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# PQ-externalBS model assumptions:

#### no_ming

##### Member
Hi, Mr. Harper, again is me. The following question is about BS model:
It ask "Which is an assumption of BSM model....."
But I have checked the notes of book 1, none of them is an assumption of BSM model, am I right?

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#### brian.field

##### Well-Known Member
Subscriber
The fact that we are looking at government bonds makes this a tricky question.

When considering stocks, I think A and C would be considered assumptions of Black-Scholes. Stock Prices are assumed to normally distributed (recall the ln(St/S0) is assumed to be normal and so, St/So assumed to be lognormal). The is equivalent to saying that the stock price follows a Geometric Brownian Motion in the more general B-S Stochastic Differential Equation (although I can't recall is GBM is necessary or if Arithmetic BM is sufficient). B-S also assumes constant variance or constant volotility.

When considering bonds, B and D are clearly inconsistent with BS but I think the questions is trying to highlight the fact that bond price volatility decreases as the bond approaches maturity as it must mature at par, generally, so I would argue that that answer to the question is B.

#### no_ming

##### Member
The fact that we are looking at government bonds makes this a tricky question.

When considering stocks, I think A and C would be considered assumptions of Black-Scholes. Stock Prices are assumed to normally distributed (recall the ln(St/S0) is assumed to be normal and so, St/So assumed to be lognormal). The is equivalent to saying that the stock price follows a Geometric Brownian Motion in the more general B-S Stochastic Differential Equation (although I can't recall is GBM is necessary or if Arithmetic BM is sufficient). B-S also assumes constant variance or constant volotility.

When considering bonds, B and D are clearly inconsistent with BS but I think the questions is trying to highlight the fact that bond price volatility decreases as the bond approaches maturity as it must mature at par, generally, so I would argue that that answer to the question is B.
Hello, Mr.field, actually, I choose D for this question as I think its an unreasonable assumption,
For B, does it mean that price volatility of bond must become zero at maturity?

#### brian.field

##### Well-Known Member
Subscriber
The question expcitlty asks for which of the options IS an assumption of B-S. D is clearly not an assumption of B-S. Since the price of a bond converges to pare at maturity, the volatility must diminish. THis is something that makes equilibrium models such as Cox Ingersoll and Vasicek Interest Rate Models (with which I have little familiarity) more challenging....

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
I think it's a great example of an imprecise question (having written many imprecise questions in my life ). To Brian's point, the question asks (emphasis mine) "Which of the following is an assumption of the BSM model that makes this recommendation inappropriate?" ... first please note that question does not exactly specify what kind of option (i.e., price option or rate option), but let's safely assume it's a bond price option, then if i consider the choices in turn:
• A. BSM assumes continuous (log) returns are normal such that it's assumption is that price is lognormal, so technically BSM does not assume "price follows a normal." If this read, "price of the government bond follows lognormal," we'd have a good candidate answer (in terms of a true BSM assumption that is inappropriate.). So, I exclude (A).
• B. This is not a BSM assumption because a key BSM assumption is constant volatility. But bond price volatility does decrease as maturity approaches. So literally this is not an answer to the question, but it could what the questions looking for b/c it gets to a key problem with BSM against bond (price) options: BSM volatility is constant, but interest rate volatility is not constant, although we often do tend to assume short-term rate vol is contant
• C. price following BSM is an assumption of BSM, and it is a reason why BSM is inappropriate for bond options because unlike BSM diffusion, the bond price pulls to par (converges) as maturity approaches. This (C) appears correct to me, therefore
• D. Technically I agree with Brian, we can dismiss. But I would not blame somebody for picking this actually because the BSM as a Weiner does assume a drift (it is a separate idea that expected return does not enter). See Hull's Chapter 13 where the Wiener process is drift + shock. There actually is an underlying assumption of price increase. And, higher risk-free rate --> higher call option, which is also a different matter, but reveals the nuance underneath this choice. If I were revising this question, I would want to simplify this choice to avoid the theoretical complications
In any case, Tuckman has been assigned on this for actually all of the time i've teaching FRM (over ten years, this LO has been in the syllabus):
"FIXED INCOME VERSUS EQUITY DERIVATIVES While the ideas behind pricing fixed income and equity derivatives are similar in many ways, there are important differences as well. In particular, it is worth describing why models created for the stock market cannot be adopted without modification for use in fixed income markets.

The famous Black-Scholes-Merton pricing analysis of stock options can be summarized as follows. Under the assumption that the stock price evolves according to a particular random process and that the short-term interest rate is constant, it is possible to form a portfolio of stocks and short-term bonds that replicates the payoffs of an option. Therefore, by arbitrage arguments, the price of the option must equal the known price of the replicating portfolio.

Say that an investor wants to price an option on a five-year bond by a direct application of this logic. The investor would have to begin by making an assumption about how the price of the five-year bond evolves over time. But this is considerably more complicated than making assumptions about how the price of a stock evolves over time. First, the price of a bond must converge to its face value at maturity while the random process describing the stock price need not be constrained in any similar way. Second, because of the maturity constraint, the volatility of a bond’s price must eventually get smaller as the bond approaches maturity. The simpler assumption that the volatility of a stock is constant is not so appropriate for bonds. Third, since stock volatility is very large relative to short-term rate volatility, it may be relatively harmless to assume that the short-term rate is constant. By contrast, it can be difficult to defend the assumption that a bond price follows some random process while the short-term interest rate is constant." -- Tuckman, Bruce; Serrat, Angel. Fixed Income Securities: Tools for Today's Markets (Wiley Finance) (pp. 225-226). Wiley. Kindle Edition.

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#### brian.field

##### Well-Known Member
Subscriber
@David Harper CFA FRM - can you clarify what you interpret as the question's correct answer? It is a bit unclear to me and I suspect it may be to others as well.

Thanks!

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@brian.field sorry! For me, the correct answer is (C). I dismiss (A) and (B) on a literal reading of the queston asked (Which of the following is an assumption of the BSM model ...?) and I am sympathetic to (D) due to the theoretical complications that it appears to admit, although I agree with you ultimately about (D).

#### brian.field

##### Well-Known Member
Subscriber
The question expcitlty asks for which of the options IS an assumption of B-S. D is clearly not an assumption of B-S. Since the price of a bond converges to pare at maturity, the volatility must diminish. THis is something that makes equilibrium models such as Cox Ingersoll and Vasicek Interest Rate Models (with which I have little familiarity) more challenging....
and my quoted post here proves that my earlier post was wrong since B is not an assumption of BS! Maybe I should read my posts before I post them!

#### no_ming

##### Member
Thanks Mr.Field & Mr. Harper, for the very clear explanation. I understand now.