Mathematical AppendixTM Last revised: March 02, 2002
Dr. Risk's "Mathematical AppendixTM" contains some of the details that Dr. Risk tries to keep out of the rest of the 'Zine. It's not everybody's cup of tea – for some it's more like hemlock. However, if you like this sort of thing, drink up!
New, this month:
- "Ask Dr. Risk!": Nada.
- Dr. Risk's Links: Nada.
- 7/28/00 Results: "The World's Simplest Model of the Credit Spread"
Table of Contents this page:
Ask Dr. Risk | Links | Results
Ask Dr. Risk!
... and if you buy the risk neutral probability, you get another probability, free, ... (4/28/00)
Dear Dr. Risk – With respect to option theory, can you tell me if an option's price implies any probability that the underlying will trade at or beyond the strike (i.e. that any intrinsic value will be realized), if only for a brief time? I guess what I'm asking is can you use implied volatility to calculate such probabilities. Is there a way to prove this mathematically? – Una
Dear Una – The Black-Scholes-Merton (BSM, 1973) equation for pricing an option contains not one, but two probabilities that the underlying price will be beyond the strike at expiration. "Beyond" means "higher than" for the call and "lower than" for the put. Both these probabilities are based on the assumption that the volatility -- i.e., the standard deviation of the change in the logarithm of the underlying price – is the implied volatility. Neither of these probabilities is the "real" probability, which nobody knows or will ever know. Nor is it a "subjective" probability, which is anybody's guess. Instead, it is a "risk neutral" probability, which is an artifact of complete markets, such that the discounted, risk neutral probability is the current, Arrow-Debreu price of a unit of future wealth in a particular state of the world.
Specifically, the BSM model says for a call option that
C = S e-dt N(d1) – K e-rt N(d2).
The “risk neutral probability” that the underlying price will exceed the strike price at maturity is N(d2). Here, the drift rate for the logarithm of the underlying price is µ = r-d-s2/2. The N(d1) is another probability that the strike price at maturity will exceed the strike price at maturity, but with drift rate µ' = r-d+s2/2. I don’t have a neat interpretation for this probability.
The BSM model says nothing about the probability that the underlying will at some point, “if only for a brief time”, trade beyond the strike. However, given the BSM assumptions, you could deduce various probabilities, and people do this to price barrier options.
How do you prove this, mathematically? You assume that the underlying price is its starting value, and that its random value in the future is a lognormal random variable, with a specific volatility and drift rate. Then you integrate the probability distribution with respect to the random underlying price, over the in-the-money region. It’s a bit of a pain, and not suitable for a family publication. – Dr. Risk
The Square Root of Three (5/28/00)
Dear Dr. Risk -- q1. looking at your derivative dictionary, the entry for asian option. you mentioed the "square root of three" rule. what is it and what is a good ref on this? -- Gnin
Dear Gnin -- Take a look at my "Derivative Games" page, focusing on "Celia in Derivativesland". That explains some basic of the arguments. I'm can't refer you to a theoretical discussion in the literature. -- Dr. Risk
Results
The World's Simplest Model of the Credit Spread (7/28/00)
Fairly obviously, the credit spread depends on the probability distribution for the possible levels of severity of default. Let's consider the simplest possible case:
- Time consists of one period that starts "now" and ends one year "later".
- The default-free and credit-risky debt mature at the end of one year.
- The tenor for the coupon is one year. Let c1 denote the coupon rate.
- The bond defaults, with risk-neutral probability p. If it defaults, the bondholder recovers the fraction r of the promised payment of interest and repayment of principal.
- The bond sells now for par.
Then we have the following binomial tree for the bond’s value:
100 (1+c1)
/
100
\
100
(1+c1) r
The pricing equation for the bond is:
100 = [p
100 (1+c1) r
+ (1-p)
100 (1+c1)] / (1+r)
Let l
= 1-r
denote the loss rate. Rearranging the pricing
equation, we get
1+r = (1+c1) [p
r
+ 1-p
] = (1+c1) [1 - p
(1-r)]
= (1+c1) [1-p
l].
Hence, the ratio of the promised wealth relative for
the credit-risk debt to the promised
wealth relative for credit-riskless debt is
(1
+ c1)
/ (1+r) = 1
/ (1 - pl).
The credit-risky coupon is
c1
= (1+r)
/ (1 - pl)
- 1 = (r + pl)
/
(1 - pl)
and the credit spread is
c1
- r = (r + pl)
/ (1 - pl)
- r = [(r + pl)
- r (1
- pl)]
/ (1 - pl)
= [pl
+ r pl)]
/ (1 - pl)
= pl
(1 + r)
/ (1 - pl).
As either the probability of default or the loss rate
go to zero,
- the ratio of wealth relatives goes to one
- the credit-risk coupon approaches the riskless rate
- the credit spread approaches zero,
all of which are believable features of the model.
We can easily extend this model in several directions:
- allowing multiple periods and coupon tenors of less than one year
- deducing the probability of default from the credit-risky coupon
and we may do that in the coming months.
Of course, the model has its limitations. For example, the binomial probability distribution for loss rate is simplistic. Unfortunately, relaxing the binomial assumption is difficult. Amazingly, the extensions of the basic model seem to be state of the art for practical, reduced form models.
Alpha
According to the Sharpe-Lintner model, in equilibrium the expected rate of
return for each asset is
E(Ri) = R + bi
[E(RM)-R],
where
Ri denotes the rate of return on the ith asset,
RM denotes the rate of return on the market portfolio,
R is the riskless rate of interest, and
bi
= Cov(Ri,RM) / Var(RM) denotes the ith asset's
systematic risk.
If the expected rate of return exceeds this equilibrium value, then the asset is
underpriced, and we would expect it to outperform the market, adjusted for
risk.
We can't directly observe expectations, but we can infer them statistically from averages
over historical data. If we regress Ri - R against
RM - R, we assume that
Ri - R = ai
+ bi
[RM -R] + ei,
where ei
is a standard normal error term with mean zero and variance s2. Taking the expectation, we get
ai
= E(Ri) - R - bi
[E(RM)-R]
so we can estimate bi
as the slope coefficient in the regression, and we can interpret our estimate of
the intercept as an estimate of the amount by which the ith asset has
outperformed the market.
Sharpe ratio
The Sharpe ratio is the outperformance of an investment, divided by its
standard deviation. A priori, it is:
SRi = [E(Ri) - R] / si,
which tells us how many standard deviation units the expected rate of
return is above the riskless rate:
E(Ri) = R + SRi si.
Typically, we use estimates of the parameters, rather than the
unobservable moments, which -- to belabor the obvious -- are not available.
The Sharpe ratio does not take systematic risk into account. Taking that into
account, we would have a modified Sharpe ratio,
SRi' = [E(Ri) - R - bi
[E(RM)-R] / si = ai
/ si
However, perhaps it would be more logical to divide the alpha by its standard
error from the regression. Thus, the new, improved "Sharpe Ratio"
would be the t-statistic for the hypothesis that ai
= 0,
Theorem: D(lS,lK) = D(S,K)
In the BSM (1973) model, call (and put) value is a linear homogeneous function of underlying spot (S) and strike (K): C(lS,lK) = l C(S,K). This is not a no brainer and might not be the case in reality, but is the case, given the specific BSM assumptions, including constant volatility.
A numerical approximation to delta is
D(S,K)
» [C((1+h)S,K)
- C((1-h)S,K)] / 2hS
= S [C((1+h),K/S)
- C((1-h),K/S)] / 2hS
= [C((1+h),K/S)
- C((1-h),K/S)] / 2h
®
D(1,K/S)
as h®0
which clearly depends only on K/S, not S and K. Hence, delta
depends only on K/S:
D(S,K)
= D(1,
K/S),
so
D(lS,lK)
= D(1, lK/lS)
= D(1, K/S) = D(S,
K).
QED
Theorem: G(lS,lK) = G(S,K)/l
Similarly, a numerical approximation to gamma is
G(S,K)
» [C((1+h)S,K) - 2 C(S,K) + C((1-h)S,K)] / (hS)2
= S [C((1+h),K/S)
- 2 C(1,K/S) + C((1-h),K/S)] / (hS)2
= [C((1+h),K/S)
- 2 C(1,K/S) + C((1-h),K/S)] / h2S
= {[C((1+h),K/S)
- 2 C(1,K/S) + C((1-h),K/S)] / h2}/ S
®
G(1,K/S)
/ S as h®0
which depends on K/S and S. Gamma has two factors. The first, G(1,
K/S), depends only on K/S. The second is inversely
proportional to S. Thus, as S and K increase proportionately, the first factor
doesn't change, but the overall gamma changes inversely proportionally to
S. Hence,
G(S,K)
=
G(1,K/S)/S,
so
G(lS,lK)
=
G(1,lK/lS)/lS
= G(1,K/S)/lS
= G(S,K)/l.
QED
Theorem: If r=d and S=K, then Pput = Pcall.(7/28/99)
As Dr. Risk explained in Derivatives Strategy (forthcoming, 1999), we can see that if
- we're in a Black-Scholes-Merton world (constant volatility, dividend yield, and interest rate);
- we price options with a Cox-Ross-Rubinstein (CRR) version of the binomial model;
- the interest rate and dividend yield are equal;
- the options have the same underlying spot price, forward, or futures price; expire at the same time; and are at the money (ATM, which means that the strike and the spot price are equal);
then the price of a European (an American) call option equals that of the corresponding put.
However, for American options and a finite number of binomial periods, this isn't true with a Jarrow-Rudd (JR) version of the binomial model.
The deeper reason for this result is that in the CRR model, every node above the strike price – with coordinates,
(t',S´a) – has its
equally valuable counterpart below the strike – with coordinates, (t',S/a). For each such pair of nodes at time
t', the payoff for exercise of the call (S´a-S)+ is larger than the payoff for
exercise of the put (S-S/a)+.
For example, (100´2-100)
= 100 > (100-100/2) = 50. However, the value at (t,S) of each dollar at
(t',S/a) – the Arrow-Debreu price,
a(t,S;t',S/a) – is larger than the value,
a(t,S;t',S´a), of a dollar received at the
(t',S´a). For each of
the nodes in the pair, if you multiply the intrinsic value at that node by the appropriate Arrow-Debreu price you get the same product.
That is,
a(t,S;t',S/a)
´
(S-S/a)+ =
a(t,S;t',S´a)
´
(S-S´a)+.
Let me be more explicit. In the following table, the middle column refers to the node above the strike and the rightmost column refers to the node. Row one defines the node as n periods after the spot date and i steps up (column two) or down (column three). A step up (down) increases (decreases) the price by the multiplicative factor, u (d). Row two defines the payoff for a call (column two) and a put (column three). Row three defines the Arrow-Debreu price, based on the risk neutral probability, q, which we'll define, later. The combinatorial factor is Cn,m = n! / (m! (n-m)!). Row four shows the product of payoff and A-D price.
|
(1) |
(2) | (3) |
| Node | (t',S´uidn-i) | (t',S´un-idi) |
| Payoff | (S´un-idi-S)
= S (uidn-i-1) = S (1 - 1 / uidn-i) uidn-i |
(S-S´un-idi) = S (1-un-idi) |
| Arrow-Debreu Price a(t,S;t',S') | Cn,i r-nqi(1-q)n-i | Cn,n-i r-nqn-i(1-q)i |
| Value = Payoff ´ a(t,S;t',S') |
S (uidn-i-1)
Cn,i r-nqi(1-q)n-i =S (1-1 / uidn-i) Cn,i r-nuiqidn-i(1-q)n-i |
S (1-un-idi) Cn,n-i
r-nqn-i(1-q)i |
The tricky algebra takes place in the bottom row, middle column. A little work with the definition and you can see that for the two products in the bottom row to be equal, sufficient conditions are:
- 1 / u = d
- Cn,i = Cn,n-i
- uq = (1-q)
- d(1-q) = q.
In the JR binomial model, u = emdt+sÖdt and d = emdt-sÖdt. The CRR assumption is a special case of this, with m=0, so u = e+sÖdt and d = e-sÖdt =1/u.
The combinatorial equality follows directly from the definition, and the fact that n-(n-i)=i.
The last two expressions take a bit more work. In the JRR model they aren't true. The risk neutral probability of a move up is q = (r-dy) / (u-d)y, and of a move down is 1-q = (uy-r) / (u-d)y. The elements of this definition are r = 1+the periodic rate of interest and y = 1+ the periodic dividend yield. In the CRR model (u=1/d), if r=y, then
- q = (r-dy) / (u-d)y = (r-y/u) / (u-1/u)y = (u-1) / (u2-1) = 1/(u+1)
- 1-q = u / (u+1)
- uq= u / (u+1) = 1-q
- d(1-q) = (1/u) u/(u+1) = 1/(u+1) = q.
As the saying goes, Q.E.D.
This implies that the put and call should have the same value at t. This argument doesn’t work in a Jarrow-Rudd version of the binomial model, because of the drift, except in the limit as time steps go to zero, because the periodic drift goes to zero much faster than the periodic volatility. However, if it works in the limit for the binomial case, then it works in continuous time.
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