# 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?

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.

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