Put pricing embedded in autocall

Question

When pricing an autocall, there are 3 parts:

1. Strip of coupon,
2. Zero coupon bond,
3. Put down and in.

Probabilities of a call is given from the trigger level on call dates. However, let's say my autocall is 1Y max and is callable on months 6 and 12, should I price 2 put D&I and multiply the price by the autocall probabilities ? Or should I just compute the 1Y Put D&I ?

The reason is: When pricing a down and in put with Heston model I'm in line with what's expected. However, when moving to autocall products, my coupons are below what are priced by bank's pricers

About my computation: Given Monte Carlo paths and trigger level, I compute probabilities of touching the trigger. This gives me a set of probabilities for each call dates. I compute then discount factors * ZC bond value and this gives me my expected ZC bond value given my probabilities. In the end, I sum up the price of the put and (1 - ZC Present value). Finally I divide the result by the ZC bond expected value. Is it correct ?

Edit: what i am looking for if given the put price, my probabilities, my discount factors, i want to get the guaranteed coupons. So this is not exactly the same as getting the payoff and computing the value of the product.. I'm solving the equation Present Value of contingent coupon (probabilized) + PV of probabilized zero coupon - Put = 0. Some help would be appreciated, thanks

Answer 1

I'm not sure I correctly understood your question. Consider this simple autocall structure. Let $T_1$ denote the observation date of the autocall feature. More specifically, if the underlying spot price at $T_1$ is higher than a given threshold $\alpha$, the structure will pay a coupon $C$ to its holder and then expire. Otherwise, the structure will deliver the same cash flow as a down and in put with maturity $T_2 > T_1$.

Assuming zero discount rates for the sake of clarity, the price of the aforementioned structure is

\begin{align} V_t &= \Bbb{E}_t^\Bbb{Q}\left[ 1\{ S_{T_1} \geq \alpha \} C + 1\{ S_{T_1} < \alpha \} \phi(S_{T_2}) \right] \\ &= \Bbb{E}_t^\Bbb{Q}\left[ 1\{ S_{T_1} \geq \alpha \} C \right] + \Bbb{E}_t^\Bbb{Q}\left[ 1\{ S_{T_1} < \alpha \} \phi(S_{T_2}) \right] \end{align}

Now, although you can always write $$ \Bbb{E}_t^\Bbb{Q}\left[ 1\{ S_{T_1} \geq \alpha \} C \right] = C \,\, \underbrace{\Bbb{Q}(S_{T_1} \geq \alpha)}_{\text{call prob.}} $$ you can on the other hand only write $$ \Bbb{E}_t^\Bbb{Q}\left[ 1\{ S_{T_1} < \alpha \} \phi(S_{T_2}) \right] = \underbrace{\Bbb{E}_t^\Bbb{Q}\left[ 1\{ S_{T_1} < \alpha \} \right]}_{1-\text{call prob.}} \underbrace{\Bbb{E}_t^\Bbb{Q}\left[ \phi(S_{T_2}) \right]}_{\text{DI put price}} $$ iff the random variables $1\{ S_{T_1} < \alpha \} $ and $\phi(S_{T_2}) $ are independent, which is not the case in most diffusion models where $S_{T_1}$ and $S_{T_2}$ are dependent.

Wrapping up, this means that in general, you cannot write that: $$ \text{autocall price} = \pi \, \text{coupon} + (1-\pi) \, \text{down and in put price}$$ where $\pi$ is here represents the autocall probability at a single observation date.