Question

Suppose that \(S\) is the price process of a dividend paying asset with dividend process \(D .\) (a) Show that the forward price \(f\left(t, x, T, S_{D}\right)\) is given by the cost of carry formula $$ f\left(t, x, T, S_{T}\right)=\frac{1}{p(t, T)}\left(S_{t}-E_{t, x}^{Q}\left[\int_{t}^{T} \exp \left\\{-\int_{t}^{s} r(u) d u\right\\} d D_{s}\right]\right) $$ Hint: Use the cost of carry formula for dividend paying assets. (b) Now assume that the short rate \(r\) is deterministic but possibly time- varying. Show that in this case the formula above can be written as $$ f\left(t, x, T, S_{T}\right)=\frac{S_{l}}{p(t, T)}-E_{l, x}^{\otimes}\left[\int_{t}^{T} \exp \left\\{-\int_{s}^{T} r(u) d u\right\\} d D_{s}\right] $$

Short answer

Question: Derive the forward price of a dividend-paying asset using the cost of carry formula, assuming a deterministic short rate. Also, provide the general formula for the forward price. Answer: The general formula for the forward price of a dividend-paying asset is given by: $$ f(t, x, T, S_T) = \frac{S_t - E_{t,x}^{Q}\left[\int_t^T \exp\left\{-\int_t^s r(u)du\right\}dD_s\right]}{p(t, T)} $$ Assuming a deterministic short rate, the forward price formula can be simplified as: $$ f(t, x, T, S_T) = \frac{S_l}{p(t, T)} - E_{l, x}^{\otimes}\left[\int_t^T \exp\left\{-\int_s^T r(u)du\right\}dD_s\right] $$

Step-by-step solution

Cost of carry formula for dividend-paying assets

The cost of carry formula for a dividend-paying asset is given by: $$ F_t = S_t \cdot e^{(r - d)(T-t)} $$ where \(F_t\) is the forward price, \(S_t\) is the asset price at time \(t\), \(r\) is the risk-free interest rate, \(d\) is the dividend yield, and \(T-t\) is the time to maturity.

Derive the forward price formula (Part A)

Using the cost of carry formula, we can express the forward price as: $$ f(t, x, T, S_T) = \frac{S_t - E_{t,x}^{Q}\left[\int_t^T \exp\left\{-\int_t^s r(u)du\right\}dD_s\right]}{p(t, T)} $$ where \(E_{t,x}^{Q}\) denotes the conditional expectation under the risk-neutral measure \(Q\), and \(p(t,T)\) is the price of a zero-coupon bond with maturity \(T\) and price at time \(t\).

Simplify the formula assuming deterministic short rate (Part B)

Since the short rate \(r\) is deterministic, we can express the forward price as: $$ f(t, x, T, S_T) = \frac{S_l}{p(t, T)} - E_{l, x}^{\otimes}\left[\int_t^T \exp\left\{-\int_s^T r(u)du\right\}dD_s\right] $$ This applies the deterministic rate nature of the problem, and shows that the forward price formula from part (a) can be written in the desired form.

Key concepts

Cost of Carry Formula

The cost of carry formula plays a key role in the financial world, especially when dealing with commodities or securities such as stocks that can generate income over time, like dividends. This formula helps us to understand the relationship between an asset's current price and its future price, known as the forward price.

For a dividend-paying asset, the cost of carry formula takes into account not just the asset's current price and the risk-free interest rate, but also the expected income from dividends. Typically, the formula is:
\[F_t = S_t \cdot e^{(r - d)(T-t)} \]
where \(F_t\) is the forward price, \(S_t\) represents the asset's price at time \(t\), \(r\) is the risk-free rate, \(d\) is the dividend yield, and \(T-t\) is the time until the forward contract expires. The exponential term, \(e^{(r - d)(T-t)}\), encapsulates the 'cost of carry' which includes both the cost of financing the asset and the dividends expected to be received during the holding period.

Dividend Paying Assets

Assets that pay dividends, such as many stocks, provide periodic payments to their holders and are thus attractive to investors seeking regular income. When modeling the expected future price of such assets, the dividends must be factored into the equation.

In the forward price formula, dividends are accounted for by subtracting the present value of expected dividend payments during the life of the forward contract. This is expressed as:
\[E_{t,x}^{Q}\left[\int_t^T \exp\left\{-\int_t^s r(u)du\right\}dD_s\right]\]
where \(E_{t,x}^{Q}\) represents the risk-neutral expectation of the dividend's present value, while the integral accounts for the continuous nature of the dividend payouts and the discounting due to the risk-free rate.

Risk-Neutral Measure

The concept of a risk-neutral measure is central to modern financial theory, particularly in pricing derivatives like forward contracts. In a risk-neutral world, investors are indifferent to risk; they expect to earn the risk-free rate of return regardless of the riskiness of the investment.

Mathematically, the risk-neutral measure, often denoted by \(Q\), is a probabilistic tool used to calculate the expected present values of future cash flows, like dividends. By using the risk-neutral measure, \(E_{t,x}^{Q}\), the complexities of risk preferences are neglected, simplifying the calculation of forward prices. Essentially, it allows us to view future cash flows as if they were as secure as a risk-free asset, and thus we only need to discount them at the risk-free rate.

Zero-Coupon Bond

A zero-coupon bond is a type of bond that does not pay periodic interest or 'coupons'. Instead, it is issued at a discount to its face value and pays its full face value at maturity. The difference between the purchase price and the face value represents the investor's earnings.

In financial formulas, such as the forward price formula, a zero-coupon bond is used to determine the present value of a future sum of money. In the forward price formula, the price of the zero-coupon bond, \(p(t, T)\), serves as a discounting factor. It shows how much one would need to invest in a risk-free bond today to receive the asset's price at the forward contract's maturity, minus expected dividends, if any.