Mixed Forward And Option Valuation Across Payment Dates

Why One Focal Date Matters

A mixed derivative position can produce cash flows at several dates: a premium may be paid when an option is entered, an option payoff may occur at expiration, an asset distribution may be paid in between, and a forward may settle later. These amounts cannot be added directly unless they are first expressed at the same date. The reliable method is to choose a focal date and accumulate or discount every cash flow to it.

This tutorial uses a deterministic effective annual interest rate and cash flows at integer-year dates. Unless stated otherwise, distributions are known amounts paid to holders of the underlying asset, and derivative contracts are cash-settled at their stated settlement dates. The forward itself does not pay the holder interim distributions. Rates and dates must be converted consistently if a problem uses a different compounding interval.

Definition: A signed cash flow $C_t$ at time $t$ is positive when received and negative when paid. At effective annual rate $i$, its value at focal date $T$, for $T\geq t$, is $C_t(1+i)^{T-t}$. If $t>T$, its value at $T$ is $C_t/(1+i)^{t-T}$. The net value at the focal date is the sum of all cash flows expressed at that date.

For a position with cash flows $C_0,C_1,\ldots,C_T$, where $C_t$ is the net cash flow at time $t$, the accumulated value at time $T$ is

V_T=\sum_{t=0}^{T} C_t(1+i)^{T-t}.

A negative $V_T$ means the position has a net accumulated cost over the stated cash flows; a positive $V_T$ means a net accumulated gain. This calculation measures the value of the specified cash flows under the given rate. It does not, by itself, establish that an option premium is fair or that a quoted derivative price is free of arbitrage.

Forward Prices When the Asset Pays Distributions

Suppose an asset has current price $S_0$ and will pay known distributions $D_j$ at times $t_j$, all before a forward maturity $T$. Under a constant effective annual rate $i$, the fair delivery price $K_0$ for a forward initiated at time 0 is

K_0=S_0(1+i)^T-\sum_j D_j(1+i)^{T-t_j}.

The distribution amounts are subtracted because an investor who owns the asset receives them, while a long forward holder does not. Each distribution is accumulated from its payment date to the forward maturity before it is deducted from the accumulated spot price.

Assumptions: The formula applies to a forward with no initial payment, known distributions, and deterministic interest at the stated rate. It presumes that the asset and distributions can be traded or replicated without relevant transaction costs, taxes, or constraints. If distributions are uncertain, their treatment requires additional assumptions; simply subtracting an expected distribution does not automatically give the exact fair price.

The formula follows by comparing ownership financed at time 0 with a forward purchase. Buy one unit of the asset for $S_0$, financing that amount, and retain each distribution. At time $T$, the financing balance is $S_0(1+i)^T$. If each distribution is accumulated at the same rate until $T$, their total value then is $\sum_jD_j(1+i)^{T-t_j}$. Thus the net cost of owning the asset at $T$, after crediting the distributions, is the right-hand side of the formula. A forward with delivery price equal to this net cost has zero value at initiation under these assumptions.

S_0(1+i)^T-\sum_jD_j(1+i)^{T-t_j} = \left(S_0-\sum_j\frac{D_j}{(1+i)^{t_j}}\right)(1+i)^T.

The two forms are equivalent: the first accumulates everything to maturity, while the second subtracts the present values of the distributions before accumulating to maturity. The focal-date form is especially useful when the question also includes option cash flows at other dates.

Worked Example: Fair Forward Price With Two Distributions

An asset costs $300 today and pays known distributions of $12 at time 1 and $8 at time 2. Find the fair delivery price for a three-year forward if the effective annual interest rate is 4%.

Method. Accumulate the spot price and each distribution to time 3. The distributions belong to an asset holder, not to the forward holder, so they reduce the forward delivery price.

The accumulated spot price is

$$ 300(1.04)^3=300(1.124864)=337.4592. $$

The time-1 distribution accumulates for two years, and the time-2 distribution accumulates for one year:

$$ 12(1.04)^2=12(1.0816)=12.9792, $$

$$ 8(1.04)=8.32. $$

Therefore,

$$ K_0=337.4592-12.9792-8.32=316.16. $$

The fair delivery price is $316.16. As a check, the total accumulated value of the distributions is $21.2992, and $337.4592-$21.2992=$316.16. The common mistake is to deduct $12 and $8 without accumulating them to time 3; distributions paid earlier have time to earn interest before the forward maturity.

Combining Option and Forward Cash Flows

An option’s payoff is not the same as its profit. For a call with strike price $X$ and underlying price $S_T$ at expiration, the long call payoff is $\max(S_T-X,0)$. The profit over a period must also account for the premium paid and the dates of both amounts. A premium paid at time 0 and a payoff received at time 2 cannot be subtracted as raw dollar amounts when the requested result is at time 3.

Likewise, a long forward settled in cash at time $T$ has settlement $S_T-K$, and a short forward has settlement $K-S_T$. These are settlement amounts, not values automatically expressed at another date. Accumulate them from settlement to the chosen focal date when necessary.

Choose the focal date.
Use the date requested in the problem, often the latest cash-flow date.

List each cash flow with its date and sign.
Record option premiums as payments, option payoffs as receipts, and forward settlements using the position’s direction.

Convert each amount to the focal date.
Accumulate earlier amounts or discount later amounts using the specified interest basis.

Add and check.
Confirm that every amount is at the same date and that distributions are credited only to the party entitled to receive them.

Worked Example: Option Premium and Forward Settlement at Different Dates

An investor pays $14 at time 0 for a cash-settled call expiring at time 2, with strike $105. The underlying price at time 2 is $115, so the call pays its intrinsic value. The investor also takes a short three-year forward on an asset with $S_0=$$100, known distributions of $4 at time 1 and $3 at time 2, and effective annual interest of 5%. At time 3, the asset price is $110. Find the accumulated net value of these positions at time 3. Assume the call premium and forward are the only cash flows in the positions apart from their stated settlements.

Method. First find the fair delivery price for the forward, then calculate each position’s signed cash flows and accumulate them all to time 3. The option holder does not receive the asset’s distributions.

The forward’s fair delivery price is

$$ K_0=100(1.05)^3-4(1.05)^2-3(1.05). $$

The three components are $100(1.157625)=115.7625$, $4(1.1025)=4.41$, and $3(1.05)=3.15$. Hence

$$ K_0=115.7625-4.41-3.15=108.2025. $$

The short forward settlement at time 3 is $K_0-S_3$:

$$ 108.2025-110=-1.7975. $$

The call payoff at time 2 is $\max(115-105,0)=10$, which accumulates one year to $10(1.05)=10.50$. The $14 premium paid at time 0 accumulates three years to $14(1.05)^3=14(1.157625)=16.20675$, a negative cash flow. Thus the net value at time 3 is

$$ 10.50-1.7975-16.20675=-7.50425. $$

Rounded to the nearest cent, the accumulated net value is -$7.50. As a check, the option contributes $10.50-16.20675=-5.70675$ at time 3, and the short forward contributes $-1.7975$; their sum is $-7.50425$. The result is not the call payoff alone: it includes the accumulated premium and the forward settlement.

A Multi-Date Portfolio Calculation

When several contracts and dates appear together, a signed cash-flow ledger helps prevent a premium, payoff, or settlement from being counted twice. A distribution can be relevant to the forward delivery price even though it is not a cash flow received by a forward holder. Keep those roles separate: use the distribution to calculate the fair forward price, then include only actual cash flows of the stated portfolio in its focal-date total.

Worked Example: Long Forwards and a Later-Premium Call

At time 0, an investor enters two long three-year forwards on an asset with $S_0=$$100. The asset pays known distributions of $3 at time 1 and $2 at time 2. The effective annual interest rate is 4%, and each forward has the fair delivery price. The investor also buys a call at time 1, paying a premium of $6 then; the call expires at time 2 with strike $110. At time 2 the underlying price is $115, and at time 3 it is $112. Find the accumulated net value of the positions at time 3. The call is cash-settled, and the investor does not own the underlying asset.

Method. Determine the forward delivery price using distributions accumulated to time 3. Then calculate the two forward settlements and the call’s premium and payoff, expressing every cash flow at time 3.

For one forward, the fair delivery price is

$$ K_0=100(1.04)^3-3(1.04)^2-2(1.04). $$

The accumulated spot price is $100(1.124864)=112.4864$. The distributions accumulate to $3(1.0816)=3.2448$ and $2(1.04)=2.08$. Therefore,

$$ K_0=112.4864-3.2448-2.08=107.1616. $$

Each long forward settles at time 3 for $S_3-K_0=112-107.1616=4.8384$. Two contracts therefore produce

$$ 2(4.8384)=9.6768. $$

The call payoff at time 2 is $\max(115-110,0)=5$, which accumulates to $5(1.04)=5.20$ at time 3. The premium is paid at time 1 and accumulates for two years, giving $6(1.04)^2=6(1.0816)=6.4896$. The net value at time 3 is

$$ 9.6768+5.20-6.4896=8.3872. $$

Rounded to the nearest cent, the result is $8.39. The forward settlement check is $2(S_3-K_0)=2(112-107.1616)=9.6768$; the option’s time-3 net value is $5.20-6.4896=-1.2896$. Adding these gives $9.6768-1.2896=8.3872$, as above. The distributions affect the fair forward price but are not added again as receipts to this investor.

Common Timing and Sign Traps

The most frequent errors arise from mixing dates or confusing a contract payoff with the net result of holding the position. For example, in the previous example, subtracting the $6 premium directly from the $5 call payoff would give $-$1$. That calculation is wrong for a time-3 result: the premium was paid at time 1, while the payoff was received at time 2. At time 3 their values are $-$6.4896$ and $+$5.20$, respectively, so their combined value is $-$1.2896$.

Using the wrong forward direction. A long forward settles for $S_T-K$; a short forward settles for $K-S_T$.
Subtracting distributions at the wrong date. For a maturity-$T$ forward, a distribution at time $t_j$ is accumulated by $(1+i)^{T-t_j}$ when working at maturity.
Crediting distributions to an option holder. An option on an asset does not itself give its holder the distributions paid to owners of the asset.
Adding cash flows before aligning dates. A premium at time 0 and a payoff at time 2 are not directly comparable as dollars at time 3.
Calling a payoff “profit.” A payoff describes the contract settlement; profit also depends on the premium and any other position cash flows.

Key takeaway: Write every cash flow with its date and sign, choose one focal date, and convert each amount to that date before adding. Forwards with known interim distributions require distribution amounts to be adjusted for their timing; option premiums and payoffs must also be carried from their actual payment dates.

Check Your Understanding

For each problem, use the stated effective annual rate and express the requested result at the specified date.

An asset costs $250 and pays a known distribution of $5 at time 1. Find the fair delivery price for a two-year forward at 4%. (Answer: $265.20)
A call premium of $8 is paid at time 0. The call pays $18 at time 2. Find the accumulated net value at time 3 if the effective annual rate is 5%. (Answer: $9.64)
An asset costs $400 and pays a known distribution of $10 at time 2. Find the fair four-year forward delivery price at 3%. (Answer: $439.59)
An investor is short a two-year forward on an asset with $S_0=$$80 and a known $2 distribution at time 1. The effective annual rate is 5%, and the time-2 asset price is $90. The investor also paid $4 at time 0 for a call that expires at time 1 with strike $85; the time-1 asset price is $90. Find the combined accumulated net value at time 2. (Answer: -$3.06)
An asset costs $120 and pays a known $4 distribution at time 1. An investor takes a long three-year forward at the fair delivery price; the time-3 asset price is $140. At time 2, the investor pays $7 for a call expiring at time 3 with strike $130; the time-3 asset price is $140. Find the combined net value at time 4 if the effective annual rate is 5%. (Answer: $8.55)

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