Question 1: Binomial Tree Calibration [30 points]
So far in this course, you were given the value of u and d in the binomial tree. This exercise will walk you
through two methods for determining u and d.
As you know, the Black-Scholes model assumes that the price of a risky asset (the stock SBS, with current
normally distributed under the risk-neutral measure Q, with mean EQ ⇥ln SBS ⇤ = ln (S ) + r 1 2 t and variance VarQ ln SBS = 2t: t 0 2
price S ) follows a lognormal distribution. Specifically, in the Black-Scholes model, for any t > 0, ln SBS is 0t
t
✓SBS◆Q ✓✓ 1◆ ◆ ln t ⇠N r 2 t, 2t ,
(1)
or, equivalently,
SBS Q t ⇠ LogNormal μ ̃, ̃2 ,
S0 2
(2) where μ ̃ := r 1 2 t, ̃2 := 2t, r > 0 is the risk-free rate, and > 0 is assumed to be known (stock
S0
volatility). 2
You are given the following formulas for the lognormal distribution:
S B S EQ t
S0
̃ 2
=eμ ̃+2 =ert andVarQ
✓ S B S ◆ 2 2 2
t =e2μ ̃+ ̃(e ̃ 1)=e2rt(e t 1).
S0
The idea behind the determination of the tree parameters u and d is to choose them so that the mean and variance (under Q) of the stock price on the tree approximately match the mean and variance of the lognormal distribution above. There is some flexibility in this procedure. The two classical methods that we will explore below are easy to implement: the first one assumes ud = 1 (Method 1) and the second one assumes qu = 1 (Method 2).
(a) General setup: (8 points)
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Consider a multi-period binomial model, with each period of length h (in fractions of a year), and assume that the model is arbitrage-free. The known time-0 price of the risky asset (stock) is S0. At time h (end of the first period), the stock price (Sh) is a random variable that can take the values uS0 or dS0, for some u and d such that u > d > 0. The risk-free rate of interest is r > 0 per year, compounded continuously. More generally, for each n 1, the stock price at the end of the nth period is given by
Snh = ZS(n 1)h,
where Z is a random variable that can take the values u or d 2 (0, u).
(i) (2 points) Compute the risk-neutral mean and variance (i.e., under Q) of the random variable Y1 :=
Sh in terms of u, d and rh. S0
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(ii) (3 points) Let T be a positive integer and ST h be the stock price at the end of the T th period in the binomial model. Show that EQ[STh] = EQ[SBS] always holds. In other words, the means of the
binomial model and the Black Scholes model always match.
Hint: The following fact is useful: ST h = QT Snh and the jump random variables Snh/S(n 1)h, n = 1,…,T, are iid (equal to Z).S0 n=1 S(n 1)h
(iii) (3 points) Suppose that we keep t = T h fixed, and let h # 0 (at the same time T ! 1). Intuitively explain why ST h is approximately lognormally distributed.
Hint: You may use the Central Limit Theorem, which says that the sum of many iid random variables is approximately normally distributed.
(b) Method 1: (12 points)
In the first method, we assume ud = 1. The choice of u and d are given by
u = e ph and d = e ph.
Our goal is to show that this formulation approximately gives us the Black-Scholes model.
(i) (2 points) Derive an expression for qu := Q [Sh = uS0] = Q [Y1 = u], as a function of the time step h.
(ii) (3 points) Show that qu ! 1/2 as h # 0.
(iii) (3 points) Show that
Th
(v) (2 points) Given the assumption of no-arbitrage, what condition(s) must the model parameter > 0 satisfy, for a given h? What if h # 0?
(c) Method 2: (10 points)
Consider the same one-period binomial model as in (a). Instead of assuming that u and d are chosen such
Th
lim VarQ(ln(Sh/S0)) h#0 VarQ(ln(SBS/S0))
= 1.
(iv) (2 points) Based on (iii), show that ln(STh) has asymptotically the same variance as ln(SBS), as
Th Remark: As a consequence, ST h has asymptotically the same distribution as SBS , since they are both
h # 0.
lognormally distributed with the same mean and the same -parameter.
h
thatud=1asinMethod1,assumenowthatuanddarechosensuchthatqu =Q[Sh =uS0]= 1,that
is, the probability of an up-jump is 1 under the risk-neutral measure. 2 2
(i) (2 points) Given the assumption that qu = 1 , show that there exists > 0 such that u = erh + and d = erh . 2
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(ii) (3points)Determine ,uanddsothattherisk-neutralvariance(i.e.,underQ)oftherandomvariable Y1 matches the risk-neutral variance of SBS/S0.
(ii) (3 points) Show that EQ[(STh)2] = EQ[(SBS)2] for any positive integer T. Th
Remark: Similarly to Method 1, because of the Central Limit Theorem and the matching moments, ST h has asymptotically the same distribution as SBS .
(iii) (2 points) In this model, does the assumption of no arbitrage imply any restriction on the choice of the model parameter > 0, for a given h?
Question 2: [30 points]
Consider two standard one-dimensional Brownian motions {WtP}t 0 and {VtP}t 0 on a filtered space space (⌦, {Ft}t 0, F, P). Suppose that the filtration {Ft}t 0 is a filtration for the two Brownian motions and that the two Brownian motions {WtP}t 0 and {VtP}t 0 have a correlation of ⇢ > 0.
Consider a market model consisting of a risky asset whose value at time t is St and a bank account worth ert at time t. The risky asset does not pay dividends. Assume that the stochastic process {St}t on the filtered space (⌦, {Ft}t 0, F, P) satisfies S0 > 0 and has the following stochastic differential equation:
dSt = ↵Stdt + StdWtP,
where ↵, 2 R+.
A newly designed financial contract (derivative) pays off at maturity an amount that depends on the value of the risky asset, as well as the level of inflation in the market. Assume that inflation is modeled by a stochastic process {It}t on the same space that satisfies I0 > 0 and has the following stochastic differential equation:
h
dIt = Itdt + ⌘ItdVtP,
At time T > 0 (the maturity of the contract), the derivative’s payoff is given by
T = max✓ST e aT, S0egT◆, IT I0
where a 0 and g > 0 are given constants. This contract guarantees a minimum payoff of S0 egT at maturity. I0
where , ⌘ 2 R+.
Th
Define the stochastic process {Zt}t by
Zt = St, forallt 0.
It
(a) (5 points) Derive an expression for the dynamics of the stochastic process {Zt}t under the probability
measure P (the stochastic differential equation that {Zt}t satisfies under P).
(b) (6 points) For a given T > 0, derive an expression for the expected value and the variance of the random variable ZT under the probability measure P, assuming that the Brownian motions {WtP}t 0 and {VtP}t 0 are independent.
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(c) (7 points) For a given T > 0, derive an expression for the expected payoff of this derivative at time T under the probability measure P, assuming that the Brownian motions {WtP}t 0 and {VtP}t 0 are independent.
(d) (7 points) Now, instead of having a stochastic inflation rate, the inflation rate is assumed constant over the lifetime of this derivative contract, so that It = I0 > 0 for t 2 [0, T ]. Consider a contract that has the following payoff at maturity: ✓ pST aT S0 gT ◆
T=max I e ,Ie . 00
Suppose also that this market model is a Black-Scholes model. Under the risk-neutral probability measure Q, the risky asset’s price follows the stochastic differential equation
dSt = rStdt + StdWtQ,
where {WtQ}t 0 is a standard Brownian motion under Q, and r, > 0. Derive an expression for
time-0 price of this financial derivative, as a function of S0, I0, a, g, r, , and T .
Hint: Show that 0 can be written in the form
A ⇥ N (d1) + B ⇥ N (d2) ,
0, the
for some functions A, B, d1, and d2 of the parameters S0,I0,a,g,r, , and T, where N denotes the standard normal CDF.
(e) (5 points) Define the Vega of a security’s price, and explain what it measures. Compute the Vega of 0, the time-0 price of the derivative above in (d).
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Formula Sheet Black-Scholes Partial Differential Equation for a Derivative
The market consists of a dividend-paying stock with price process {St}t and continuous dividend yield 0, and a risk-free bank account with price process {Bt}t, where Bt = ert for each t 0, and r > 0 is the continuously compounded interest rate. The price V (t, St) at time t of a European derivative on the stock, with payoffXT attimeT,issaidtosatisfytheBlack-ScholesPDEif
( V t + ( r ) S t V S + 1 2 S t2 V S S = r V ( t , S t ) , 2
with boundary condition V (T , ST ) = XT , whereV = @V (t,S ),V = @V (t,S ),andV = @2V (t,S ).
S @St t t @t t SS @St2 t Call and Put Prices in the Black-Scholes framework
The Black-Scholes formula for the price at time zero of a European call option on a non-dividend-paying stock
(i.e., = 0) is given by
where N is the standard normal cumulative distribution function,
c(0,S0,K,T) = S0N(d1) Ke rTN(d2), ln(S0/K) + (r + 2/2)T p
d1= pT and d2=d1 T.
The Black-Scholes formula for the price at time zero of a European put option on a non-dividend-paying stock is
given by
The following equation might be useful:
p(0,S0,K,T) = Ke rTN( d2) S0N( d1). S0N0(d1) = Ke rT N0(d2),
where N0(x) = d N(x) is the standard normal density function. dx
Some Results on Normal and Lognormal Distributions
Let X be a Gaussian (normal) random variable with mean m and variance 2 (this distribution is denoted by N (m, 2)). Then eX is log-normally distributed with parameters m and 2, and for any K 2 R+,
E[K1{eX>K}] = KN✓m lnK◆
In particular, this implies that
✓ ◆✓◆ X 2 m+ 2 lnK
E[e1{eX>K}]=expm+2 N . The moment-generating function of X is given by
E[etX ] = emt+ 2t2/2. X m+ 2
E[e]=e 2. 6
Table entry
z .00 .01
z
Table entry for z is the area under the standard normal curve to the left of z.
.02 .03 .04 .05 .06 .07 .08 .09
Standard Normal Probabilities
0.0 .5000
0.1 .5398
0.2 .5793
0.3 .6179
0.4 .6554
0.5 .6915
0.6 .7257
0.7 .7580
0.8 .7881
0.9 .8159
1.0 .8413
1.1 .8643
1.2 .8849
1.3 .9032
1.4 .9192
1.5 .9332
1.6 .9452
1.7 .9554
1.8 .9641
1.9 .9713
2.0 .9772
2.1 .9821
2.2 .9861
2.3 .9893
2.4 .9918
2.5 .9938
2.6 .9953
2.7 .9965
2.8 .9974
2.9 .9981
3.0 .9987
3.1 .9990
3.2 .9993
3.3 .9995
3.4 .9997
.5040 .5080 .5438 .5478 .5832 .5871 .6217 .6255 .6591 .6628 .6950 .6985 .7291 .7324 .7611 .7642 .7910 .7939 .8186 .8212 .8438 .8461 .8665 .8686 .8869 .8888 .9049 .9066 .9207 .9222 .9345 .9357 .9463 .9474 .9564 .9573 .9649 .9656 .9719 .9726 .9778 .9783 .9826 .9830 .9864 .9868 .9896 .9898 .9920 .9922 .9940 .9941 .9955 .9956 .9966 .9967 .9975 .9976 .9982 .9982 .9987 .9987 .9991 .9991 .9993 .9994 .9995 .9995 .9997 .9997
.5120 .5160 .5517 .5557 .5910 .5948 .6293 .6331 .6664 .6700 .7019 .7054 .7357 .7389 .7673 .7704 .7967 .7995 .8238 .8264 .8485 .8508 .8708 .8729 .8907 .8925 .9082 .9099 .9236 .9251 .9370 .9382 .9484 .9495 .9582 .9591 .9664 .9671 .9732 .9738 .9788 .9793 .9834 .9838 .9871 .9875 .9901 .9904 .9925 .9927 .9943 .9945 .9957 .9959 .9968 .9969 .9977 .9977 .9983 .9984 .9988 .9988 .9991 .9992 .9994 .9994 .9996 .9996 .9997 .9997
.5199 .5239 .5279 .5319 .5596 .5636 .5675 .5714 .5987 .6026 .6064 .6103 .6368 .6406 .6443 .6480 .6736 .6772 .6808 .6844 .7088 .7123 .7157 .7190 .7422 .7454 .7486 .7517 .7734 .7764 .7794 .7823 .8023 .8051 .8078 .8106 .8289 .8315 .8340 .8365 .8531 .8554 .8577 .8599 .8749 .8770 .8790 .8810 .8944 .8962 .8980 .8997 .9115 .9131 .9147 .9162 .9265 .9279 .9292 .9306 .9394 .9406 .9418 .9429 .9505 .9515 .9525 .9535 .9599 .9608 .9616 .9625 .9678 .9686 .9693 .9699 .9744 .9750 .9756 .9761 .9798 .9803 .9808 .9812 .9842 .9846 .9850 .9854 .9878 .9881 .9884 .9887 .9906 .9909 .9911 .9913 .9929 .9931 .9932 .9934 .9946 .9948 .9949 .9951 .9960 .9961 .9962 .9963 .9970 .9971 .9972 .9973 .9978 .9979 .9979 .9980 .9984 .9985 .9985 .9986 .9989 .9989 .9989 .9990 .9992 .9992 .9992 .9993 .9994 .9994 .9995 .9995 .9996 .9996 .9996 .9996 .9997 .9997 .9997 .9997
.5359 .5753 .6141 .6517 .6879 .7224 .7549 .7852 .8133 .8389 .8621 .8830 .9015 .9177 .9319 .9441 .9545 .9633 .9706 .9767 .9817 .9857 .9890 .9916 .9936 .9952 .9964 .9974 .9981 .9986 .9990 .9993 .9995 .9997 .9998
7