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18

Consider reading Lorenzo Bergomi's excellent book -- or at least the first chapter available here for download --, it will help you clarify things. Some remarks as to your original question: It is well known that, under a pure diffusion assumption, the total P&L of a delta hedged European option (i.e. an option whose payoff only depends on the value of ...


8

The payoff $\max\{a\cdot S_t - K,0\}$ can be re-written as $a\cdot\max\{S_t - K/a,0\}$. Therefore it can be priced as a regular call option with the strike $K/a$.


8

Yes, it can work. However, keep in mind that: you'll be safer if you don't share any objects between threads; see my answer here, in particular the last point; even if you use different seeds, there's no guarantee that the generated sequences won't overlap. If you're willing to change the engine code so that you can pass the relevant parameters, a safer ...


6

Whenever you use any model to price anything, all you need to do is make sure you model the underlying dynamics that the product you're pricing actually depends on. Any product will be dependent on numerous facets, to varying degrees - this is the same with modelling anything. The modelling that happens in pricing financial derivatives is an integration ...


6

Any non path dependent European type payoff $f(S_T)$ can be replicated in a model independent way with vanilla calls and puts provided $f$ is twice differentiable (in the distribution sense). This is a consequence of the Carr-Madan formula.


5

Sigh. I'm not sure that there's a best way to do multi-threaded MC in QuantLib. I'm afraid that you're underestimating the amount of development you'd need for option 2. You're not going to get away with some OpenMP code as you suggest, because calculations on different paths are not trivially parallel: the RNGs we have are not parallel, and even if you ...


5

I agree with your computations. The problem is that the initial price of the asset-or-nothing call of 38.66 can't arise within the Black-Scholes framework. This seems to be an inconsistency/error in the question. Below you see a plot of asset-or-nothing call price as a function of the implied volatility. Note that "1" means 100% implied volatility. Let $...


5

Adding to Luigi's answer, second point: The issue of overlapping Mersenne Twister sequences can be addressed with dynamically created Mersenne Twister Generators, cf. http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/DC/dc.html. I created a wrapper for the dcmt library so that it fits more easily into the QuantLib library, see https://github.com/pcaspers/...


5

Let $f_0(S_T) =f(S_T|S_0)$ be the risk-neutral PDF for the underlying asset price at time $T$ (conditional on the price $S_0$ at present time $t=0$). The probability that the price is above a strike price $K$ at time $T$ is $$P(S_T \geqslant K) = \int_K^\infty f_0(x) \, dx.$$ This is just definitional regardless of the shape of the distribution (eg. ...


4

I don't know if these are the most commonly traded or most popular (for your definition of popular) but here are a few exotic products that I recall being supported by the flagship product at my former employers. Exotic Options: Asian - Strike price is dependant on average price throughout the deal, not just at expiry Bermudan - So called because it's ...


4

In short answer, Yes: the backward PDE solution with $v(t,L)=0$ and the expectation coincides under the Black-Scholes market. In the one dimensional case, this topic is mathematically treated in the theory of the scale function and the spead measure. See Revez-Yor 3rd.ed. Ch.VII.3 for details. I don't know whether there are some rigorous theories on the ...


4

Although the answer of @SRKX is right on spot, I was already writing a solution along the lines of how you had specifically approached the problem. I think it might still be useful to you, so here it goes The price of the chooser option, as seen of today $t=0$ is by definition \begin{align} V_0 &= \underbrace{e^{-r T_2}}_{\text{Payoff dicount factor}} ...


4

The difference is that the barrier option is weakly path dependent while the lookback option is strongly path dependent. In case of a knock-out barrier option, conditional on the option being alive at the pricing time you don't need to carry any additional state variables except for the current asset price. The payoff doesn't directly depend on the level of ...


4

The price is, under the risk-neutral measure, $$ P_t = e^{-r(T-t)}\mathbb E[S_T^1 \mathbb 1(S_T^2\le K)\mid \mathcal F_t].$$ Since the risk-neutral asset processes are independent geometric brownian motions, $S_T^1$ and $S_T^2$ are conditionally independent given $\mathcal F_t.$ So the conditional expectation factors and you get $$ P_t = e^{-r(T-t)}\mathbb ...


4

"pricing" am autocallables is simply working out what it's worth. This is done by having some model (Google local vol / stochastic local vol) which is calibrated to the market (ie listed vanilla options, broker quotes for less liquid tenors, and some light exotics (ie American barriers, cliquetes, etc.)), and then simulating the underlying many times, ...


4

It depends very much on the individual option you are pricing. Sometimes you can get a Black-Scholes PDE with some extended state and boundary conditions. up-and-out barrier option will have virtually the same pricing PDE, and zero boundary condition at $S=0$ and $S=B$ (barrier level). The up-and-in barrier option can be priced by $$C_\text{up-and-in} + C_\...


3

If you are looking for derivatives on weather (temperature, heating degree days, cooling degree days) and a financial "index", I think your best bet would be to look for hybrid weather/commodity derivatives.


3

You can refer to one of my previous answers here for a detailed development. There are actually two ways you can price this: - the price of a call plus a put with adjusted strike (like above) - a put plus the price of a call with an adjusted strike (like in my answer). The only difference is whether you do $\max( a, b ) = b + ( a - b )^+$, or $\max( a, b ...


3

The proof is fine. For example, $D(t)S(t)$ is a martingale and then \begin{align*} E\big(D(t)S(t)\big) = S(0). \end{align*} Regarding the function $C(1, T-T_0, K)$, it is the value, at time $T_0$, of the option payoff \begin{align*} \left(\frac{S(T)}{S(T_0)} - K \right)^+. \end{align*} Here, you can treat $\frac{S(T)}{S(T_0)}$ as the normalized value or ...


3

You can derive these formulae by tweaking the black scholes derivation. If you are using PDE method, you will use different boundary conditions. If you are using integration over the risk neutral probability , you will use a different payoff function but the same risk neutral density. Alternatively , you can observe that these payoffs are combinations ...


3

I know this may sound extreme, but in my experience for these kind of payoffs you do need a LMM and possibly with at least 8 factors. There is no shortcut unfortunately, even a 3 factor gaussian model, which you can use to price faster using trees, will not capture the possible dynamics of the curve implicit in an 8 factor LMM. Just my modest opinion ...


3

It is possible to price CRAs using the LMM using a Brownian bridge technique. You simulate to each coupon date and then infer the expectation of the coupon given the values of the rates at the start and end of the accrual period. http://ssrn.com/abstract=1461285 Interpolation Schemes in the Displaced-Diffusion LIBOR Market Model and the Efficient Pricing ...


3

The easiest way is to use single-expiry volatility that you would get from your volatility surface. It is usually good enough for government work (e.g. to get a sense if you are getting raped by a dealer or to understand your vega risk). A better way is to use local volatility model and the whole volatility surface up to the date of expiry. There is also a ...


3

You're using a wrong tool for the job. Write your Monte Carlo in a faster language (Java would probably suffice, if not than C++ which is standard for such things). Then you will be able to efficiently generate more than 1000 paths. In fact, doing Monte Carlo derivatives pricing with 1000 paths is worthless. Your results are, most probably, very inaccurate. ...


3

Delta-hedging can be seen from banks as "manufacturing the product". Banks are product manufacturers, so they delta-hedge. Exotic options are options which are not volatility-only products. They depend on volatility dynamics. Delta-hedging exotic options need most of the time to use vanilla options in the hedging portfolio.


3

the payoff is max(X-Y-K,0). so this option pays you the most if X goes up and Y goes down. So you need X and Y to move in opposite directions. The more X and Y move in the same direction (high rho) the less you get paid.


3

Gap risk contracts. These are daily-restriking putspreads that pay & cancel only if the underlying drops more than (say) 20% as measured vs yesterday's closing level. Contracts can range from as short as 6 months to 10 years. Cannot replicate that using Europeans.


3

The floating strike lookback call options has zero gamma only on the day it is issued (and only assuming an homogeneous model for the underlying). Afterwards it has non zero gamma. The payoff on maturity $T$ is: $$ \text{payoff} = S_T - \min \{S_u | u \in [0, T]\} $$ Now assume your model for the underlying is homogeneous with degree 1, that is when viewed ...


3

You can only replicate the digital part with call spreads. To be clear, let's decompose the Autocall as: the first part made of digital options (paying the coupons in case of early redemption) the second part made of a down and in put (this is the part responsible for the fact that, at maturity, you benefit from a conditional capital protection as long as ...


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