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11

Interesting question. Unfortunately for you, the answer is no, it cannot be done. The principal difference between a basket of options and an option on the basket (or index) is correlation risk. In fact, there is a systematic difference between the implied volatility of the basket and the (properly weighted) sum of implied volatilities on the components. ...

9

The main problem in your code is this line: rowSums(coef(model) * frame[, -1]) I'm not sure exactly what is does, perhaps some matrix multiplication, but definitely not what you expect it to do. Try to replace it with manual multiplication spread <- frame[,1] - (coef(model)[1]*frame[,2] + coef(model)[2]*frame[,3] + coef(model)[3]*frame[,4] + coef(...

8

No offense but it will be much more complicated than what you think... I'm not even sure that you are familiar with risk-neutral pricing in the first place? I'll try to give you some clues. This security is called a basket option. On top of the multi-asset feature, there are non-trivial mechanisms embedded in the contract you mention: an auto-callable ...

6

I think you should not just ask what the implied vol is of a basket of equity derivatives but you should aim to generate a volatility surface. A spot implied vol gives you nothing to work with. What you need is an implied vol surface in order to understand the smile and skew effects when you quote basket options in the market and/or as price taker. Take a ...

5

Let us consider a basket $B$ with components $S_1,\dots,S_n$ : $$B(t) = \sum_{i=1}^nw_iS_i(t)$$ At time $t$, each component has standard deviation $\sigma_i$, $i \in \{1,\dots,n\}$, and pairwise correlations are $\rho_{ij}$, $i \not= j$. Thus: $$\sigma_B^2=\sum_{i=1}^nw_i^2\sigma_i^2+2\sum_{i=1}^n\sum_{1=j}^iw_iw_j\sigma_i\sigma_j\rho_{ij}$$ The implied ...

4

Once you have slogged through all the relatively useless theoretical literature, this paper is a rediscovery (and pretty good write-up) of how basket option pricing is really done in serious quant packages at the big banks.

4

You may find my recent paper helpful. Choi (2018) Sum of All Black-Scholes-Merton Models: An Efficient Pricing Method for Spread, Basket, and Asian Options (arxiv) The method can handle the options on any linear combination of assets such as spread, basket and Asian options. You can obtain fairly accurate deterministic (i.e., not Monte Carlo) values with ...

3

The problem in your code is that the correlation is completely ignored. I would replace the loop by the following piece of code: for i in range(0,M,1): Rand1 = scipy.random.randn(1) Rand2 = scipy.random.randn(1) growthFactor1 = drift1 * exp(v1 * sqrt(dt) * Rand1) S1next = S1 * growthFactor1 growthFactor2 = drift2 * exp(v2 * sqrt(dt) * (0....

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

Let $\tau_{(1)} = \min(\tau_1, \ldots, \tau_K)$ be the first-to-default time. Moreover, for $1< m \le K$, let \begin{align*} \tau_{(m)} = \min\left(\tau_k: k=1, \ldots, K, \tau_{k} > \tau_{(m-1)}\right). \end{align*} be the $m^{\rm th}$-to-default time. In particular, $\tau_{(K)} = \max(\tau_1, \ldots, \tau_K)$. Note that, for $t \ge 0$, \begin{align*} ...

3

The formula works for total variance, not "strike specific" variance that you need to construct basket vol surface from components, because single historical correlation (or correlation matrix) just does not provide enough information to uniquely reconstruct expected distribution of basket returns (unless for a trivial case where all components are gaussian, ...

3

CBOE has something with limited capacity. Yahoo Finance also gives the current option chain. But historical option data is not free. The most affordable I saw is here. I don't know about its validity but their structure seems good and almost clean. More importantly, data seems reliable. p.s. I am not sure if providing the paid data link is within T&C ...

3

Perhaps this paper by Hyun Woo Byun and coauthors is what you're looking for: Using a Principal Component Analysis to develop Multi-Currency Trading algorithms in the FX market They apply principal component analysis to a currency basket of 9 pairs with a 2 month rolling window. In a second step, various techniques (logistic regression, decision trees, ...

3

To add (and contradict a bit) to what Brian B said. The exo desks that have multiple positions in basket options frequently price and manage these positions using the moment matching models (for efficiency reasons). For baskets with a lot of stocks, most desks would use a single vol, usually using a proxy like a liquid index with a spread or a multiplier.

3

@Sergey correctly identified the problem. The explanation is that coef(model) is a vector, frame is a data.frame, and element-by-element multiplication takes place in column-major order. The shorter vector (coef(model)) is recycled along the longer vector (each column in frame). For example: frame <- data.frame(V1=1:5) frame$V2 <- 2 frame$V3 <- ...

2

Is it possible to replicate the option of a custom index? Yes and you can find OTC market-makers who will make a price. They use portfolio replication to mimic the payoff of the option with a position in the underlying (Black-Scholes, '73). Even though the underlying custom index is not traded it can be perfectly constructed via its traded constituents. So ...

2

Freddy has already answered it and my answer had an assumption in it so clarifying - If payoff of basket with underlined securities A,B and C are $$P_b = C_1*P_A + C_2*P_B + C_3*P_C$$ Where $$C_1 , C_2 ,C_3$$ are contants then portfolio delta is $$\delta_b = C_1*\delta_a+C_2*\delta_b+C_3*\delta_c$$ In short as Freddy Said , and I assumed if the ...

2

Please note that this is subjective, but I hope it can help. I was told that Frozen Concentrated Orange Juice forward contracts (FCOJ) are used to have a proxy for weather risk. https://www.theice.com/products/30/FCOJ-A-Futures you can imagine have a look at other agricultural forwards, since for these kind of market, demand is linked to economy level (=...

2

You may want to look at the literature on index tracking; perhaps you find useful ideas there. I would write and solve it as an optimisation model. Since you seem most interested in the number of assets that are required to closely replicate your basket (i.e. the cardinality of the replicating basket), you could solve the model for different cardinalities ...

2

There are a few issues that need to be separated here. Issue "zero" is whether your MC is able to correctly represent the dynamics you've chosen for your assets. If you implement your MC properly, by construction it should converge in distribution to the postulated dynamics. No bias there. Variance yes potentially, because of discretisation, but no ...

2

I just wanted to say that I solved the problem using the symbolic/analytical features of Mathematica and Matlab to perform the inverse Fourier transform and then I used high-order numerical integration to solve the smoothing integral.

1

1

QuantLib is what you are looking for. It is free/open source library written in C++, it is available in Python as well (via SWIG): https://www.quantlib.org/install/windows-python.shtml Examples are shipped with QuantLib and among them some show how to price options. To get a feel for what it's like, you can check this blog post, explaining how to price an ...

1

He says the following: Let's use a multi-asset local volatility model calibrated for each stock on its market smile of maturity $T$ (a one-maturity smile), and with the Brownian motions correlated through a correlation matrix $\rho$ Then there exists a local volatility for each asset such that: (1) the smile of maturity $T$ for each asset is recovered, (2) ...

1

Another approach would have been to use some projection as in Pooley and Vetzal Convergence remedies for non-smooth payoffs in option pricing. In your case, it may be a projection of the initial condition to the RBF space (I have read your paper, and it looks interesting). I wonder a bit how the two approaches compare.

1

This is not a direct answer to your question as I am not sure whether the instrument you described exists, but OP would probably find the mathematics behind transmission congestion contracts very interesting. Transmission congestion contracts enable the hedging of fluctuations in electricity prices across the power grid, and are auctioned off by regional ...

1

To develop it from scratch, you could simulate the portfolio of the security combination, and utilize the portfolio's notional value, volatilities into Black Scholes Merton for fair values of ATM options.

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