# Valuations in Coupa Treasury

## Basics

This document contains information on the calculation of present values for various instruments in Coupa Treasury. For this purpose, the valuation date is time 0 and the instrument is split into its future cash flows.

### Interest methods and time factors

The period between the points in time s < t is measured depending on the day count convention of the deal or currency. This convention is referred to below as z_{s,t} or z_{t} = z_{0,t}.

### Discount factors

#### Within a year

The discount factor D_{t} is calculated from the money market rate r_{t} for the point in time t and the time factor z_{t}.

This is based on the usual simple interest method for deals within a year.

#### Not within a year

The discount factors for n = 2, …, N years are calculated from the capital market rates r_{n} using the bootstrapping method. This is based on the discount factor D_{1} for one year (see above).

To calculate the discount factor D_{n}, all previous discount factors must already have been calculated.

Missing interest rates for complete fiscal years are interpolated in a linear manner before the discount factors are calculated.

### Forward interest rates

For the points in time s < t, the forward rate f(s,t) is calculated based on the discount factors D_{s} and D_{t}, interpolated in a linear manner if applicable.

Depending on the interest method, the time period between s and t is measured using the time factor z_{s,t}.

For use with simple interest rates:

For use with compound interest rates:

### Forward exchange rates (FX)

For two currencies W_{1} and W_{2}, the time t, the interest rates r_{1} and r_{2} or the discount factors D_{1} and D_{2}, and the FX spot K in price notation from the perspective of currency 1 (i.e. with the unit W_{1}/W_{2}), the forward rate T is calculated as

For simple interest, this corresponds to

where z_{1} and z_{2} measure the period up to the date t, depending on the interest method of the currencies.

### Fixed payments valuation

A cash flow C in currency 1 for future time t has the following value in currency 2

where K is the spot rate, T is the forward rate, and D_{t,I} are the discount rates of the two currencies.

### Variable payments valuation

A cash flow at time t whose value is unknown at the valuation date, but which is fixed at the time s < t, depending on a reference interest rate r known at that time (for the period of s to t), is evaluated using arbitrage considerations. This uses discount factors D_{s} and D_{t} or the forward rate f(s,t).

If the nominal amount N at any time t is paid for the payment rz_{s}_{,t}N, then the present value of the payment is

### Effective interest rate

To determine the effective interest rate of an interest-bearing deal, all actually flowing payment flows C_{n} at the points in time t_{n} between creditors and debtors are needed (n = 0, …, N). The time factors z_{n} are calculated from t_{n} for the interest method act/act. All types of payments (loan disbursement, repayments, interest payments, fees, etc.) are treated equally. The two cash flow directions are distinguished by different signs. Each sign must occur at least once.

Then, the effective interest rate r is a solution to the following equation:

In general, this equation cannot be solved algebraically. In Coupa Treasury, an approximate solution is determined using the Newton method.

Note: If the payments change their signs more than once over time, then this equation may have several solutions.

## Deal valuation

### FX forwards

The purchase at the date t of W_{1} currency units in currency 1 is valued against W_{2} currency units in currency 2 (i.e. at the deal rate K = W_{2}/W_{1}). For the valuation in currency 3, which may be identical to currency 1 or 2, the forward rates T_{3/1} and T_{3/2} (in price quotation from the perspective of currency 3) and the discount factor D_{3} of currency 3 for time t are used.

Then the present value of the forward is

Without discounting, the factor D3 is omitted. If currency 2 is the valuation currency, the calculation is simplified to

### Commodity forwards

In contrast to FX forwards, the forward commodity price is not calculated from the cash prices and the interest curve but is linearly approximated from a price curve defined by the market.

In all other respects, this valuation equals the valuation of FX deals.

### Money market, loans, interest rate swaps and leasing deals

Each cash flow generated by the deal is valued as a fixed or variable cash flow in accordance with the sections Fixed payments valuation and Variable payments valuation. The market value of the deal is the sum of the cash flow values.

### FX options

#### Historical volatilities

Historical volatilities in Coupa Treasury are calculated from the performance of the currency pair in the year prior to the cut-off date. If for N days of this year, the rates K_{1}, …, K_{N} are available, then the historical daily volatility is calculated as

The annualized daily volatility renders the annual volatility .

#### Implied volatilities

In Coupa Treasury, implied volatilities can be maintained in the market data depending on the strike and the remaining term of the option. To valuate an option with strike E and remaining time to maturity t, the volatility is interpolated from the existing volatility as follows:

If E_{1} is the largest strike with E_{1} ≤ E, and E_{2} is the smallest strike with E_{2} > E, t_{1} the largest remaining term t_{1} ≤ t and t_{2} the smallest remaining term t_{2} > t, so that for the four support points (E_{1}, t_{1}), (E_{1}, t_{2}), (E_{2}, t_{1}), (E_{2}, t_{2}), the volatilities are

, , , and

then

is used to calculate the interpolated volatility σ as

#### Plain vanilla options

Plain vanilla options are valuated using the Black-Scholes method modified by Garman and Kohlhagen. For the valuation of an option for one unit of currency 1 using currency 2 as the counter currency, the following parameters and functions are used:

The continuous compounded interest rates are calculated from the corresponding factors as

All rates are quoted in relation to currency 2.

With

and

the resulting value of a call is

and the value of a put is

#### Single barrier options

For the valuation of the different single-barrier options, the Up And In Call (UIC) and the Down And In Call (DIC) are valued as basic types. The barrier is B. The directions up and down rates in price quotations are from the perspective of currency 2. All other types of single-barrier options can be valuated from this and from the plain vanilla call as a linear combination or through permutation of the two currencies.

For valuation of the UIC and DIC, the following parameters are used:

and N_{i} = Ф(d_{i}) for i = 3, …, 8.

Thus, the value of a UIC is

For E > B, the value of a DIC is

and for E < B, it is

In addition to the options values, the Greeks are also calculated (i.e. various partial derivatives of the option value):

For plain vanilla options, the Greeks are calculated exactly. For single-barrier options, they are approximated using appropriate difference quotients.

### Interest rate options

In Coupa Treasury, caps, floors, and collars are valuated on the Black-Scholes method extended by Black (Black 76) or based on the Bachelier method. For the valuation of an individual caplet or floorlet, whose interest rate was fixed at point in time s_{1} and whose compensation may take place at time s_{2}, the following parameters are used:

#### Valuation according to the Black-Scholes Model

The period of the option’s reference interest rate is derived from the payment period of the option.

With

and

the market value of a caplet is

and the market value of a floorlet is

_{}.

The value of the whole cap (or floor) is the sum of the values of the individual caplets (or floorlets). The value of a collar is the difference between the values underlying the cap and floor.

The intrinsic value of a caplet is

D ⋅ max(*f − x*, 0) *Nz*

and the intrinsic value of a floorlet is

D ⋅ max(*x − f*, 0) *Nz*.

#### Valuation according to the Bachelier Model

If negative forward rates or strikes are involved, the Black-Scholes formula is no longer applicable due to the logarithm. In this case, the Bachelier model can be applied in the normal volatilities-based valuation.

The period of the option’s reference interest rate is derived from the payment period of the option.

With

the market value of the caplet is

and the market value of the floorlet is

_{}.

The value of the whole cap (floor) is the sum of the values of the individual caplets (floorlets). The value of a collar is the difference between the values of the underlying cap and floor.

The intrinsic value of a caplet is

D ⋅ max(*f − x*, 0) *Nz*

and the intrinsic value of a floorlet is

D ⋅ max(*x − f*, 0) *Nz*.

#### Interpolation of the Bachelier volatilities

The volatilities for evaluation with the Bachelier model are interpolated bilinearly with strike and time till exercise. A bilinear interpolation is done as follows:

Given four points on the volatility surface σ(t,X): (t_{1}, X_{1}, σ_{1}), (t_{1}, X_{2}, σ_{2}), (t_{2}, X_{1 },σ_{3}), (t_{2}, X_{2}, σ_{4}),

where t = time till exercise of the option, and X = its strike value. Then for an option with the time till exercise t_{option} and with the strike X_{option} to which: X_{1 }≤ X_{option}_{ }≤ X_{2} and t_{1 }≤ t_{option} ≤ t_{2}, its volatility value σ_{option} is computed via bilinear interpolation as follows:

where _{}and _{}

For estimation of the volatilities outside of the delivered market data, the constant extrapolation is used.

### Commodity options

Call and put options on commodities are typically traded as Asian options. This means that the option price is determined on the basis of the arithmetic average prices instead of closing rates. In this case, Coupa Treasury also relies on an extended Black-Scholes model, the so-called Modified Geometric Approximation model of Lévy and Turnbull (1992). It is based on the assumption of a log normal distributed stochastic commodity price curve (Geometric Brownian Motion), and in special cases (i.e. when only one closing rate is used) it corresponds exactly to the standard Black-Scholes model.

To evaluate a commodity option at the time T with maturity t, the following parameters will generally be used:

If the valuation date T is within the period for average calculation (T > t_{1}), some adjustments to various parameters need to be made. These can be summarized as follows:

For valuations during the corresponding period, all cut-off dates can be considered for the final option value. This is essential to allow a correct calculation of the settlement profile of Asian options, which depends on the average of all period rates.

#### Valuation of call and put options

Two special cases are worth looking at first. Only in the third case, the familiar Black-Scholes formula is needed for evaluating calls and puts.

- All period rates are known (i.e. T > t
_{n}) and therefore m = n:

The expected value is omitted, and the value of the option is derived directly from the settlement profile using the average exchange rate A.

- The adjusted strike drops below zero, so :

**Call**: The portion A_{fix}of the known rates in the overall average exceeds the agreed strike. In consequence, it is clear before the expiry of the option exercising it will be beneficial. Thus, the option value is calculated directly from Commodity forwards using the linearly approximated forward prices.

**Put**: Already becomes worthless before the expiry of the option, as a higher resale value is guaranteed to be achieved without the option.

- From the still unknown rates, an arithmetic average is calculated. This is also the first moment of the normal distribution used in the Black-Scholes formula.

The standard deviation from the normal distribution is also adjusted to fit the period of the unknown rates, based on the (implicit) commodity volatility . The formula is based on the assumption of log normally distributed specified prices X(t_{i}) and using a calculation of the moments of the arithmetic mean A, cf. Lévy and Turnbull (1992).

For valuations of the total average period:

For valuations of the closing rate (i.e. only one rate affects the valuation):

Now d_{1} and d_{2} can be calculated directly:

The option values will then be calculated as follows: