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As far as I understand, a compounded swap rolls up individual payments into one final payment which becomes: $$ V(t_n) = N \prod_{i = 0}^{n-1}(1 + d_i L_i)-N $$

where $d_i$ is the day fraction for period $t_i$ to $t_{i+1}$ and $L_i$ is the index for the same period and where $N$ is deducted at the end because we assume no exchange of notional.

Now, to value this we need to calculate the expectation of $V(T)$ under some appropriate numéraire and measure, but we are dealing with products of various $L_i$'s which are, in general, not mutually independent, so it's not a simple matter of replacing with them forwards.

How is this then done? An internet search only revealed simple formulas using forwards. A good reference text would be welcome.

Add 1

Following suggestions in the comments, if I use the adjusted forward numéraire with maturity equal to the payment date $t_n$ and using $P(t_i, t_{i+1}) = \frac{1}{1 + d(t_i,t_{i+1}) L(t, t_{i+1})}$, then I get: $$ V(t) = P(t, t_n) \Bbb{E}^{Q^T} [V(t_n)|F_t] = N P(t, t_n) \left(\Bbb{E}^{Q^T} \left[\prod_{i=0}^{n-1} \frac{1}{P(t_i, t_{i+1})} | F_t \right]-1\right) $$

but I'm not sure that this gets me anywhere.

As far as I understand, a compounded swap rolls up individual payments into one final payment which becomes: $$ V(t_n) = N \prod_{i = 0}^{n-1}(1 + d_i L_i)-N $$

where $d_i$ is the day fraction for period $t_i$ to $t_{i+1}$ and $L_i$ is the index for the same period and where $N$ is deducted at the end because we assume no exchange of notional.

Now, to value this we need to calculate the expectation of $V(T)$ under some appropriate numéraire and measure, but we are dealing with products of various $L_i$'s which are, in general, not mutually independent, so it's not a simple matter of replacing with them forwards.

How is this then done? An internet search only revealed simple formulas using forwards. A good reference text would be welcome.

Add 1

Following suggestions in the comments, if I use the adjusted forward numéraire with maturity equal to the payment date $t_n$ and using $P(t_i, t_{i+1}) = \frac{1}{1 + d(t_i,t_{i+1}) L(t, t_{i+1})}$, then I get: $$ V(t) = P(t, t_n) \Bbb{E}^{Q^{t_n}} [V(t_n)|F_t] = N P(t, t_n) \left(\Bbb{E}^{Q^{t_n}} \left[\prod_{i=0}^{n-1} \frac{1}{P(t_i, t_{i+1})} | F_t \right]-1\right) $$

but I'm not sure that this gets me anywhere.

As far as I understand, a compounded swap rolls up individual payments into one final payment which becomes

$$ V(T) = N\prod_{i=1}^n(1+d_iL_i)-N $$

where $d_i$ is the day fraction for period $t_i$ to $t_{i+1}$ and $L_i$ is the index for the same period and where $N$ is deducted at the end because we assume no exchange of notional.

Now, to value this we need to calculate the expectation of $V(T)$ under some appropriate numeraire and measure, but we are dealing with products of various $L_i$'s which are, in general, not mutually independent, so it's not a simple matter of replacing with them forwards.

How is this then done? An internet search only revealed simple formulas using forwards. A good reference text would be welcome.

Add 1

Following suggestions in the comments, if I use the adjusted forward numeraire with maturity equal to the payment date $T$ and using $P(t_i,t_{i+1}) = \frac{1}{1+d(t_i,t_{i+1})L(t,t_{i+1})}$, I get

$$ V(t) = P(t;T) \Bbb{E}^{Q^T} [V(T)|F_t] = NP(t;T) \left(\Bbb{E}^{Q^T} \left[\prod_{i=1}^n \frac{1}{P(t_i;t_{i+1})} | F_t \right]-1\right) $$

but I'm not sure that this gets me anywhere.

As far as I understand, a compounded swap rolls up individual payments into one final payment which becomes: $$ V(t_n) = N \prod_{i = 0}^{n-1}(1 + d_i L_i)-N $$

where $d_i$ is the day fraction for period $t_i$ to $t_{i+1}$ and $L_i$ is the index for the same period and where $N$ is deducted at the end because we assume no exchange of notional.

Now, to value this we need to calculate the expectation of $V(T)$ under some appropriate numéraire and measure, but we are dealing with products of various $L_i$'s which are, in general, not mutually independent, so it's not a simple matter of replacing with them forwards.

How is this then done? An internet search only revealed simple formulas using forwards. A good reference text would be welcome.

Add 1

Following suggestions in the comments, if I use the adjusted forward numéraire with maturity equal to the payment date $t_n$ and using $P(t_i, t_{i+1}) = \frac{1}{1 + d(t_i,t_{i+1}) L(t, t_{i+1})}$, then I get: $$ V(t) = P(t, t_n) \Bbb{E}^{Q^T} [V(t_n)|F_t] = N P(t, t_n) \left(\Bbb{E}^{Q^T} \left[\prod_{i=0}^{n-1} \frac{1}{P(t_i, t_{i+1})} | F_t \right]-1\right) $$

but I'm not sure that this gets me anywhere.

As far as I understand, a compounded swap rolls up individual payments into one final payment which becomes

$$ V(T) = N\prod_{i=1}^n(1+d_iL_i)-N $$

where $d_i$ is the day fraction for period $t_i$ to $t_{i+1}$ and $L_i$ is the index for the same period and where $N$ is deducted at the end because we assume no exchange of notional.

Now, to value this we need to calculate the expectation of $V(T)$ under some appropriate numeraire and measure, but we are dealing with products of various $L_i$'s which are, in general, not mutually independent, so it's not a simple matter of replacing with them forwards.

How is this then done? An internet search only revealed simple formulas using forwards. A good reference text would be welcome.

As far as I understand, a compounded swap rolls up individual payments into one final payment which becomes

$$ V(T) = N\prod_{i=1}^n(1+d_iL_i)-N $$

where $d_i$ is the day fraction for period $t_i$ to $t_{i+1}$ and $L_i$ is the index for the same period and where $N$ is deducted at the end because we assume no exchange of notional.

Now, to value this we need to calculate the expectation of $V(T)$ under some appropriate numeraire and measure, but we are dealing with products of various $L_i$'s which are, in general, not mutually independent, so it's not a simple matter of replacing with them forwards.

How is this then done? An internet search only revealed simple formulas using forwards. A good reference text would be welcome.

Add 1

Following suggestions in the comments, if I use the adjusted forward numeraire with maturity equal to the payment date $T$ and using $P(t_i,t_{i+1}) = \frac{1}{1+d(t_i,t_{i+1})L(t,t_{i+1})}$, I get

$$ V(t) = P(t;T) \Bbb{E}^{Q^T} [V(T)|F_t] = NP(t;T) \left(\Bbb{E}^{Q^T} \left[\prod_{i=1}^n \frac{1}{P(t_i;t_{i+1})} | F_t \right]-1\right) $$

but I'm not sure that this gets me anywhere.