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In an incomplete market, vanilla options are independent assets like stocks or bonds. So the best way of thinking about how they are priced is the same way equilibrium prices in those markets occur: If too many people try to buy an option at a given strike then they push the price of those options up and we see that as the implied volatility increasing. The ...

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If you are predicting lower one year volatility than the options are pricing in, sell one year options on the underlying that you think will be lower and hedge the delta. If you are predicting higher one year volatility than the options are pricing in, buy one year options on the underlying that you think will be higher and hedge the delta. Your hedging ...

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In simple terms an option on one stock will have a delta between 0 and 1 meaning it will have the equivalente directional risk between 0 (or very close to zero, for example for a very OTM option close to maturity with a very low vol) and 1 (for example for a very ITM option). So you will never need more quantity of the underlying to hedge the option. As ...

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Is there a faster way to calculate the option price? With a recombining binomial tree, the terminal asset price has a binomial distribution -- as you might have expected. For a tree with $n$ steps, the probability of reaching price $S_{n,k}$ where $k$ is the number of up moves is $$P_{n,k} = \frac{n!}{k!(n-k)!}q^k(1-q)^{n-k}$$ The option price is the ...

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Why not? You can back out implied vol from the times you do have for the underlier price and then use that to price the options for the times you do not have. (This is assuming you are taking about pricing one particular options, not using options of one strike and expiry to price options at an other time, strike, and expiry.) You could even do a linear ...

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Investors buy (and hold) more puts and pay up more for them for a few reasons. First, people fear downside more than they like upside as shown by Kahneman and Tversky (1979, 1992). Second, people may not be able to recover easily (or at all) from downside in the macroeconomy. In classical finance terms, if we think crises are different from times of stable ...

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It's the interest rate component. That is more meaningful in the formula. Note that the call becomes more expensive. Think about it this way. You could buy the call and sell the put instead of being long the stock. This gives you a synthetic long position. You need to pay the market the cost of borrow (r). That makes the calls more expensive and the ...

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Usually, you would use the volatility from a fitted volatility curve or surface. Those are based on implied volatilities. You can use historical volatility, but then your valuation is likely to be off because the volatility curve/surface is not constant and at historical vol. You should use a yield curve to present value nodes. This is unlikely to make a big ...

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First of all, option contracts normally specify 100 (or in some cases, 1000 contracts of the underlying instrument) to the one option contract, so you are unlikely to encounter the 1:1 scenario you mention. Let's ignore the ratios for now, and look at the risk profile portfolio you described (1 long underlying and 1 short call). The portfolio you have is a ...

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I think you have a small misunderstanding. The hedge for a call (or put) is rarely 1:1 with stock. When you are selling this option you are actually selling the future movements of the stock and specifically the future price jumps that will happen. To hedge an option you would enter into a position that will offset the movements in price of the option ...

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I've been working on this problem a little bit lately. Unfortunately in the FX context, it's not quite as straight-forward as in the equities case, for two reasons: FX options trade OTC instead of on exchange, so you need access to broker screens to trade them (eg. on BBG) FX Options are quoted by (delta, tenor, vol) instead of (strike, tenor, price) so we ...

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This is a Structured Product. Desks issuing these products make money by selling the product for X when it is really worth X-Y (where X is usually 100%, and Y will be some number of the order 1%, depending on the complexity and maturity of the product). In the case you reference, you can see from the link that Y was 1.75%. This means that, unhedged, the ...

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I know its been a while but I would like to answer this question for all the people that arrives from now on. I hope that is okay. Let's divide the problem in two main parts. The first one is the computation of the zero coupon bond $P(t, T)$. In this case, you are using a short rate model given by the factor dynamics $dy(t)$ and the short rate dynamics $r(t)$...

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As @Lliane explains, you are actually describing a position in which the underlying is rebalanced everyday, hence the compounding effect of the leveraged ETF vanishes. Maybe a bit of modelling can be helpful to illustrate the relationship between leveraged ETFs and volatility. Let $S_t$ be the value of the underlying and $V_t$ the value of a leveraged ETF ...

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I disagree that these products are convex*. At any point in time, the ETF exposure to the underlying is linear, it's just that it changes through time. A 2x ETF will just have 2x exposure to the underlying - where the exposure is based on the nav at the point of rebalancing. Say the nav is \$100 per share, then it will hold \$200 of exposure to the ...

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I think you are missing an important point regarding who initiates options positions. We know that put options are more expensive than theory would indicate as discussed in Bondarenko (2014). Simply: put option buyers are especially motivated to initiate positions, more so than put sellers. Thus put open interest is a measure of put buyers initiating ...

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In both of your wealth equations, there is no need to subtract 1.2 from your investment in money market because you already own the option. Solve it you will get $\Delta=-\frac{1}{2}$ and $X_0=0$. Now, it means you need to short sell $-\frac{1}{2}$ shares, get $2\\\$$, and put 2\\\$$ into your investment account to guarantee that at$t=1$, you will get$1.5 ...

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Both products actually have positive convexity, they will buy more underlying (SP500) when the price goes up and sell it when it goes down. However, if you hedge every day, you will just cancel out that gamma convexity. You have to let the position run a few days if you want to trade the gamma, because it is generated by the daily hedging of the 3x etf, not ...

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I came up with a semi-equivalent solution in case this helps anyone else. If one buys/shorts the stock using a margin loan, while buying option calls/puts to protect against a rise/drop in stock price it appears provide a similar solution, allowing one to hold the position for longer amounts of time, while adding leverage and limiting exposure.

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What are common methods to compute implied volatility index? One could use VIX method on other underlying. Yes, the CBOE offers this for Apple, Google, Amazon, Goldman Sachs and IBM (see here). In my working paper here I use the CBOE VIX methodology on a sample of 268 individual equities in the same way. It also includes a comprehensive derivation of the ...

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I was thinking of simply limiting set of options that go into computation to K0 strike+ 1 option on each side (cboe.com/micro/vix/vixwhite.pdf). Not sure if this is a good idea though. Hence the question If you have a full/complete options chain then naturally to calculate the VIX you should use the VIX formula, which should be interpreted as a definition. ...

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Pathwise finite difference Gamma formula is indeed: $$\Gamma(S_0,T, dS; Z) = (dS)^{-2} \left[ (S_T^{up} (Z) - K)^+ -2 (S_T (Z) - K)^+ + (S_T^{dn} (Z) - K)^+ \right],$$ where $Z$ is a standard normal rv, and $$S_T (Z) = S_0\eta (Z),$$ $$S_T^{up} (Z) = (S_0+dS)\eta(Z) = (S_0+dS)S_0^{-1} S_T (Z)$$  S_T^{dn} (Z) = (S_0-dS)\eta(Z) = (S_0-dS)S_0^{-1} S_T (...

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Perhaps it might help if we define the difference between Brownian Motion (BM) and Geometric Brownian Motion (GBM). BM has independent, identically distributed increments while GBM has independent, identically distributed ratios between successive factors. The definition is inherited from that of arithmetic random walks, which are modelled as sums of random ...

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The short answer? In the absence of a dividend, call premium exceeds put premium by the carry cost (look at the explanation of conversions and reversals). If you're short the stock, your receive interest on the proceeds. However, you pay a borrow cost to the lender of the stock, thereby "reducing the interest rate by the borrowing costs required to ...

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I'm not a pro and my experience is with equity options not futures so take this with a grain of salt. You have the right idea to defend when the underlying's price touches a strike price. However, I would adjust before that, even long before that if these are wide strangles. With equities, you can delta neutral hedge with the appropriate number of shares ...

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To verify this, you could go all option pricing equation on this. Or you could take the path of least resistance and use an online option pricing calculator and enter zero for the dividend and zero for the interest rate and lo and behold, the time premium for the same series put and call will be identical.

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This is going to get ugly :->) There are 6 basic synthetic positions relating to combinations of put options, call options and their underlying stock (the Synthetic Triangle): Synthetic Long Stock = Long Call + Short Put Synthetic Short Stock = Short Call + Long Put Synthetic Long Call = Long Stock + Long Put Synthetic Short Call = Short Stock + Short ...

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If MSFT's share price <130, then as p→130−, the 110 call's price rockets whilst the 130 call stays OTM. So "the risk is larger" that your 100 call is assigned, while the 130 call expires worthless. Perhaps this is just poor wording. The $100 call will be assigned if it is ITM at expiration. There is no larger risk that it will be assigned. ... 0 I think this is a language issue. "Larger" means in Scenario 2 compared to Scenario 1. The phrase does not mean "the risk is larger if the stock keeps heading higher above the upper strike than if it stops at the upper strike". It means "assuming the stock keeps heading higher, the risk is larger for Scenario 2 than for Scenario 1&... 1 Instead, why not buy a B put, and sell an A put? That is a different strategy, a bear/debit put spread, equivalent to a bear/credit call spread, which is shown in your other question. If you are comparing the value of the position for a given$p$, you must take into account that a debit spread costs cash up front, whereas a credit spread pays cash up front. ... 1 Instead, why not buy an A call, and sell a B call? That is a different strategy, a bull/debit call spread, equivalent to a bull/credit put spread, which is shown in your other question. If you are comparing the value of the position for a given$p$, you must take into account that a debit spread costs cash up front, whereas a credit spread pays cash up ... 0 These are just suggested guidelines. The 75% comes from tastytrade (your link) which was founded by Tom Sosnoff. He was a CBOE market maker for 20 years and he created the option platform Thinkorswim which he sold to Ameritrade for$700 million. At a minimum, this guideline comes from his decades of experience and I wouldn't be surprised if he did ...

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