P2 Instructional Video: Bodie, Chapter 24

CFO2013

New Member
Hello,

Can you please explain how you arrive at the formula using the call option approximation for the market timing effect. I believe this formula appears at 20 minutes and 36 seconds after the start of the video. Thanks in advance for your help.
 

David Harper CFA FRM

David Harper CFA FRM
Subscriber
Hi @CFO2013 I think you refer to Bodie's formula 24.6:
"Because the ability to predict the better-performing investment is equivalent to holding a call option on the market, in any given period, when the risk-free rate is known, we can use option-pricing models to assign a dollar value to the potential contribution of perfect timing ability. This contribution would constitute the fair fee that a perfect timer could charge investors for his or her services. Placing a value on perfect timing also enables us to assign value to less-than-perfect timers. The exercise price of the perfect-timer call option on $1 of the equity portfolio is the final value of the T-bill investment. Using continuous compounding, this is $1 * exp(rt). When you use this exercise price in the Black-Scholes formula for the value of the call option, the formula simplifies considerably to: MV(Perfect timer per $ of assets) = C = 2N[0.5*σ(M)*sqrt(T)] - 1"

So *apparently* if we assume S = 1 and X = e(rt), then BSM call option (c) = S*N(d1) - K*exp(-rT)*N(d2) simplifies to C = 2N[0.5*σ(M)*sqrt(T)] - 1. (I haven't checked it but appears reasonable given the canceling it implies). I hope that helps, thanks,
 

CFO2013

New Member
This is indeed correct, d1 = (0.5*σ(M)*SQRT(T)) and d2 = d1-σ(M)*SQRT(T) = -0.5*σ(M)*SQRT(T) = -d1. Therefore, C = N(d1) - N(-d1) = 2N(d1)-1
Many Thanks again David for your help.
 

Shau_2207

Member
I found an example of MV(Perfect timer per $ of assets) in GARP book see attached but i am not able to figure out the working or calculation 2N, do we need to multiple N value , is the N value right tail or left tail?

Can you please help?
 

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gsarm1987

FRM Content Developer
Staff member
Subscriber
@Shau_2207 This expression is a result of simplification. Please see David's comment above (april 17 and 18, 2014). In short, equity is a call option on asset.

starting point: S(V,F,T,t) = VN(d1) - Pt(T)FN(d2).
on page 238/340 of your ebook (BT notes), you will see it is using an analogy that market timing is just like holding a call option
 
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