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Understanding of Black-Scholes-Merton Model - 2


New Member
Dear David,

Following you patient explanation for my previous post (http://www.bionicturtle.com/forum/viewthread/2148/), I'm just wondering:

1) when you said that "let me remind you something Hull says: the BSM does not depend on the expected return of the stock", does it contradict with the fact that BS model is assuming risk neutrality and risk-free growth rate for all securities

2) On page 145 of FRM (5th), Merton model for call valuation is: c = SoExp(-r*t)N(d1) - KExp(-rt)N(d2), does the r* refer to dividend yield for stocks underlying and foreign rate of interest for currency underlying? I've noticed that r* doesn't appear in the call valuation formula used in the video cast "Importance of d2 in Black-Scholes to Merton Model in Credit Risk " (http://www.bionicturtle.com/learn/article/importance_of_d2_in_black_scholes_to_merton_model_in_credit_risk_10_min_scr/ ). In fact, only SoN(d1) is used. Is this one of the main difference between Merton as a default prediction model versus BS as a option pricing model?

3) Do we derive this call valuation formula by the following general steps:
i. take dividend-paying stocks as an example, that has a continuous dividend yield of r*. St = SoExp[(r-r*)T]
ii. If we refer to formula of a call option value, with volatility incorporated and discounted by risk-free interest rate, the current call value is then [StN(d1) - KN(d2)]*exp(-rT), replacing St with SoExp[(r-r*)T] from step i), it follows that c = SoExp(-r*T)N(d1) - Kexp(-rT)N(d2).
iii. I suppose this reasoning applies to option with currency and futures as underlying as well.

4) If the above steps are correct, is my following interpretation of its application to option on futures correct as well?
i. since futures income yield is riskless interest rate. My interpretation is that a person going long a futures contract can afford to short sell the underlying for So and invest the proceeds at risk-free interest r. And this risk-free interest is equivalent to the "income" brought to the person. Is this correct?
ii. like step i in 3), Ft = FoExp[(r-r)T] or Ft=Fo
iii. therefore for call option on future, c = FoN(d1) - KN(d2)]*exp(-rT)

5) I forgot to mention in my previous post that when you calculated d2 in your videocast, you used the volatility of asset (or sigma) directly as the denominator,instead of multiplying it by squareroot(T).

Thank you very much!