# Tag Info

22

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

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 ...

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 ...

7

Your statement of the problem is not very detailed. Are $\mu_{S/X}$ constant ? What about interest rates? In the classic exchange option problem, where the payoff is $(X_T - S_T)^+$, they actually do not matter since all the risk is related to $S$ and $X$, no cash is involved (up to selling your replicating portfolio at maturity to pay immediately the ...

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.

6

$\frac{1}{S_t}$ is log-normal If $S_t$ is a geometric Brownian motion, so is $\frac{1}{S_t}$ and indeed any power $S_t^\alpha$. Simply use Itô's Lemma and set $f(t,x)=\frac{1}{x}$, \begin{align*} \mathrm{d}f(t,S_t) &= \left(0-\mu S_t\frac{1}{S_t^2}+\frac{1}{2}\sigma^2S_t^2\frac{2}{S_t^3}\right)\mathrm{d}t-\sigma S_t \frac{1}{S_t^2}\mathrm{d}W_t \\ &=-...

5

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 ...

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

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 ...

5

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}} ...

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. ... 5 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 $$C_\text{up-and-in} + C_\... 5 Typically structures like this are traded as notes. They will be sold at a face value of 100%, where that is normally the combination of a zcb (ie 1y usd, say 97.5%), expected coupon (say +10%), short Knock In put (also knocked out by the autocall feature, say -8%), and some profit for the issuer (in this case, 100%-97.5%-10%+8%=0.5%). Sometimes these are ... 5 I image you want to calculate the following payoff:$$\pi_T = \max\left[ \max(S_T^1, S_T^2) - K, 0 \right]$$If dynamics are expressed with the following dynamic (from your code, it should be the case):$$dS_t^i = rS_t^idt + \sigma S_t^idW_t^i$$where W = (W^1, W^2) is two dimensional SBM, then, I guess you could simplify your code a bit! From your ... 5 As @noob2 noted, nobody is going to quote you a price unless you're a customer. And when I say "customer", I mean "customer of the desk", not just of the bank. Would require an ISDA covering the specific product area + some commitment to minimum 'spend' with that business line. 4 The difference is that the barrier option is weakly path dependent while the lookback option is strongly path dependent. In case of a knock-out barrier option, conditional on the option being alive at the pricing time you don't need to carry any additional state variables except for the current asset price. The payoff doesn't directly depend on the level of ... 4 The price is, under the risk-neutral measure,$$ 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 For Q1 in order to create a negative delta product you would have to offset it by selling a positive delta product to someone else, which is certainly possible. Q2 I agree with the proxy solution , but it is not very reliable since the fund manager can change the volatility of the fund by changing the composition of the assets. This cannot be avoided ... 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 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 ... 4 I believe something is wrong with your analytical pricing formula: I have provided an R script for the analytical pricing formula specified on (p. 211) and it gives a call price of 0.2853. library(pbivnorm) maxassets_analytical <- function(r, T, K, sigma1, sigma2, S1, S2, b1, b2, rho){ y1 <- (log(S1/K) + (b1 + sigma1^2/2)*T)/ (sigma1*sqrt(T)) y2 ... 3 The proof is fine. For example,$D(t)S(t)is a martingale and then \begin{align*} E\big(D(t)S(t)\big) = S(0). \end{align*} Regarding the functionC(1, T-T_0, K)$, 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'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 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

You can use: Antithetic variates and; Control variates. Both are variance reduction techniques which will allow you to use fewer paths/simulations. Usually antithetic variates are very efficient on their own. Combining both can be a bit tricky. You could start by simulating the value of a plain vanilla call. Then include antithetic variates and/or control ...

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