Conditional Expectation Approximation for Basket Call Options
Summary
The document presents a question about applying Beisser’s conditional expectation technique to price a basket call and determine its adjusted forwards. The basket consists of weighted stock prices, with deterministic rates and dividend yields defining each stock’s forward price. The quoted setup conditions on a common Gaussian variable and uses Jensen’s inequality to produce a lower-bound style approximation expressed as a weighted sum of artificial European call values with adjusted strikes and forwards.
The key construction is to choose a conditioning threshold at which the conditional basket value equals the option strike, then use the resulting conditional component values to define adjusted strikes. The question asks how to obtain the adjusted forwards, but supplies no answer or worked derivation. Consequently, the excerpt is useful for understanding the structure and motivation of the approximation, but it does not provide enough information to implement it fully. Assumptions about the joint stock distribution, weights, correlations, and the choice of conditioning variable would need to be established from the underlying method.
Key ideas
- The basket call is approximated by conditioning on a shared Gaussian variable.
- Jensen’s inequality motivates a lower-bound style estimate based on conditional basket values.
- A conditioning threshold is selected so the conditional basket value matches the strike.
- The resulting component thresholds act as adjusted strikes in artificial option terms.
- The excerpt poses, but does not resolve, how to calculate the adjusted forwards.
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Full text
# Method of conditional expectations for basket
# Method of conditional expectations for basket
I am reading paper "An analysis of pricing methods for baskets options".
Unfortunatly, I can not find the working paper "Beisser, J. (1999): Another Way to Value Basket Options, Working paper, Johannes Gutenberg-Universität Mainz" in order to understand how to calculate adjusted forwards $\tilde{F}_i^T$ in the "Beisser’s conditional expectation techniques" Section.
> Beisser’s conditional expectation techniques By conditioning on the random variable $Z$ and using Jensen’s inequality the price of the basket call is estimated by the weighted sum of (artificial) European call prices, more precisely $$ \mathbb{E}([B(T) - K]^+) = \mathbb{E}(\mathbb{E}([B(T) - K]^+|Z)) \geq \mathbb{E}(\mathbb{E}([B(T) - K|Z]^+)) = \mathbb{E}(\sum^n_{i=1} w_i \mathbb{E}[S_i(T)|Z] - K)^+ $$ where $$Z := \frac{\sigma_z}{\sqrt{T}}W(T) = \sum^n_{i=1} w_i S_i(0)\sigma_i W_i(T) $$ with $\sigma_z$ appropriately chosen. Note that in contradiction to $S_i(T)$, all conditional expectations $\mathbb{E}[S_i(T)|Z]$ are log-normally distributed with respect to one Brownian motion $W(T)$. Hence, there exists an $x'$, such that $$\sum^n_{i=1} w_i\mathbb{E}[S_i(T)|W(T) = x'] = K. $$ By defining: $$ \tilde{K_i} :=\mathbb{E}[S_i(T)|W(T) = x'] $$ the event $$ \sum^n_{i=1} w_i \mathbb{E}[S_i(T)|Z] \geq K $$ is equivalent to $$\mathbb{E}[S_i(T)|Z] \geq \tilde{K}_i, \forall i = 1,2, \ldots, n.$$ Using this argument we conclude that $$\mathbb{E}(\sum^n_{i=1} w_i\mathbb{E}[S_i(T)|Z] - K)^+ = \sum^n_{i=1} w_i\mathbb{E}([\mathbb{E}[S_i(T)|Z] - \tilde{K}_i]^+) = \sum^n_{i=1} w_i[\tilde{F}^T_i N(d_{1i}) - \tilde{K}_i N(d_{2i})],$$ where $\tilde{F}^T_i$, $\tilde{K}_i$ adjusted forwards and strikes and $d_{1i}$, $d_{2i}$ are the usual terms with modified parameters.
Question. How to apply the conditional expectation techniques in order to calculate adjusted forwards $\tilde{F}_i^T$?
The price of a basket of stocks as the weighted average of the prices of $n$ stocks at maturity $T$: $$ B(T) =\sum^n_{i=1} w_iS_i(T), $$ $S_i(\cdot)$ - stock prices, $w_i$ - weights. $T$-forward price of stock $i$ $$ F_i^T = S_i(0) \exp\left( \int_0^T (r(s) - d_i(s))ds \right), $$ where $r(\cdot)$ and $d_i(\cdot)$ are deterministic interest rates and dividend yields.
I have found the Lecture Notes with the method of conditional expectations but can not apply this method to the problem.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
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