Pricing Options with Cash or Physical Deferred Delivery

# Handling delayed settlement / delivery in option pricing


# Handling delayed settlement / delivery in option pricing












Can you please shed some light on how settlement/delivery delay impacts option pricing? Assume settlement/delivery happens 2BD after option expiry date and the price is computed using Black Forward model where

$$d_1 = \frac{\ln\left(\frac{F}{K}\right) + \frac{v^2}{2} T}{v \sqrt{T}},$$

and $F$ is the forward’s price, $K$ is the strike price, $v$ is the volatility, and $T$ is the time to expiration.

Should $F$ be computed as of expiry date or delivery date? Also, assume there's a cash dividend during the option's life. Will greatly appreciate if someone can give a worked-out example. I have noticed conflicting information online and in a trading system that my firm uses.

## Answer by river_rat (score 5)

https://quant.stackexchange.com/a/83984

So the answer depends on a few things, and is remarkably more complicated if you have stochastic rates. But assuming deterministic rates you have two cases - cash deferred and physically deferred settlement.

We need a few things. Let $T_e$ be the expiry time and $T_d > T_e$ be the delivery time. We have an implied vol surface $\sigma(K, T)$ and some lognormal asset $X(t)$. Let $P(t,T)$ be the zero bond ie discount factor and we assume zero dividends, storage costs and convenience yields. The asset forward curve is then given by $F(t,T)=X(t) / P(t,T)$.

Finally we consider a call with strike $K$ which we can price via Black-76 with price $$V(t)=P(t,T) \times Black(F(t,T), K, \sigma(K,T),T) = P(t,T)\times \mathbb{E}(\left(X(T)-K\right)^+|\mathbb{F}_t)$$ for a given expiry $T$ and strike $K$

If we are considering cash deferred delivery then we know that at time $T_d$ we will receive $X(T_e)-K$ as cash. This has value $P(T_e, T_d)\times\left(X(T_e)-K\right)$ at expiry so we exercise the call under the standard assumptions for cash delivery. ie. We are looking at $$V(t)=P(t,T_e)\times\mathbb{E}(P(T_e, T_d)\left(X(T_e)-K\right)^+) = P(t,T_d)\times Black(F(t,T_e), K, \sigma(K,T_e),T_e)$$ So cash deferred delivery is just priced normally but discounted to the future delivery date.

For physically settled deferred delivery we have that we receive 1 unit of asset at time $T_d$ which we pay $K$ for and sell for $X(T_d)$. This has value $X(T_e)-P(T_e,T_d)\times K$ at time $T_e$ as it is the NPV of a forward contract. This implies that our exercise decision is different for deferred physical settlement as we are concerned rather with the fact that the price needs to be higher than some shifted strike value. Once again we see that $$V(t)=P(t,T_e)\times\mathbb{E}((X(T_e)-P(T_e,T_d)\times K)^+)$$ $$= P(t,T_e) Black(F(t,T_e), P(T_d,T_e)\times K, \sigma(P(T_d,T_e)\times K,T_e),T_e)$$ $$=P(t,T_d)Black(F(t,T_d),K,\sigma(P(T_d,T_e)\times K,T_e)$$ So for physical deferred settlement we price with the delivery forward date and with the vol of a shifted strike. For completeness (and as an exercise for the reader) you can show that if you have dividends, storage and convenience yields then your pricing equation ends up being $$V(t) = P(t,T_d) Black(F(t,T_d),K,\sigma(\frac{F(t,T_e)}{F(t,T_d)}\times K,T_e),T_e)$$

## Answer by Enrico Schumann (score 0)

https://quant.stackexchange.com/a/83978

It is discussed in Haug (2007), section 6.1, including a numeric example. The author adds a discount factor to the payoff (but essentially reuses the standard Black-Scholes-Merton formula). The question is also discussed at Delayed Settlement Option- how will values in Black Scholes change .

```
@BOOK{,
  title     = {The Complete Guide to Option Pricing Formulas},
  publisher = {McGraw-Hill},
  year      = 2007,
  author    = {Espen Gaarder Haug},
  edition   = {2}
}
```