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MFE 406 – Problem Set 2

Due: W 08 Feb 2023, 11:55 pm PDT

1. Convergence of the Binomial Model to the Black-Scholes Formula (Notes Part 1, §3.3)

Consider the ∆t = T/n ↘ 0 limit (as n ↗ ∞) of the Binomial model discussed on pages 65 & 67 of Part 1 of the notes.

In what follows, please assume that we wish to value a European call option with strike price K and maturity T on a non-dividend paying stock with initial stock price S0 and constant volatility σ. The riskless interest rate is r.

You should assume that, in log space, the up and down moves over each timestep ∆t √

are given by: α ∆t ± σ ∆t, respectively. You are free to choose the lattice drift rate α according to your convenience, since each of the choices illustrated in the notes has its own advantages and disadvantages.

  1. (a)  What are the risk-neutral probabilities {qu, qd = 1 − qu} for your choice of α?
  2. (b)  How do {qu,qd} behave as ∆t ↘ 0? Obtain expansions of both valid to at least O(∆t)1; i.e., higher order (neglected) terms are of o(∆t)1.

Now, consider the valuation formula shown at the bottom of page 67 of the notes:

−rT m Xm mqu k  (2k−m)σ√∆t+mα∆t 

C0(S0,0)=e qd k q k=0

CT Sk =S0e ,m∆t d

(c) Identify, for a given m, the smallest node k′ for which the call finishes in-the-money at time T. Thereby, reduce the valuation formula to a sum over k from k′ to ∞.

  1. (d)  Change co-ordinates from k to xk = log(Sk/S0) so that the sum is over xk ranging from xk′ (which depends on m) to ∞.
  2. (e)  Next, consider m large and use the exponential formula: limm↗∞(1 ± γ/m)m = e±γ , √ nn as well as Stirling’s approximation: n! ∼ 2πn e , to approximate the powers of the q probabilities and the binomial coefficients in the large m limit. Hint: Don’t worry about whether the approximations hold in the cases where k or m−k remains small: remember that the contributions of those terms vanish in the m↗∞ limit.

(f) Finally, consider the m↗∞ limit of the sum as a Riemann sum or integral over xk. Zy12 r2π1b2√b

Usingtheformulæ: dxe−2(ax +bx+c) = e2(4a−c)N ay+ √ (fora>0) Z∞−∞ a2a

1 −x2/2
and √ dxe = 1 − N(y) = N(−y), express your results in terms of the

2π y
standard normal cdf N (·) and – hopefully – obtain the Black-Scholes formula for the call option’s value.

1

2. The Trinomial Model (Notes Part 2, §1.1)

Suppose that, in the spirit of constructing a Brownian motion as the limit of a sum of Bernoulli (binomial, random walk) increments on page 5 of the notes, we attempt the construction using trinomial (random walker plus a “stay put” outcome) increments. As discussed in class, a natural place to start is by matching the moments of the trinomially- distributed step over the interval dt with those of a standard Brownian increment dWt.

To make things more concrete, let us define the set of possible trinomial moves as:

√

u = +β dt with probability pu √ 0 with probability p0

d = −β dt with probability pd where β is (an as-yet unspecified) positive, real, O(1) constant.

  1. (a)  What constraints do the 0th moment (conservation of probability) and 1st moment (martingale) conditions place on the 3 probabilities? How many degrees of freedom are left?
    If β could be an adjustable parameter, how many degrees of freedom are there?
  2. (b)  If we now impose a 2nd moment condition, i.e., require that the variance of the trinomially-distributed increment equal that of a standard Brownian increment dWt, how do your answers to (a) change? Are there any resulting constraints on the value of β?
  3. (c)  What additional information could you potentially use to remove any degrees of freedom remaining in (b)?
    What values would the 3 probabilities and β take?

3. Absolute Moments of the Standard Normal Distribution and SBM Variation Properties (Notes Part 2, §1.2)

Calculate E[|z|α],α ∈ R for the standard normal variate z ∼ N(0,1).

  1. (a)  What restrictions apply to α for the expectation to converge?
  2. (b)  Can you express your results in terms of a well-known function?
  3. (c)  How do your results change if, instead of z ∼ N(0,1), you calculate E[|Wt|α],α ∈ R for the SBM Wt ∼ N(0, t)?

Now, consider the process of constructing Bt as the n ↗ ∞ limit of the sum of n indepen- dent Gaussian increments.

  1. (d)  Using your results from part (c), obtain expressions for the mean and variance of Vnm(t) ≡ n-increment approximation to the mth variation V m(t) of Bt, thereby either confirming or refuting the scaling relationships and coefficients asserted in section 1.2 of Part 2 of the lecture notes. Hint: You might wish to consider m=0 in addition to m=1,2,3.
  2. (e)  Under what circumstances do the mean and variance of Vnm(t) each either diverge, converge to 0, or remain finite (but non-zero) as n↗∞?
    Hint: summarize your results in a table with columns for the mean and variance and rows corresponding to values or ranges of m.

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