# Tag Info

7

Assume you buy a plain vanilla call option at the price $V$ and the spot $S$. You immediately delta hedge buy selling $\partial V / \partial S$ units of the underlying asset. The underlying asset now instantaneously jumps form $S$ to $S' = S + \Delta S$. The new value of the call option is $V'$. Your total p&l is \text{P&L} = V' - ...

6

Yes, in the sense that it is assumed that the delta will be passed between participants at time of execution. Not necessarily. A non delta neutral trade may be used for speculation , or for hedging.

4

Using Taylor polynomials of 2nd order:$$V(r+h)\approx V(r) + \frac{\partial{V}}{\partial{r}}h +\frac{1}{2}\frac{\partial^2{V}}{\partial{r}^2}h^2$$ $$V(r-h)\approx V(r) - \frac{\partial{V}}{\partial{r}}h +\frac{1}{2}\frac{\partial^2{V}}{\partial{r}^2}h^2$$ The sum of the previous 2 equation will give us gamma as: $$Gamma = \frac{\partial^2{V}}{\partial{r}^2} ... 4 The key here is to observe that the volatility at the time the option is written is not exactly equal to the volatility that the markets actually experience during the option's lifetime. The seller will price the option according to her best estimate of future volatility over its lifetime, but will always prove to have been too high or too low. More ... 3 One way to check your hedging strategy would be to calculate its PNL distribution (histogram) of ( hedging strategy + option) . Mean PNL should be around 0, and shape should look like gaussian. Also you can check https://www.pricederivatives.com/en/simple-example-simulation-of-delta-hedging-with-python/ 3 Lets give it a rough go then. Two assumptions. (1) We disregard repo (to lend the stock you may want to short) or financing on your hedged position. And (2) We assume no trading of the gamma on the option. Then I would assume the break-even is equal to the expiry should be equal to... (CALL) paidPremium/(1-hedgedDelta) + callStrike (PUT) putStrike - ... 2 You could, for a particular vol, time to maturity, spot and strike have a delta neutral position by buying a certain amount of the put. But consider what would happen if the spot goes up: The delta of the call will increase and the delta of the put will decrease. Plus, you just spent an additional premium compared to delta hedging with the underlying. ... 2 Your portfolio composition is not clear. To simplify, we assume that it consists of units of a stock and options on this stock. What you can do is to sell 4000 units of options that will bring it to gamma neutral, and then to balance the delta, you can buy 2,400-450=1,950 units of the stock. 1 You are generally correct in thinking that this strategy should make money most of the time, however I would warn you to be very careful with strategies that earn a small amount of money most of the time but lose many multiples of that when things go wrong. These are examples of “picking up pennies in front of a steamroller”. The options market is full of ... 1 Chapter 5.5.2 (Hedging with One Stock) paragraph that includes (3) does not assume that the value of the payoff at any t is Markovian, that is, it is a function of S(t) only, so there is no "delta" as in (1) to use. Summary 6.7 has the answer to what Shreve does want to say about formula (3) (in particular the three paragraphs I highlighted; ... 1 Under your hypotheses, the implied volatility at which you close the trade out will be the forward volatility \sigma_3 where \sigma_3<\sigma_2, so you will make a loss on that. This loss will offset the theoretical gains you have made for the first 15 days of gamma hedging. 1 This discussion has also confused me slightly, so I will add something that is possibly clarifying, although most likely will not be. It is also a reminder that I need to stop programming and brush up on options pricing theory. The Black Scholes hedge portfolio is given by:$$ \Pi_t = \frac{\partial V}{\partial S}(t,S_t)S_t + \left[1 - \frac{\partial V}{\...

1

So @Alex, I think answered your question, but I'll give a go as to further clarify. First of all, there are the mechanics of trading a delta neutral portfolio. The assumptions of Black-Scholes are continuous hedging. This implies an obvious constant attention and in addition, access to fractional share (or contract) trading. And, of course, no inter- or ...

1

The two formulations seem to be exactly the same. If I take the equations from the first method: $w_{D1}*\Delta_{D1} + w_{D2}*\Delta_{D2} = -2$ $w_{D1}*\Gamma_{D1} + w_{D2}*\Gamma_{D2} = -3$ And substitute for delta and gamma of the two options: $-w_{D1}+ 5 w_{D2}= -2$ $2w_{D1} -2w_{D2} = -3$ which after shifting the constants to the left becomes ...

1

Assuming that, after buying the options, you will no longer trade in the underlying or options, it only makes sense to buy a put, in addition to the call, if you want to bet that the price of the underlying (PU) will experience a big increase or a big decrease. Note that you will be paying the premiums of both the call and of the put. Therefore, the increase ...

1

I'll put here the answer provided in a comment by @dm63 (thanks by the way): The requirement that the portfolio earns the risk free rate is something we are imposing in order to calculate the option price.

1

On top of my head, there are two reasons. One is that the underlying may not be accessible, while you can buy/sell options on exchange or OTC. The other (more important) reason is your position on other greeks (gamma, vega, etc). With appropriate options you can adjust both your delta and other greeks.

1

Commonly used procedures are to hedge; when a 1 SD move has happened, or when your delta position exceeds some risk limit, or once a day, or based on your desired delta position. All are used. I personally prefer (2).

1

let $\frac{\partial C}{\partial S}=\delta_c$ let $\frac{\partial^2 C}{\partial S^2}=\Gamma_c$ let $\frac{\partial C_0}{\partial S}=\delta_0$ let $\frac{\partial^2 C_0}{\partial S^2}=\Gamma_0$ we want $\frac{\partial V}{\partial S}=\frac{\partial C}{\partial S}=\delta_c$ and $\frac{\partial^2 V}{\partial S^2}=\frac{\partial^2 C}{\partial S^2}=\Gamma_c$ ...

1

Rev., I am an options trader that is rarely, if ever, delta-neutral because I am using the options to make directional bets on the move of the underlying instrument. I expend virtually 99% of my analysis on the potential direction, magnitude and resistance points of the price-move in the underlying. About 1% of my analysis is on the behavior of the ...

1