Pricing a Barrier-Window Call with a Copula and Numeraire Change
Summary
The document considers a contingent claim that pays the positive excess of one stock’s terminal price over a strike, provided that a second stock finishes inside a specified price interval. The question proposes pricing it from the two terminal marginal distributions and a copula density, including when the underlying stocks follow non-flat volatility models. It begins with a discounted joint-density integral over the payoff region, though the proposed manipulation of the cumulative distribution function is not validated in the answer.
The response suggests splitting the payoff into two pieces: the joint event probability for both price conditions, and the same event weighted by the first stock’s price. The first piece can be evaluated using the joint distribution. For the price-weighted piece, it proposes a change of numeraire to recast the expectation in a similar form, then apply the same joint-distribution method. The answer is brief and gives no worked derivation or numerical evidence. Practical use therefore depends on specifying risk-neutral distributions, the relevant measure change, and a sufficiently accurate copula model.
Key ideas
- The claim combines a call-style payoff on one stock with a terminal price window on another.
- A joint distribution model can represent dependence between the two terminal stock prices.
- The payoff can be decomposed into an event-probability term and a price-weighted event term.
- A change of numeraire is proposed to handle the price-weighted expectation.
- The answer omits derivation details, so measure assumptions and model accuracy require further verification.
Tags
Full text
# Price a contingent claim with payoff $(S_T-K)1_{\{S_T>K\}}1_{\{L\leq X_T\leq U\}}$
# Price a contingent claim with payoff $(S_T-K)1_{\{S_T>K\}}1_{\{L\leq X_T\leq U\}}$
I'd like to price the following contingent claim using a copula model. $$V_T = (S_T-K)1_{\{S_T>K\}}1_{\{L\leq X_T\leq U\}}$$
where $S$ and $X$ are two stock price processes which follow a non-flat vol model. In particular, $S$ and $X$ are not standard GBM models with flat vols. Suppose you are given the distributions of $S_T$ and $X_T$ as a black box. Moreover, suppose you are given a copula density function, $f_{(S,X)}$, which is very good at approximating the joint distribution. I would like to derive a semi-analytic formula for the price, $V_t$. Below is my attempt. Does it look correct? What should I do next? Can I simplify further? \begin{align*} V_t & = e^{-r(T-t)} E[(S_T-K)1_{\{S_T>K\}}1_{\{L\leq X_T\leq U\}}] \\ &= e^{-r(T-t)}\int_{L}^{U}\int_{K}^{\infty}(a-K)f_{S,X}(a,b)dadb \\ &= e^{-r(T-t)}\int_{K}^{\infty}(a-K)\int_{L}^{U}f_{S,X}(a,b)dbda \\ &= e^{-r(T-t)}\int_{K}^{\infty}(a-K)\left(F_{S,X}(a,U) - F_{S,X}(a,L)\right)da \\ &= e^{-r(T-t)}\left( (a-K)\int_{K}^{\infty}F_{S,X}(a,U) - F_{S,X}(a, L) da - \int_{K}^{\infty} \int_{K}^{\infty}F_{S,X}(a,U) - F_{S,X}(a, L)dada \right) \end{align*}
## Answer by NN2 (score 1)
https://quant.stackexchange.com/a/59981
I don't know whether you are studying in the same class with the author of this question Copula analytic formula for $max(S_T^1−K,0)1_{L<S_T^2<U}$.
The idea is to decompose the payoff into 2 parts. The first part is $1_{S_T>K}1_{L<X_T<U}$ and the second one is $S_T 1_{S_T>K}1_{L<X_T<U}$ .
The value of the first part is equal to $P(\{(W_T^1-W_t^1) > d_1 \}\cap \{ d_2<(W_T^2-W_t^2))<d_3 \}) $ and you can use the density function $f_{S,X}$ to have its closed form solution.
And the second part can be transformed into the first part form with the change of numéraire. After that, you use the same method to have its closed form solution.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.