Actuarial Mathematics for Modelling · Financial instruments and insurance contracts as cashflow models

Forwards, Futures and Options Cashflows Explained

Updated 11 October 2026 · Fact-checked

A derivative is a contract whose cashflows depend on an underlying asset's price. A forward fixes a delivery price paid at expiry. A future is a standardised, margined forward. A call pays max(S − K, 0) and a put pays max(K − S, 0). To solve, find the expiry price, then the payoff.

Understand Derivatives: Forwards, Futures and Options Cashflows

A derivative is a contract whose value comes from an underlying asset, such as a share, a bond or an index. You do not need to own the asset. You only need to know what cashflow the contract creates and when.

A forward contract is an agreement to buy or sell the underlying at a future date, called the expiry or maturity, at a price fixed today, called the delivery price (K). It costs nothing to enter at the start if K is set at the fair forward price. At expiry, the long party pays K and receives the asset worth S. The long payoff is S − K. The short payoff is K − S. Either can be negative, so a forward carries a real risk of loss.

A future is like a forward, but it is traded on an exchange and is standardised. It is marked to market, usually daily. Gains and losses are settled through a margin account as prices move, so cashflows occur before expiry. This reduces default risk. A forward is private, settled once at expiry, and so carries more counterparty risk.

An option gives the holder a right, not an obligation. A call is the right to buy at the strike price K. A put is the right to sell at K. The holder pays a premium at the start. At expiry the holder exercises only if it helps. So a call payoff is max(S − K, 0) and a put payoff is max(K − S, 0). The writer (seller) receives the premium and has the opposite payoff, which can be a large loss. Subtract the premium (accumulated with interest, if asked) to get the profit.

The forward price comes from no-arbitrage. Buy the asset now with borrowed money, hold it until expiry and deliver it. The forward price must equal the cost of doing that. Income from the asset lowers the forward price, and the cost of borrowing raises it.

Key rules to remember

Forward price, asset with no income: K = S₀ × (1 + i)^T, or K = S₀ × e^(δT); S₀ is the spot price now, T is the time to expiry in years, i is the effective annual risk-free rate and δ the force of interest. Assumes no income, no storage costs and no arbitrage.
Forward price, asset with known income: K = (S₀ − I) × (1 + i)^T, where I = present value of income paid before expiry; Discount each income payment to time 0 at the risk-free rate before subtracting.
Forward price, continuous dividend yield: K = S₀ × e^((δ − d)T); d is the continuously compounded dividend yield. Use only when the question gives a yield.
Long forward payoff at expiry: S_T − K; Short forward payoff is K − S_T.
Long call payoff: max(S_T − K, 0); Writer's payoff is −max(S_T − K, 0).
Long put payoff: max(K − S_T, 0); Writer's payoff is −max(K − S_T, 0).
Option profit for the holder: Payoff − premium × (1 + i)^T; Accumulate the premium only if the question asks for profit including interest.
Value of an existing forward at time t: (F_t − K) × v^(T − t); F_t is the current forward price for the same expiry. This is the value of the long position, discounted at the risk-free rate.

How to solve Derivatives: Forwards, Futures and Options Cashflows questions

Use this order for any question on forward, future or option cashflows.

1Identify the contract type and whether you are long or short, or holder or writer.
2List the inputs: spot price S₀, delivery or strike price K, expiry T, interest rate and any income on the asset.
3If the question asks for a forward price, apply no-arbitrage. Accumulate S₀ and subtract the accumulated value of income.
4Find the underlying price at expiry S_T. Use the value given or the scenario in the question.
5Compute the payoff using the correct formula. For options, apply the max function and decide whether it is exercised.
6Add the timing. A forward pays once at expiry. A future pays on each marking date. An option has the premium at the start and the payoff at expiry.
7Convert to profit if asked by including the premium and any interest. Check the sign for the writer or short party.
8State the final cashflow with its time, its direction (received or paid) and units in rupees.

Quickest way: Payoff-first shortcut

When to use it: Use this for multiple-choice questions where S_T, K and the position are given and you only need the payoff.

Write S_T and K side by side.
For a forward, subtract: long is S_T − K, short is the reverse.
For a call, ask if S_T > K. If yes, payoff is S_T − K. If no, it is 0.
For a put, ask if K > S_T. If yes, payoff is K − S_T. If no, it is 0.
Flip the sign for the writer or short position.
Only then subtract the premium if profit is asked.

Common mistakes in Derivatives: Forwards, Futures and Options Cashflows

Giving an option a negative payoff for the holder.
You apply the forward formula S − K without the max function.
Fix: The holder will not exercise at a loss. The payoff is at least 0. The premium is the maximum loss.
Confusing delivery price with the spot price at expiry.
Both are prices of the same asset and are labelled K and S in different places.
Fix: K is fixed at the start in the contract. S_T is the market price at expiry. The payoff compares the two.
Forgetting to deduct income from the forward price calculation.
You use the no-income formula out of habit.
Fix: Check the question for dividends or coupons. Subtract their present value from S₀ before accumulating.
Saying that futures and forwards have the same cashflow timing.
Both fix a price for future delivery, so they look alike.
Fix: A forward settles once at expiry. A future settles through daily margin payments. The total gain is similar but the timing differs.
Ignoring the writer's side or mixing up long and short.
Questions often ask for the counterparty's cashflow, and you reuse the holder's sign.
Fix: The writer's payoff is the negative of the holder's. Write long or short next to each answer.
Using the wrong time in the accumulation factor.
You use the time to the next event instead of the time to expiry, or use months as years.
Fix: Convert T to years and use the same time unit as the rate.

Worked examples

Example 1

A share has a spot price of ₹500 and pays no dividends. The risk-free effective annual interest rate is 6%. Find the no-arbitrage price of a forward contract to buy the share in 2 years, and the long party's payoff if the share price in 2 years is ₹610.

Show the solution

There is no income, so K = S₀ × (1 + i)^T.
K = 500 × 1.06².
1.06² = 1.1236.
K = 500 × 1.1236 = ₹561.80.
Long payoff = S_T − K = 610 − 561.80 = ₹48.20.
The short party pays out ₹48.20, so the short payoff is −₹48.20.

Answer: Forward price is ₹561.80. The long party receives a payoff of ₹48.20 at expiry.

Example 2

An investor buys a 1-year European call on a share with strike ₹800 for a premium of ₹40, and a 1-year European put on the same share with strike ₹800 for a premium of ₹30. The effective annual rate is 5%. Find the total profit at expiry if the share price is ₹900, and then if it is ₹700. Include interest on the premiums.

Show the solution

Total premium paid at start = 40 + 30 = ₹70.
Accumulated premium at expiry = 70 × 1.05 = ₹73.50.
Case 1, S_T = 900. Call payoff = max(900 − 800, 0) = ₹100. Put payoff = max(800 − 900, 0) = ₹0.
Total payoff = ₹100. Profit = 100 − 73.50 = ₹26.50.
Case 2, S_T = 700. Call payoff = max(700 − 800, 0) = ₹0. Put payoff = max(800 − 700, 0) = ₹100.
Total payoff = ₹100. Profit = 100 − 73.50 = ₹26.50.

Answer: The profit is ₹26.50 in both cases.

Exam tips

Write the payoff formula first, then substitute. This earns method marks even if the arithmetic slips.
Read the question for dividends, coupons or storage costs before choosing a forward price formula.
State clearly whether you are giving a payoff or a profit, and whether premium interest is included.
For futures questions, describe the timing difference: margin settlement before expiry versus one payment at expiry.
In computer-based papers, build the payoff in a single formula using a MAX function and check one case by hand.

Practice questions from Financial instruments and insurance contracts as cashflow models

Derivatives: Forwards, Futures and Options Cashflows in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Derivatives: Forwards, Futures and Options Cashflows: frequently asked questions

What is the difference between forwards and futures?

A forward is a private contract settled once at expiry. A future is standardised, traded on an exchange and marked to market with margin payments before expiry. Futures carry less default risk because the exchange clears the trades.

How do I calculate a forward price in actuarial questions?

Accumulate the spot price at the risk-free rate to the expiry date. If the asset pays income before expiry, subtract the accumulated value of that income, or subtract its present value from the spot price first. This follows from no-arbitrage.

What is the payoff of a call and a put option?

A long call pays max(S_T − K, 0). A long put pays max(K − S_T, 0). S_T is the asset price at expiry and K is the strike. The writer's payoff is the negative of these.

Does a forward contract cost anything to enter?

If the delivery price is set at the fair forward price, the contract has zero value at the start and no premium is paid. Its value changes later as the market price and interest rates move. An option, in contrast, always requires a premium.