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?


2 Answers 2


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).


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.

  • $\begingroup$ The regular Black Scholes price is considered for t=0 to t=2? $\endgroup$
    – assf
    Jul 27, 2021 at 9:37
  • $\begingroup$ No, from t=0 to t=1. $\endgroup$
    – emot
    Jul 27, 2021 at 9:45

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