# Value a structured note with Black-Scholes

Apologies in advance if this seems like a straight forward question but I'm really unsure how to go about it. Say I have the payoff for a structured note benchmarked against an index and I have a figured out a combination of two different options will essentially provide the same payoff. When I use the Black-Scholes-Merton model to value the options, the value that I get is significantly lower than the par value of the note. e.g. par value is 1000 and the options are at 200 in total. Is that possible? What is the general approach when it comes to calculating the value of a structure note?

Also given: volatility & risk free rate.

Thanks!

• does the structured note include a principal? this will need to be discounted appropriately. – Mark Joshi Mar 28 '15 at 3:43
• As in whether the structured note is principal guaranteed? All it says was the the par value is 1000 and if the index's return is negative then the note's total payoff will be 1000 x (1 - R). So the principal is not guaranteed. – PLui Mar 28 '15 at 4:10
• it doesn't sound like an option at all. – Mark Joshi Mar 28 '15 at 20:49
• Given the example I gave, wouldn't a short put with strike at 1000 have essentially the same payoff? – PLui Mar 29 '15 at 0:32
• and what happens if the return is positive? – Mark Joshi Mar 29 '15 at 8:01

You said:

• if the index's return is negative then the note's total payoff will be 1000 x (1 - R)

• When the return is positive, then the payoff is 1000 + 1000 * 2.5 * max{R - 0.1, 0}.

Hence, your payout function should be as follows:

v(T)= Indicator {Index(T) < 1000, Index(T); 1000 < Index(T) < 1100, 1000; Index(T) > 1100, 1000 + 2.5*(Index(T)-1100) } and indeed you can re-write as

V(T) = Index(T) - max(Index(T)-1000,0) + 2.5*max(Index(T)-1100,0)

A) If the underlying process that drives the Index price does not correspond with the assumed process by Black Scholes then you cannot value this note via Black Scholes. But you can run a monte carlo simulation based approach to value such note. Here are couple steps to get you started

• You essentially need to to decide on the type of model you need to apply. For that you need to know which model best describes the price dynamics of the underlying index. Be mindful in case there are any correlating brownian motions or correlating processes which would make it a little more exciting to deal with.

• Next, you would need to simulate different index price paths to evolve the index price.

• Then, you derive the final payoff, using the index value at each price path at the time that coincides with the expiration of your note.

• You then properly discount each payoff.

• At the end you average the discounted payoffs to get to your expected discounted future value of the note which is the price anyone would want to trade at if he/she believed that your model and variable inputs properly described the evolution of the index.

B) If the underlying processes correspond then you can simply value the first portion of this note via "no-arbitrage argument" and treat the other payoff components as separate call options.

• Thanks for the detailed explanation Matt! I would have run Monte Carlo as well but I'm being asked to use Black Scholes. Maybe I'm totally confused but how do I apply Black Scholes to what you suggested? – PLui Mar 30 '15 at 23:34
• Can you please first confirm that the payoff function is correct? – Matthias Wolf Mar 31 '15 at 2:17
• Yes, the payoff function is correct. – PLui Mar 31 '15 at 14:57
• Plui, to answer your question knowledge of what drives the price of the index is essential. Unless of course this is a homework or take home exam and the lecturer asked you to assume that the payoff can be modeled via BS. – Matthias Wolf Mar 31 '15 at 16:11
• Yes, this is part of a homework question. I had asked the prof and the hint he gave me was think of it as a combination of multiple options. I wasn't sure if my line of thinking is correct because when I used BS to price the options, the total value is very far from 1000. Maybe I'm missing something very fundamental here... – PLui Mar 31 '15 at 17:49

I think the problem lies in the principal. Normalize $S_0=1000.$ Essentially you get $$S_T \text{ if } S_{T} < 1000,$$ $$1000 \text{ if } 1000 < S_T < 1100,$$ $$2.5 (S_T - 1100) \text{ if } 1100 < S_T$$ I'd write this as $$S_T - \max(S_T-1000,0) + 2.5 \max(S_T-1100,0).$$ Since $S_0=1000$ you'll get something close to $1000.$

• If I'm also given the volatility and the risk free rate, would I be able to value it with Black-Scholes? – PLui Mar 30 '15 at 23:40
• you would also need the dividend rate for $S_t$ but then yes. – Mark Joshi Mar 31 '15 at 0:19
• @MarkJoshi, when you say "you'll get something close to 1000", do you mean the price of the note should be close to 1000? I do not follow the rational if that is the case because unless you know the "risk free rate", dividends (or other yields, after all it is an index not a stock) as well as the volatility it would be hard to tell, imho. Also, should the payoff (if ST > 1100) not be 1000 + 2.5*(ST-1100)? – Matthias Wolf Mar 31 '15 at 2:34
• the two functions are the same. I said something close not exactly. – Mark Joshi Mar 31 '15 at 3:44
• @MarkJoshi How do I apply the Black-Scholes if I have the dividend rate? I'm failing to link the payoff and how I can use Black-Scholes to price it... – PLui Mar 31 '15 at 15:07