Topic 3 – Discounting and forwards — Desk–Quant Dialogue Playbook

1. Discount factors: the basic object

We want a way to say: “1 unit of cash at time T is worth how much today?”

That number is the discount factor $D(0,T)$ :

$\text{PV of 1 at time } T \text{ today} = D(0,T)$

So:

Future cash flow $X$ at $T$ → present value = $X \cdot D(0,T)$ .
Discount factor is always between 0 and 1 (for positive rates).

Different rate conventions are just different ways to parameterise $D(0,T)$ .

2. Compounding conventions and their relation to discount factors

2.1 Simple interest

Given a simple rate $r_{\text{simple}}$ (per year):

$\text{Future value of 1} = 1 + r_{\text{simple}} T$

So:

$D(0,T) = \frac{1}{1 + r_{\text{simple}} T}$

Used for short-term money-market instruments and some quotes.

2.2 Periodic compounding (e.g. annual, semi-annual)

Rate $R$ , compounded $m$ times per year:

$\text{Future value} = \left(1 + \frac{R}{m}\right)^{mT}$

So:

$D(0,T) = \left(1 + \frac{R}{m}\right)^{-mT}$

Examples:

$m = 1$ : annual compounding.
$m = 2$ : semi-annual.
$m = 4$ : quarterly, etc.

2.3 Continuous compounding

Continuously compounded rate $r$ :

$\text{Future value} = e^{rT} \quad\Rightarrow\quad D(0,T) = e^{-rT}$

This is what we’ve used in the previous topics.

2.4 Yield vs discount factor

All these can be inverted:

Given $D(0,T)$ , you can define:

Simple yield:

$r_{\text{simple}} = \frac{1}{T}\left(\frac{1}{D(0,T)} - 1\right)$
m-times compounding rate:

$R_m = m\left(\left(\frac{1}{D(0,T)}\right)^{1/(mT)} - 1\right)$
Continuous rate:

$r = -\frac{1}{T} \ln D(0,T)$

Practitioner takeaway:

The economic object is $D(0,T)$ . “5% simple vs 4.88% continuous” are just different labels on the same $D(0,T)$ . If you get your $D(0,T)$ wrong, all pricing is wrong.

3. Forwards from discounting: clean frictionless case

Assume an asset with:

Spot price $S_0$ ,
No income (no dividend / convenience yield),
You can borrow/lend at risk-free curve $D(0,T)$ ,
You can short/hold the asset freely.

You want the forward price $F_{0,T}$ for delivery at $T$ .

3.1 Cost-of-carry argument (no-arb)

Strategy 1 – Buy-and-hold:

At time 0: borrow $S_0$ , buy the asset.
At time T: deliver the asset, repay the loan:
- Loan repayment = $S_0 / D(0,T)$ .
Your “effective forward sale price” is $S_0 / D(0,T)$ .

Strategy 2 – Enter a forward:

At time 0: no cash.
At time T: deliver asset via forward, receive $F_{0,T}$ .

No-arbitrage ⇒ both strategies should give same result in PV terms, so:

$F_{0,T} = \frac{S_0}{D(0,T)}$

Under continuous compounding with constant rate $r$ :

$F_{0,T} = S_0 e^{rT}$

Under simple rate $r_{\text{simple}}$ :

$F_{0,T} = S_0 (1 + r_{\text{simple}} T)$

Key idea:

Forward price is spot grown at the funding curve.

4. Commodities twist: storage, income, convenience yield

In equities, you often add:

Dividends (income),
Repo / funding.

In commodities, similar idea:

Storage cost: paying to hold inventory.
Income / yield: e.g. collateral yield, or quality premia.
Convenience yield: benefit of having the physical commodity (security of supply).

Generic formula (continuous rates):

$F_{0,T} = S_0 e^{(r + u - y)T}$

where:

$r$ = funding rate,
$u$ = storage cost rate,
$y$ = income / convenience yield rate.

For power specifically:

Often you don’t think of “spot grown at rate”. Instead, the forward curve is the primary price object; “spot” is just the delivery settlement.

5. Curve building concepts (very high level)

You almost never have a neat analytic curve in practice. You have quotes:

Interest-rate curve: deposits, futures, swaps → bootstrap discount factors $D(0,T)$ .
Commodity forward curve: monthly, quarterly, yearly forward prices → build a smooth term structure $F(0, T)$ .

5.1 Interest-rate curve (for discounting)

Pipeline:

Start from cash instruments for very short maturities (O/N, 1W, 1M).
Use futures / FRAs for short-to-medium tenors.
Use swaps for long end.
Bootstrap discount factors $D(0,T_i)$ sequentially.
Interpolate in a chosen space:
- DF space,
- zero rates,
- instantaneous forward rates.

Technical details (day count, business-day rules, OIS vs Libor) matter in production, but the mental model is:

“Solve for the curve of discount factors that makes the PV of each traded instrument equal its market price.”

5.2 Forward curve for commodities

Similar but with forwards:

You have traded products: Cal-27, Q1-26, Jan-26, etc.
Each is a volume-weighted average of underlying hourly/daily forward prices.
You choose a parametric or spline model for the underlying continuous forward curve $F(0, t)$ .
Fit parameters so that integrals/averages over delivery periods match market quotes.

Systems consequence:

For any cash flow at time bucket tit_i:
- Use discount curve to PV (finance).
- Use forward curve to project expected price (commodity).

6. Connecting back to options (you’ll need this instinct)

In the no-arbitrage / Black–Scholes world:

Discounting uses $D(0,T)$ .
Forwards use $F_{0,T} = S_0 / D(0,T)$ (or adjusted for carry).
European option on the forward is priced off:
- that forward,
- that discount factor,
- and a vol surface.

So when they ask, “What inputs does your pricer need?” you say:

“For a European option I need: the discount curve, the forward curve for the underlying, and the volatility surface. Discount curve gives me PV, forward curve gives the $F$ s, vol surface gives me the distribution shape.”

7. Interview soundbites

Short things you can say:

On discounting: “I treat discount factors $D(0,T)$ as the primitive. Rates are just parametrisations of $D$ . Any cash flow at $T$ gets multiplied by $D(0,T)$ to give PV.”
On forwards: “In frictionless markets, forwards are just spot grown at the funding curve: $F = S / D$ . In commodities we add carry and convenience yield, but the principle is the same: the forward curve is locked to spot and funding by no-arbitrage.”
On curve building: “We bootstrap discount factors from deposits/futures/swaps, and a commodity forward curve from traded forwards. Then we interpolate in a way that preserves no-arbitrage and is stable for hedging.”