Pricing Stock Forwards with Stochastic Interest Rates
Summary
The document derives the no-arbitrage forward price for a non-dividend-paying stock when interest rates are stochastic. The forward delivery price equals the current stock price divided by the price of a zero-coupon bond maturing on the delivery date. That bond price is the risk-neutral conditional expectation of the discount factor over the remaining life of the contract, so it incorporates uncertainty in future short rates.
The explanation first uses discounted risk-neutral valuation of the forward payoff and then checks the result with a replicating portfolio: short the stock and buy enough zero-coupon bonds to offset it, alongside the forward. This argument assumes an arbitrage-free setting and a stock with no dividends. It clarifies why simply substituting a random rate into a constant-rate exponential formula is not generally the right pricing step; the relevant discount factor is summarized by the bond price.
Key ideas
- For a non-dividend-paying stock, the forward delivery price is spot divided by the matching zero-coupon bond price.
- With stochastic rates, the bond price is the risk-neutral conditional expectation of the accumulated discount factor.
- The relationship follows from discounted valuation of the forward payoff under the risk-neutral measure.
- A replicating portfolio of a short stock position and zero-coupon bonds gives an arbitrage-based check.
- The derivation assumes no dividends and an arbitrage-free market.
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Full text
# Stochastic Interest rate spot forward relationship
# Stochastic Interest rate spot forward relationship
For a stock that pays no dividends, and has constant interest rates, I know that the relationship can be described with:
$F(t) = s(t). e^{r(T-t)}$
F(t) = Forward price, S(t) = Stock price, r = risk free rate, T = maturity date, t = time now
How does the equation change, when rates are stochastic?
edit
From another textbook, it appears the relationship can be modeled as:
$ F(t) = s(t) / P(t) $
where $ P(t) = e^{-\int_{t}^{T} r(s) ds} $
does this look right?
## Answer by Gordon (score 5)
https://quant.stackexchange.com/a/33291
The forward price $K$, determined at time $t$, is the amount such that the payoff at time $T$ is $S_T-K$, while the value at time $t$ is zero. That is, \begin{align*} B_t E\left(\frac{S_T-K}{B_T}\mid \mathcal{F}_t \right)= 0, \end{align*} Where $E$ is the risk-neutral expectation operator. Then, \begin{align*} K&=\frac{E\left(\frac{S_T}{B_T}\mid \mathcal{F}_t \right)}{E\left(\frac{1}{B_T}\mid \mathcal{F}_t \right)}\\ &=\frac{E\left(\frac{S_T}{B_T}\mid \mathcal{F}_t \right)}{\frac{1}{B_t}E\left(\frac{B_t}{B_T}\mid \mathcal{F}_t \right)}\\ &=\frac {\frac{ S_t}{ B_t} }{\frac{1}{B_t}P (t,T)}\\ &=\frac{S_t}{P(t,T)}, \end{align*} where \begin{align*} P (t,T) &= E\left(\frac{B_t}{ B_T}\mid \mathcal {F}_t \right)\\ &=E\left(e^{-\int_t^T r_s ds}\mid \mathcal {F}_t \right) \end{align*} is the price at time $t$ of a zero-coupon bond with maturity $T$ and unit face value.
Alternatively, at time $t$,
- enter into a forward contract with forward price $K$, which has zero cost at time $t$,
- short one share with income $S_t$, and
- long $\frac{S_t}{P (t,T)}$ units of zero-coupon bond with maturity $T$.
The net cost at time $t$ is zero. At maturity $T$,
- the forward contract has value $S_T-K$,
- the short position of one share has value $-S_T$, and
- the zero-coupon bond has value $\frac{S_t}{P (t,T)}$.
Assuming arbitrage free, the value at time $T$ is then \begin{align*} S_T-K - S_T + \frac{S_t}{P (t,T)} = \frac{S_t}{P (t,T)}-K =0, \end{align*} that is, \begin{align*} K=\frac{S_t}{P (t,T)}. \end{align*}Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.