Let $D(t,T)$ denote the discount factor for the discount rate and $D_f(t,T)$ the discount factor for the forecast rate. The currency of the discount rate must be the same as the settlement currency (Currency). Note that the currency of the forecast rate may in future be different to the settelment currency but is currently not implemented.

The Distrubution Type on the volatility price factor can only be set to Lognormal currently. This assumes that the price factors are log-normally distributed (hence have implied Black volatilities). Note that this can be extended to Normal resulting in the price factor having implied Bachelier volatilities.

## Cashflows

### Fixed Interest Cashflows

A cashflow with

• principal $P$
• fixed interest rate $r$
• accrual start date $T_1$
• accrual end date $T_2$
• accrual day count convention with $\alpha$ the accrual year fraction from $T_1$ to $T_2$
• payment date $T$
• Fixed Amount $C$

has a standard payoff:

with value at time $t$ of $G(r)D(t,T)$.

### Floating Interest Cashflows

In addtion to the Fixed Interest Cashflow, a Floating Interest Cashflow also has

• reset date $t_0$
• reset start $t_1$
• reset end $t_2$
• margin rate $m$

The cashflow dates must satisfy $T_1\le T_2, t_0\le t_1\lt t_2$ and $t_0\le T$ and the payoff is $P(L(t_0)+m)\alpha$ with $L(t)$ is the simply-compounded forward rate at time $t$ given by

where $\alpha_2$ is the accrual year fraction from $t_1$ to $t_2$ using the rate day count convention.

The value of a standard floating interest rate cashflow at time $t\lt t_0$ is,

If $T=t_2, \alpha=\alpha_2$ and the discount and forecast rate are the same, then the value is

For $T\neq t_2$, the valuation needs a convexity correction but this is yet to be implemented. The standard payoff is:

where

• $r$ is a simply-compounded forward rate
• $\eta$ is the swaplet multiplier
• $\kappa$ is a caplet multiplier, with $K_c$ the caplet strike
• $\lambda$ is a floorlet multiplier, with $K_f$ the floorlet strike

### Caplets/Floorlets

A caplet/floorlet is a call/put option on a simply compounded rate. The option payoff at time $T$ is:

where $K$ is the strike and $\delta$ is either $+1$ for caplets and $-1$ for floorlets. If $T=t_2$ then the option value at time $t\lt t_0$ is

where $\mathcal B_\delta(F,K,v)$ is the Black function and $\sigma$ is the volatility of the forecast rate at time $t$ for expiry $t_0$, tenor $t_2-t_1$ and strike $K$. Note that if $T\neq t_2$ then the above formula is still used as no covexity correction has been applied.

#### Averaging

A cashflow with averaging depends on a sequence of simply compounded rates $r_1,...,r_m$ with the same nominal tenor $\tau$. Each rate $r_k$ jas a positive weight $\omega_k$. Let $t_{0,k}$ be the reset date of $r_k$. The average rate $R$ at time $T$ is

### Cashflow Lists

Consider a fixed or floating cashflow list with payment dates $t_1\le ... \le t_n$ and notional principal amounts $P_1,...,P_n$. If $U_i(t)$ denotes the value of the $i^{th}$ cashflow at time $t$, then the value of the cashflow list is $U(t)=\sum_{i=1}^n [t_i\ge t]U_i(t)$

There may also be an optional Settlement Date $T$ and Settlement Amount $C$. If not specified, then $T=-\infty$. Cashflow payment dates must be after the settlement date ($T\lt t_1$).

The time $t(>T)$ value of the deal is

,

where either $\delta=1$ for a Buy, else $\delta=-1$ for a Sell. If $t\le T$, the deal is treated as a forward contract on the underling cashflow list. If $A$ is the accrued interest up to $T$, then $K=C+A$ if Settlement Amount Is Clean else $K=A$. The value at time $t\le T$ is

For cash settled deals, the valuation profile terminates at $T$ with a corresponding cashflow $V(T)=(U(t)-K)\delta$. If physically settled, then the cashflow is $-K\delta$ at $T$ and the profile continues until $t_n$.

#### Fixed Compounding Cashflow Lists

For cashflow lists with the interest frequency greater than its payment frequency with payment dates $T_1\lt ...\lt T_c$,let $n(k)$ be the index of the last cashflow with payment date $T_k$ (with $n(0)=0$). For groups of cashflows with the same payment date, interest is compounded as follows: the $i^{th}$ cashflow at time $T_k$ pays $K_i+C_i$ with

where

• $I_i=r_i\alpha_i$
• $P_i$ is the principal amount\$
• $r_i$ is the fixed rate
• $\alpha_i$ is the accrual year fraction
• $C_i$ is the Fixed Amount of the $i^{th}$ cashflow.

The final value of the cashflows with payment date $T_k$ is

#### Floating Compounding Cashflow Lists

Similar to the Fixed Compounding Cashflow Lists, $I_i=G_i(r_i(u_i))\alpha_i+m_i\alpha_i$ where $u_i$ being the reset date. Let $V_i(t)$ be the value at time $t$ of an amount $I_i$ paid at the accrual end date $t_i$ (as opposed to the actual payment date $T_k$). The estimated interest is $I_i=\frac{V_i(t)}{D(t,t_i)}$ when $t otherwise $I_i(t)=I_i$ .

The compounding method can be:

Include Margin where the $i^{th}$ cashflow pays $P_i I_i(1+I_{i+1})...(1+I_{n(k)})+C_i$ at time $T_k$ with a value at $t\le T_k$ of

Flat where the $i^{th}$ cashflow pays $P_i I_i(1+J_{i+1})...(1+J_{n(k)})+C_i$ at time $T_k$ with $J_i=I_i-m_i\alpha_i$. Its value at $t\le T_k$ is

None in which case the $i^{th}$ cashflow pays $P_i I_i+C_i$ at time $T_k$.

## CFFixedInterestListDeal

A series of fixed interest cashflows as described here

## CFFloatingInterestListDeal

A series of floating interest cashflows as described here

## FixedCashflowDeal

The time $t$ value of a fixed cashflow amount $C$ paid at time $T$ is $D(t,T)C$.

## MtMCrossCurrencySwapDeal

This currency swap adjusts the notional of one leg to capture any changes in the FX Spot rate since the last reset. At each reset, the principal of the adjusted leg is set to the principal of the unadjusted leg multiplied by the spot FX rate. MtM cross currency swaps are path dependent.

The unadjusted leg is either a fixed or floating interest rate list and is valued as such, however, the floating adjusted leg is valued as

where

• $A_i(t)=P_i(t)-P_{i+1}(t)$ for $i\lt n$ and $A_n(t)=P_n(t)$
• $P_i(t)$ is the expected principal $P_i(t)=F(t,t_{i-1})\tilde P_i$,
• $\tilde P_i$ is the unadjusted leg principal for the $i^{th}$ period.
• $F(t,T)$ is the forward FX rate for settlement at time $T$.

## SwaptionDeal

Let $t_0, T_1$ and $T_2$ be the Option Expiry Date, Swap Effective Date and Swap Maturity Date respectively of the swaption deal ($t_0 \le T_1 \lt T2$). If the deal is cash settled, then let $T$ be the Settlement Date.

The value of the underlying swap is

where $V_{float}(t)$ is the value of floating interest rate cashflows, $V_{fixed}(t)$ the value of fixed interest cashflows and $\delta$ is either $+1$ for payer swaptions and $-1$ for receiver swaptions.

If the fixed leg has payments at times $t_2,...,t_n$, then the Present value of a Basis Point is

where $P_i$ is the principal amount and $\alpha_i$ is the accrual year fraction for the $i^{th}$ fixed interest cashflow. The forward swap rate is

Define the effective strike rate as

Note that presently only zero-margin floating cashflow lists are supported (but this can be extended). The value of the underlying swap is given by $U(t)=\delta(s(t)-K(t))F(t)$. If both fixed and floating cashflows have the same payment and accrual dates, then $K(t)=r$ where $r$ is the constant fixed rate on the fixed interest cashflow list.

#### Physically Settled Swaptions

If the Settlement Style is Physical and $U(t_0)\ge 0$, then the option holder receives the underlying swap and the value of the deal for $t\ge t_0$ is $U(t)$. Note that physical settlement has significant path dependency.

#### Cash Settled Swaptions

If the Settlement Style is Cash, then the option holder receives $\max(U(t_0),0)$ on settlement date $T$. The value of the deal at $t\lt t_0$ is $F(t)\mathcal B_\delta(s(r),K(0),\sigma\sqrt{(t_0-t)})D(t_0,T)$. Note that this assumes a lognormal distribution of the forecast rate and uses the Black Model as usual.

#### Swap Rate Volatility

Forward starting (where the effective date of the underlying swap is several months or years after the option expiry) and amortizing swaptions are not currently supported. This can be extended as needed. Otherwise, $\sigma$ is the volatility of the underlying rate at time $t$ for expiry $t_0$, tenor $\tau=T_2-T_1$ and strike $K(0)$