Module 5 · Derivatives

Pricing and Valuation of Forward Contracts

EN: No-arbitrage forward price and value during the contract's life.
VN: Giá forward không-arbitrage và giá trị qua thời gian.

1. Forward Price (No Income) Core

About: F = S(1+r)^T. The simplest no-arbitrage formula. Increases with spot, rate, and time.Tóm tắt: F = S(1+r)^T. Công thức no-arbitrage đơn giản nhất. Tăng theo spot, rate, time.

\[ F_0(T) = S_0 \cdot (1 + r)^{T} \]

Practice problem

Spot $80, r = 5% (annual), T = 6 months. Compute forward.

Show solution

F = 80 × (1.05)$^{0.5}$

= 80 × 1.0247

F ≈ $81.98

2. With Income (Dividends, Coupons) Core

About: Subtract PV of income (dividends, coupons) before compounding. Income lowers the forward (lazy way: 'short the spot you'd otherwise own → less benefit').Tóm tắt: Trừ PV income (cổ tức, coupon). Income làm giảm forward.

\[ F_0(T) = (S_0 - PV_0(I)) \cdot (1 + r)^{T} \]

Or with continuous yield

$F_0(T) = S_0 \cdot e^{(r - q) T}$ where q = continuous dividend yield.

Practice problem

Spot $100, r = 5%, T = 1 year, $4 dividend in 6 months. Compute forward.

Show solution

PV(div) = 4/1.05^0.5 ≈ 3.904

F = (100 − 3.904)(1.05)

F ≈ $100.90

3. With Storage Costs Core

About: Add PV of storage costs (commodities). Storage raises the forward — like a negative dividend.Tóm tắt: Cộng PV chi phí lưu kho (hàng hóa). Như cổ tức âm — đẩy forward lên.

\[ F_0(T) = (S_0 + PV_0(\text{costs})) \cdot (1 + r)^{T} \]

Practice problem

Commodity spot $50, r = 4%, storage cost PV = $1, T = 1 year. Compute forward.

Show solution

F = (50 + 1)(1.04)

F = $53.04

4. Value of a Forward During Its Life Core

About: V_t(long) = (F_t − F_0) / (1+r)^(T−t). Mark-to-market of an existing forward. Long benefits when current forward > original.Tóm tắt: V_t(long) = (F_t − F_0)/(1+r)^(T−t). MTM forward đang mở. Long lời khi F_t > F_0.

\[ V_t(\text{long}) = \frac{F_t(T) - F_0(T)}{(1 + r)^{T - t}} \]

Components

F_t(T) Forward rate at time t for delivery at T (current market quote).
F₀(T) Original locked-in forward price.
T − t Remaining time to delivery.

Long benefits when current forward > original; short is the negative of this.

Practice problem

Original forward locked at $100 (1 year). 6 months later, current 6-month forward = $103, r = 5% annual. Value to long?

Show solution

V = (103 − 100) / (1.05)$^{0.5}$

= 3 / 1.0247

V ≈ $2.93 to long

Practice problem

Spot price of a non-dividend stock is $100. Risk-free rate 5% (annual compounding). 1-year forward price?

Show solution

F = 100(1.05)¹

= $105