Pricing Options with Delayed Cash Settlement under Black–Scholes
Summary
The document explains how to value a European option whose payoff is determined at expiry but paid later. It distinguishes the option expiry date from the cash settlement date and expresses the value at expiry as the payoff multiplied by the price of a zero-coupon bond covering the delay. Taking conditional expectations then carries that adjusted payoff back to the valuation date.
Under a freezing assumption for the future bond price, the resulting expression is the standard Black–Scholes value multiplied by a ratio of discount bond prices. A second answer gives the non-stochastic-rate interpretation: discount the ordinary option value over the period between expiry and payment using the forward risk-free rate for that interval. The simplification depends on assumptions about interest rates and, in the first derivation, the bond-price freezing approximation. The document does not address more general stochastic-rate models or show a numerical example.
Key ideas
- The option payoff is fixed at expiry, while delayed settlement adds discounting from expiry to payment.
- The delay adjustment can be represented using the price of a zero-coupon bond over the settlement interval.
- With a frozen future bond price, the ordinary Black–Scholes value is adjusted by a discount bond price ratio.
- The simple forward-rate discounting result assumes non-stochastic interest rates.
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# Delayed Settlement Option- how will values in Black Scholes change
# Delayed Settlement Option- how will values in Black Scholes change
If there is an option that expires a year from now, but is settled after 2 years, how would the Black Scholes formulation for such a situation look like? Will the risk free rate now be for 2 years or one?
What I think is: since we can break down an option as a fixed interest rate investment + volatile stock, the only thing that will have any effect on the change in price will be the stock volatility. So the risk free rate we take should be for a year only. Is this the right way to think?
## Answer by ir7 (score 3)
https://quant.stackexchange.com/a/66205
For an option with delayed cash settlement, expiry time $T$ and settlement time $T_p(\geq T)$, paying $(S_T-K)^+$ at $T_p$, the present value of this payment at $T$ is:
$$ E_T\left[\beta_T \beta_{T_p}^{-1} (S_T-K)^+ \right] = P(T,T_p)(S_T-K)^+,$$
with $\beta_t = \exp \left(\int_0^t r_u du \right) $, $r$ risk-free interest rate, $P$ associated zero-coupon bond price, $P(u,U)= E_u\left[\beta_u \beta_{U}^{-1} \right]$.
So, due to conditional expectation tower property, the present value of the option at $t(\leq T)$ is:
$$E_t\left[\beta_t \beta_{T_p}^{-1} (S_T-K)^+ \right] = E_t\left[E_T\left[\beta_t \beta_{T_p}^{-1} (S_T-K)^+ \right]\right] $$ $$ =E_t\left[\beta_t \beta_{T}^{-1} E_T\left[\beta_T \beta_{T_p}^{-1} (S_T-K)^+ \right] \right] $$ $$= E_t\left[\beta_t \beta_{T}^{-1} P(T,T_p) (S_T-K)^+ \right]. $$
If we make a 'freezing' assumption on the, otherwise stochastic, $P(T,T_p)$:
$$ P(T,T_p) = \frac{P(t,T_p)}{P(t,T)},$$
we get:
$$\frac{P(t,T_p)}{P(t,T)}E_t\left[\beta_t \beta_{T}^{-1} (S_T-K)^+ \right] $$
(the standard BS formula just gets a sort of cash 'transportation' adjustment multiplier).
## Answer by dm63 (score 2)
https://quant.stackexchange.com/a/66192
Just to be clear we are talking about an option that pays $max(0,S_1-K)$ paid at time $t=2$. Then the only difference between this and a standard option is the extra discounting from $ t=1$ to $t=2$ . So the price $P$ must satisfy $$ P=BS/(1+r)$$ where BS is the regular Black Scholes price and $r$ is the forward risk free rate from $t=1$ to $t=2$.
The above technically assumes interest rates are non stochastic.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.