Pricing claims of parties in a fund

Question

I'm working on the following problem and would appreciate some input because I'm stuck.

Consider a fund that works as follows. The fund starts with $S_0$ worth of assets following a Geometric Brownian Motion. At each time of the schedule $\{t_i\}_{1 \leq i \leq N}$, party $B$ gets a payment worth $\alpha S_{t_i}$. This goes on for $N$ dates, then a party $A$ fills the fund back to $S_0$ if $S_{t_N} < S_0$. If $S_{t_N} > S_0$ the difference is paid to party $C$.

Now I need to value the claims of all three parties. At maturity the claim of $C$ is worth $\max(S_{t_N}-S_0, 0)$, while the claim of $A$ is worth $\min(S_{t_N} - S_0, 0)$. Now let $Q$ be the risk-neutral measure. I thought the initial value of these claims to be $e^{-rt_N}E^Q(\max(S_{t_N}-S_0, 0))$ and $e^{-rt_N}E^Q(\min(S_{t_N}-S_0, 0))$ respectively. I approximated this expectation by simulating under $Q$ (so simulate $S_{t_i}$ under $Q$ with drift term $r$, where $r$ is the risk free rate). However at each period I reset as follows: $$S_{t_i^+} = (1-\alpha)S_{t_i^-}$$ where: $$S_{t_i^-}=S_{t_{i-1}^+}e^{(r-\frac{\sigma^2}{2})(t_i-t_{i-1})+\sigma W_{t_i-t_{i-1}}}$$ 1) Can I still use this pricing method? Doesn't taking a cut of $S_{t_i}$ every period ruin it?

2) The second question I have is what is the claim of $B$ and how to price it? My intuition: After $N$ periods $B$ has received $\sum_i \alpha S_{t_i}$. But how to price this?

3) And lastly: there is supposed to be a parity relationship between the values of the three claims. The fund transforms $S_0$ and the contribution of $A$ into payments to $B$ and $C$ and $S_0$. So I thought, somehow the price of the claim of $A$ should be equal to the prices of $B$ and $C$. Is this correct? If so why? If not what should I do?

Answer 1

To be rigorous, I have modified your original question to define:

$$ S_{t_i^+}=(1-\alpha)S_{t_i^-}$$

where $t_i$ is a payment date for $B$. At $t_i$, $B$ receives the amount $\alpha S_{t_i^-}$. This way we distinguish $S_{t_i}$, $S_{t_i^+}$ and $S_{t_i^-}$ where:

$$ S_{t_i} = S_0e^{(r-\frac{\sigma^2}{2})t_i+\sigma W_{t_i}} $$

1) To me there is no particular issue. Letting $0<t_0<t$, note that:

$$ \begin{align} S_t&=S_0e^{(r-\frac{\sigma^2}{2})t+\sigma W_t} \\[9pt] &=\left(S_0e^{(r-\frac{\sigma^2}{2})t_0+\sigma W_{t_0}}\right)e^{(r-\frac{\sigma^2}{2})(t-t_0)+\sigma (W_t-W_{t_0})} \\[6pt] &=S_{t_0}e^{(r-\frac{\sigma^2}{2})(t-t_0)+\sigma W_{t-t_0}} \end{align}$$

Letting:

$$ \begin{align} S_{t_0^-} & = S_0e^{(r-\frac{\sigma^2}{2})t_0+\sigma W_{t_0}} \tag{1} \\[6pt] S_{t_0^+} & = (1-\alpha)S_{t_0^-} \tag{2} \end{align}$$

We have:

$$ S_{t_0^+}e^{(r-\frac{\sigma^2}{2})(t-t_0)+\sigma W_{t-t_0}} = (1-\alpha)S_{t_0^-}e^{(r-\frac{\sigma^2}{2})(t-t_0)+\sigma W_{t-t_0}} = (1-\alpha)S_t$$

If we have a 2nd date $t_1>t_0$:

$$ \begin{align} S_{t_1^+}e^{(r-\frac{\sigma^2}{2})(t-t_1)+\sigma W_{t-t_1}} & = (1-\alpha)S_{t_1^-}e^{(r-\frac{\sigma^2}{2})(t-t_1)+\sigma W_{t-t_1}} \\[6pt] & = (1-\alpha)S_{t_0^+}e^{(r-\frac{\sigma^2}{2})(t_1-t_0)+\sigma W_{t_1-t_0}} e^{(r-\frac{\sigma^2}{2})(t-t_1)+\sigma W_{t-t_1}} \\[6pt] & = (1-\alpha)S_{t_0^+}e^{(r-\frac{\sigma^2}{2})(t-t_0)+\sigma W_{t-t_0}} \qquad \\[6pt] & = (1-\alpha)^2S_t \end{align}$$

So you can simulate your asset and then multiply by $(1-\alpha)^N$ posteriorly. From $\text{(1)}$ and $\text{(2)}$ you see that:

$$ S_{t_i^-} = (1-\alpha)^{i-1}S_0e^{(r-\frac{\sigma^2}{2})t_i+\sigma W_{t_i}} = (1-\alpha)^{i-1}S_{t_i}$$

2) The price of a claim is its discounted risk-neutral expectation, hence letting $\pi_B$ be the price of $B\text{'s}$ claim we have:

$$ \begin{align} \pi_B&=E^Q\left[\sum_{i=1}^Ne^{-rt_i}\alpha S_{t_i^-}\right] \\[6pt] & = \sum_{i=1}^N\alpha e^{-rt_i}E^Q\left[S_{t_i^-}\right] \\[6pt] & = \sum_{i=1}^N\alpha e^{-rt_i}(1-\alpha)^{i-1}E^Q\left[S_{t_i}\right] \\[6pt] & = \alpha S_0\sum_{i=1}^N e^{-rt_i}(1-\alpha)^{i-1}e^{rt_i} \\[6pt] & = \alpha S_0\sum_{i=0}^{N-1} (1-\alpha)^{i} \\[6pt] & = S_0(1-(1-\alpha)^N)\end{align}$$

3) First, note that:

$$ \min(S_{t_N}-S_0,0)=-\max(S_0-S_{t_N},0) $$

Hence party $A$ has sold a put with strike $S_0$ to the fund. Additionally, vis-à-vis party $C$ the position is equivalent to the fund having sold a call with strike $S_0$. As a result, combining the positions of $A$ and $C$, the fund is short a forward contract on the asset $S_t$.

Moreover, $B\text{'s}$ payments can be interpreted as dividends: the asset pays a proportional dividend with rate $\alpha$ at times $\{t_i\}_i$. After each payment the asset depreciates by $\alpha S_{t_i^-}$.

Assuming the asset is effectively dividend-paying, we conclude that to replicate all payments the fund only needs to hold one unit of the asset: it will pay $B$ with dividends received and at maturity $t_N$ it will sell it for a price equal to $S_0$ to either $A$ (if $S_{t_N} \leq S_0$) or $C$ (if $S_{t_N} > S_0$).

Answer 2

This is an addendum to the answer provided by Daneel Olivaw.

Let's assume that the payment at $t_i$, for $i=1, \ldots, N$, is $\alpha_i S_{t_i}$. We define the step function \begin{align*} M_t = \sum_{i=1}^N \alpha_i \pmb{1}_{\{t_i \le t\}}. \end{align*} Moreover, we assume that the fund value process $\{S_t, \, t \ge 0\}$ satisfies an SDE of the form \begin{align*} dS_t = S_{t-}\big(rdt-dM_t + \sigma dW_t \big). \end{align*} Then \begin{align*} S_t = S_0 e^{(r-\frac{1}{2} \sigma^2)t + \sigma W_t}\Pi_{t_i \le t} (1-\alpha_{i}). \end{align*} The valuation of the respective payoffs can now be followed as the answer of Daneel Olivaw.