call option with the aid of the U-Bond and the T-Bond. I could go in to more details on how to do this in particular models, but I would basically just write the same things as in this book: Interest ... 2 As Brian B states above the short answer includes Money market accounts, swaps and zero coupon bonds among other instruments. Lets say we have an interest rate derivative that we need to value via replication. Now if we think of what we mean by a replicating portfolio its clear that the main ingredient needed is to match the pay structure\payout of the ... 2 The concept of replication is indeed applied to IR products, after all they are also hedged in practice. However, in the equity world we start with the replicating portfolio and then arrive to the pricing formula. In contrast, for IR products we employ a convenient numeraire which helps us to arrive at the pricing formula directly (in a non-constructive and ... 2 You are correct that showing the self-financing condition for the BS-portfolio is not as straightforward as one may think: A portfolio$V_t(\alpha_t,\beta_t)$(for stock$S_t$and zerobond$B_t$) is self-financing iff: $$V_t=\alpha_tS_t+\beta_t B_t$$ It further implies $$dV_t=\alpha_tdS_t+\beta_tdB_t$$ To replicate a derivative$C(S_t,t)$by a self-... 2 Regarding the dividends: In order to avoid jumps on ex-dividend date, you can make the simplifying assumption that dividends are paid continuously and adjust the returns of the assets. The size of dividends could be estimated from historical data or can be set proportionally to the asset price. 2 The state price vector are the prices of securities which pay \$1 if and only if that state of the world occurs. This is just a question of being able to replicate the payoffs $$\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}$$ with payoff vectors $\vec{b} = [1,1,1]^T$ and $\... 2 That's impossible. Since neither the vanilla options nor the underlyings have any exposure to the correlation, no portfolio of these instruments can either. 1 I think the title here is misleading. Let's go back to the BS world with$r=0$to$a(S_t)=S_t \sigma.$In that case, all you are saying is that you can replicate a call option by holding$N(d_1)$units of stock at time$t.$What does this have to do with the second equation? I am guessing that this is the price process of an asset of nothing option with ... 1 if you let the implied vol depend on K you get two terms the first is$N(d_2) $but you get a correction term which is the slope times the vega $$\frac{\partial C}{\partial \sigma} \frac{\partial \sigma}{\partial K}.$$ (see eg my book) 1 these kinds of questions usually require careful attention to details: if it's a hw question of some kind, consult shreve's lecture notes, he has a whole section on this precise topic in all its glory. as for intuition, since holding$\frac{\partial \xi}{\partial S}$at any point in time eliminates the dW term, in the context of a discrete time period model ... 1 the reason the derivative is worth that much is because it is replicating it with stock. in your example, you have a security that in 99% cases gives you \$10 and else \$0. However, what is the underlying that you use to hedge? Nothing. Therefore it must be worth the expected value or 9.90 1 For forward libor models, one can hedge interest rate options by using bonds. (Note that forward libor is a tradeable security under the forward measure). See http://www.columbia.edu/~mh2078/market_models.pdf For affine yield models (like Vasicek or CIR models) the inverse problem is the most useful. Given an interest rate process, I can compute a ... 1 It is better to use a factor model, if one is available. Are you asking this question because you don't have access to one? Also, what is the nature of the asset you want to track? Is it an index or a single security? What asset class? What risk factors is it exposed to (e.g. interest rate and credit risk vs. stock market volatility and other equity ... 1 definition of a variance swap is$ \int^{T+\Delta}_T \mathbb{E}_t[v_s] ds $where$v_s$is the variance and$\mathbb{E}_t[v_s]$is the expectation of the variance of time s at time t. therefore, pnl is:$ (\int^{T+\Delta}_T \mathbb{E}_t[v_s] ds - \int^{T+\Delta}_{T} \mathbb{E}_{t-\delta}[v_s] ds)*d\delta \$

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A variance swap has a set of fixing times, and the volatility between those times has no specified effect. Therefore you end up wanting to apply a model. For a model-free approximation, though, your formula works up to a constant.

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