# Tag Info

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The proof is relatively long, so I focus on displaying the reasoning and major steps. We work on a Black-Scholes model. Without loss of generality, we focus on an option with strike $P$ to buy at $t_e$ a European call option expiring at $T$, written on a stock $S$. Expectations are always taken with respect to the risk-neutral measure $Q$ unless otherwise ...

6

You're right. Euler's equation states $$p_t=\mathbb E^\mathbb P_t[M_{t+1}X_{t+1}],$$ that is pricing under $\mathbb P$ requires you to know the stochastic discount factor (SDF, aka pricing kernel) $M$. $M$ is (typically) found in a general equilibrium setting, depending on the marginal utility of investors. (Note: a strictly positive $M$ exists if the market ...

6

Assuming that the only things that can happen on the period are $100$ and $50$, and we can buy a stock and a call option with strike $90$, even without knowing the probabilities of these moves we can relate the price of the stock $S$ and the option $C$ If we buy $0.2 S$ and sell one call option $C$, we have a portfolio that will be worth $10$ in either end-...

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OTM. If you buy deep ITM calls, your delta is 1. You will essentially own the stock with no leverage. With deep OTM calls, you will have gamma working for you. Your delta will increase as the stock rallies and are highly leveraged to the movement of the stock.

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I think you are referring to a compound option. It's valuation under Black-Scholes assumptions is given in the link. The option was first derived by Geske (1978), see here for the original paper .

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The risk of a credit spread is the difference in strikes less the premium received. In your first graph, the credit is \$7.50 and a strike difference of$11 so the risk is -\$3.50 as noted. In your second graph, the worst case scenario is a \$2.50 gain which means that the credit must exceed the difference in strikes and that is not possible. It has ...

4

It sounds to me that they just mean that each bound can be seen as a function of the parameter(s) in the parametrization and this function is Lipschitz continuous. An example: Consider the XY-plane. Let $Y(x)$ be a function of $x$. This function can be seen as describing the upper bound of the area below the graph. This function can then have the Lipschitz ...

4

This question will probably get closed soon, but I'll take a stab at answering anyway. I think, for an undergraduate, an interesting topic would be the FX-credit hybrids, that is, FX options (or even linear products like FX forwards and xccy swaps) with kick-in or kick-out on a credit event. For example - I want (the right) to exchange USD into EUR at some ...

3

For option pricing in the classical Black-Scholes model, you assume the underlying stock follows Geometric Brownian Motion: $$S_t = S_0 + \int_{h=0}^{h=t} S_h \mu dh + \int_{h=0}^{h=t} S_h \sigma dW_h = S_0 \exp \left( \mu t + 0.5 \sigma^2 t + \sigma W(t) \right)$$ Take the log of the solution above and you get: $$\ln\left( \frac{S_t}{S_0} \right) = \mu t + ... 3 The following does not necessarily apply solely to options market maker (MM) desks but a general MM. MMs are usually focused on running an on-going portfolio. A minority of MMs will run a daily book (with no outstanding risk at end-day) but generally this is inefficient or impossible and will therefore run a portfolio on an ongoing/indefinite trade basis. An ... 3 In your example, I believe it's assumed that the exercise date is after the dividend date. If the dividend date is after the exercise date, nothing happens. The value would decrease, consider the following timeline: t=0: You have a call option worth 4.73 and a stock worth S_0 t=1: The increase in dividend is announced but dividends are not paid out ... 3 Under the risk-neutral measure by application of Ito:$$ dS^3_t = 3 \left[ (r + \sigma^2)S^3_t dt + \sigma S^3_t dW_t \right] $$The risk-neutral drift is not the risk-free rate and hence S_t^3 \; \forall t cannot be the price of a claim or any other tradable asset. So basically along the same lines as your proof, but without calculating expectations etc. ... 3 Nice question. My interpretation is via the concept of a risk premium (i.e. risk adversity of market participants). Let me introduce the concept of a risk premium first via US corporate bonds: one can observe that the credit spread of these bonds increases as the credit quality decreases. However when looking at actual historical realized defaults of ... 2 I disagree with the author's premise. The primary reason that options are exercised early is because of discount arbitrage (an ITM option trades below its intrinsic value) or there is a dividend arbitrage in an ITM put (the time premium is less than the dividend, NOT the call). No one with a lick of sense would exercise an option that has time premium ... 2 On a conceptual level, I guess he means that, referring to American call options, for hard-to-borrow stocks (thus with high borrow cost), the yield you can generate from lending the stock has higher value than the continuation value from sticking with the option, so it’s worth it to exercise now and immediately lend it out. 2 There is a relationship and it comes about two seperate ways. A live price of an option (so one traded without delta) will be wider in an illiquid market then a liquid one as the market maker would have to buy the delta hedge and cross that wide underlying spread. That is the easy case to see, however remember that an options value is in some sense ... 2 Let \sigma_J be the volatility of the index J. Assume that J(0)\leq B. Consider the following 2 extreme cases: \sigma_J=0 \Rightarrow \forall x\in[0,T],J(x)=J(0)\leq B: hence you will always be paid I(T) at expiry. \sigma_J=\infty \Rightarrow \exists\epsilon>0, J(\epsilon)>B: hence you will almost immediately be paid I(0)\approx I(\... 2 OMMs run a position that comes from the trades that they are making throughout the day. For risk management and hedging they of course need some kind of pricing and greeks model for this overall portfolio which they calibrate intraday, and use to do PnL-Explains on PnL coming from the overnight moves in the market. Beyond pure inventory management though, ... 2 The collar strategy combines one unit of stock with a (long) put option with strike K_1 and a (short) call option with strike K_2. The payoff of this strategy is exactly K_1 if S_T\leq K_1, S_T if K_1<S_T\leq K_2 and K_2 if S_T>K_2. The easiest way to see that the statement is false is by comparing the payoff profiles of the collar and ... 1 For "Classic Autocall" or "Athena", the coupons are indeed accumulated and paid on the event of autocall, either pre-maturity or at maturity (but for the later it will not be called autocall). So only one barrier level (without considering the down-and-in put), the one of the autocall, or we could say the autocall barrier and coupon ... 1 A call option on zero coupon bonds P(t, T) imply that its price, at time t, is given by$$ V(t) = E_t^Q \left[ D(t, T) \cdot \max \left(P(\tau, T) - K, 0 \right) \right], $$with t \leq \tau \leq T. P(t, T) is the discount bond or zero coupon bond and it is given by:$$ P(t, T) = E_t^Q \left[ \exp \left( - \int_t^T r(s) ds \right) \right]. $$... 1 In short, because 1) the assumption of lognormal returns does not hold in real life--the markets have more skewness and kurtosis and 2) writers of protection want to be compensated more for writing insurance on low probability but high cost events. 1 Bear in mind that the IV you see quoted is Black Scholes IV. The only takeaway can be that the BS model is not the correct model to ACCURATELY price options. Differing IVs are the "fudge" to get better pricing and that option quoting (at the market maker level) really occurs through IV and is just expressed as price. When you look at the ... 1 1 according to http://www.cboe.com/education/getting-started/quick-facts/options-marketplace Are there requirements for individual stock issues to underlie exchange-listed equity options? Yes, option exchanges have certain eligibility requirements for stocks to become "optionable." Among the criteria are share price, number of shares outstanding, ... 1 Contact the CBOE or have your broker do it. They will list options if there is sufficient interest. Sufficient interest can often be one trade. Same as #1. Also, you can trade these deep OTM options OTC. If the expiry is longer than the LEAPs, they will most likely have to be done OTC. 1 Let's denote the option you need to hedge by C_1, which I am assuming you have sold (if you bought it then just turn the signs around). Under Heston you will need to hedge both its delta and its vega. You can use the underlying S to hedge the delta, but not to hedge vega. The most straightforward way to hedge the vega of C_1 is to buy another option in ... 1 For corporations, it is pretty common and also easy to look at the history of the implied volatility of the out of the money puts on the common equity (note that far out of the money puts are OTC not exchange-traded) the Z-spread of the bonds, or the CDS spread. When you see their relationship differing from what it's been historically, you can try to ... 1 The most risk free way to hedge FX risk is using a forward. So if you will receive 1 USD in the 1 Year, and you wanted to protect the EUR value of this receivable, you would sell USD/Buy EUR 1 year forward. If you sell the 1 Yr forward at 1.20USD/1EUR, when you receive 1 USD in a year, you would deliver it to your counterparty and receive 0.8333 EUR. If ... 1 Some of the assumptions here are wrong. The issue here is that$$S_0 \neq e^{-rT} E[S],$$but$$F = E[S]. And thus Z should be Z=V-theta*(VC-exp(-rT)*F). If you output mean(VC) it's very clear. It suggests that the choice of parameters for the Schwartz model are not consistent with the interest rate r, unless a non-zero convenience yield is expected.

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