Choosing Discount Rates for Collateralized and Uncollateralized Puts

# What Is the correct discounting, risky or riskless?


# What Is the correct discounting, risky or riskless?












Suppose I can sell a European put in two ways: 1) in a mark to market collateralized market with collateral rate equal to the riskless rate $r$; 2) in a noncollaterized market where I get the payment for the put up front $0$ and invest in either a risky market account or a risky (defaultable) bond at rate $r+\lambda$.

1) I think the European put price is

$$P_1(t) = \mathbf E\big[e^{-\int_t^T r\,ds}\big(K-S(T)\big)_+\big|\mathcal F_t\big].$$

2) I am of two minds. On the one hand, a riskless instantaneous portfolio could be formed by the put with price $P_2(t)$ and the stock with price $S(t)$, with the portfolio price being $P_2(t)-\frac{\partial P_2(t)}{\partial S}S(t)$, as in the usual Black-Scholes argument. Then the growth rate of this riskless portfolio should be just the riskless rate $r$ and we should just have the same price as in 1). On the other hand, the default probability of the risky bond/money market should impose the credit spread $\lambda$ on the interest rate, so the put price should be $$P_2(t) = \mathbf E\big[e^{-\int_t^T (r+\lambda)\,ds}\big(K-S(T)\big)_+\big|\mathcal F_t\big]$$

Which is the correct discount factor? If neither is true, what is the correct answer?

## Answer by dm63 (score 1, accepted)

https://quant.stackexchange.com/a/34201

The discount rate for 2) should be a risky rate $r + \lambda$, although we must talk about how to determine $\lambda$. If you have sold an uncollateralized put option, the put option is an unsecured obligation of yours. Hence it should carry a similar discount rate to other unsecured obligations that you have issued. Thus, $\lambda$ is your credit spread. There is one exception to this: if the buyer of this put option currently owes you money, the premium paid could be considered a reduction of the amount owed, in which case $\lambda$ is HIS credit spread. In no case does $\lambda$ depend on the riskiness of some investment you plan to make with the premium paid. If you are wondering why the Black-Scholes argument breaks down in this case, I believe that it is because the hedged portfolio is risk-free with regard to the stock, but not risk-free with regard to the possibility of default by the issuer of the put option.

## Answer by cykor21 (score 0)

https://quant.stackexchange.com/a/34689

In risk-neutral pricing environment both cases would be discounted by the risk-free rate; the riskless portoflio in BSM formula earns the risk-free rate because the portfolio return is unaffected by changes of the price of the underlying, i.e. it is immune to market risk.

However, point 2) introduces credit risk which typically would be incorporated by adding/subtracting some amount on the top of the risk-free price of the option.

But for the option writer who already received the premium upfront the credit risk remains zero.