Basket Option weight sensitivity calculation

Question

I am looking to find/estimate the "greeks"/option price sensitivities/derivatives for a basket option situation. In specific the change in price of a put option associated with a change in weight of a given asset in a portfolio. I think a finite difference method (FDM) with MCMC simulations would work for a two asset situation but wasn't sure how to carry it out for more than two assets.

Say you have three assets/stocks (with no dividends): A,B,C in a basket with their respective estimated mean returns, estimated standard deviations, and estimated correlations/covariance matrix. Instead of three strikes for each asset, you have a single portfolio strike that is 10% below the 'current' portfolio. The 'current' portfolio weights are presented/given by the 'client'.

You can relatively simply MCMC simulate the three stocks, and assign the given weights from initiation to calculate portfolio terminal value (assuming no rebalancing), and estimate various 'standard' greeks. And for a two asset portfolio, the weight of asset 2 is just 1 - weight of asset 1, so in terms of finding the change in option price due to a change in weight of each asset, which is what originally looked like a two variable problem, (i.e. to find the change in option price by changing the weight of asset 1 and asset 2), becomes a one variable problem, as any change in the weight of asset 1 will implicitly change the weight of asset 2, making simulation and calculation of the sensitivities easier. But this simplification/dimension reduction does not exist for a three or more asset situation, so I was wondering how would one do it for a three or more asset situation?

Answer 1

I would define the weights $w_1,\ldots,w_n$ as whatever number you want and the basket given by $$ B_t = \sum_{i=1}^n \frac{w_i}{W}S_t^{(i)}\ , \qquad W = \sum_{i=1}^nw_i $$ so the weights always sum to one.

This doesn't make much sense, however, because you are changing the product, not a market variable. This meaning that when the weights change, the basket is discountinuous and you cannot replicate this.

The only way to replicate the basket would be to keep the past performance fixed as $$ B_{t} = \frac{B_{t_1}^1}{B^1_{t_0}}\times \cdots \times \frac{B_{t_k}^k}{B_{t_{k-1}}^k}\times \frac{B_t^{k+1}}{B_{t_k}^{k+1}} $$ Where each $t_i$ is a time where you rebalance the weights, and each $B^i$ is a baseket with the weights defined during the period $[t_{i-1},t_i)$.

Answer 2

This is perhaps not a concrete solution to your problem but the space in the comments is limited :)

In your setupt you are not actually pricing an option on a basket but on a dynamically allocated portfolio. Thus conventional pricing and hedging approaches won't apply.

Also you are underestimating porfolio optimization algarithms. To find an optimal strategy you actually have to deal with weight sensitivity !

Here an easy example:

Let's say you want to have a static allocation at $t=0$ that gives you the highest expected porfolio value at $T$. So you set up your porfolio once and let it rest until maturity - thus your weights don't change.

Now let's assume the expected porfolio value at $T$ is given by some continuous function $\mathbb{E}[P_T]=f(w_1,w_2,w_3,\vec{a})$ where $\vec{a}$ is a vector of model parameters for your capital market model (e.g. $\mu, \sigma$ in the B&S-case) and the $w_i$ are weights of your three stocks (with $w_1+w_2+w_3=1$)

Let's say your model parameters don't change over time. To optimize $\mathbb{E}[P_T]$ you have to run a multidimensional optimization algorithm on $f(w_1,w_2,w_3,\vec{a})$ with respect to $(w_1,w_2,w_3)$ Thus a three dimensional problem. Algorithms like gradient decent or levenberg marquardt explictily use derivatives to find the solution. For a concrete application of gradient descent to a portfolio selection problem confer the following paper.

Some further thoughts: Your approach could also be seen as a way to analyse how options on equity-funds behave under changes in the fund composition. I would argue that this is already partially covered by the option's sensitivity to changes in volatility. If you change your portfolio composition by adding more of a volatile stock the effective volatility of your fund-value-process will increase.