Exercises 3.1 Volatility swap A Volatility swap is a contract that pays at a maturity date T the realized volatility realized (which is the square root of the realized variance) of a stock or index minus a strike K. By denition, the strike K is set such that the value of the volatility swap at t (i.e., today) is zero. Therefore, we have K = EP[

p VrealizedjFt]

with Vrealized 

1 N

NX????1 i=0 ln



Si+1 Si

2 We recall that a variance swap K is K = EP[VrealizedjFt]

Assuming a diusion model for the stock price St, when we take the limit N ! 1, the realized variance between t and T is Vrealized 
1 T ???? t Z T t 2 sds Here s is the instantaneous volatility dened by dSt = tStdWt We assume zero interest rate. We have seen in this chapter that the value of a variance swap (i.e., K) is completely model-independent. Indeed, there exists a static replication where the value of the variance swap can be decomposed as an innite sum of calls and puts which are priced on the market. In this problem, we propose to show that a similar model-independent solution exists for the volatility swap if we assume that the stock St and the volatility t are independent processes. This problem is based on the article [72]. Below, we note VT = Vrealized.
1. As the processes t and Wt are independent by assumption, prove that conditional on the ltration F
T generated by the volatility ftgtT , XT = ln(ST St ) is a normal process with mean ????1 2 (T ???? t)VT and variance (T ???? t)VT .
2. As an application of the tower property, we have 8p 2 C, EP[epXT ] = EP[EP[epXT jF
T ]]
By using question 1. and the equality above, prove that EP[epXT ] = EP[e(p)(T????t)VT ] (3.23)
with (p)  p2 2 ???? p 2 . Equivalently, we have p() = 1 2 
q 1 4 + 2.
3. Let f(t) be a real function, locally summable on R+. The Laplace transform of

f, noted L(z), z = x + iy 2 C, is dened as L(z) = Z 1 0 f(t)e????ztdt If L(z) is dened for all x  Re[z] > x0 and if L(z) is a summable function 8x > x0, then the inversion formula (called the Bromvitch formula) giving f(t) is 1t0f(t) = 1 2i Z a+i1 a????i1 L()exd (3.24)
where a > x0. By using the inversion formula (3.24), prove that the Laplace transform of the square root function is p t = 1 2 p 
Z 1 0 1 ???? e????t 
3 2 d (3.25)

4. By using (3.25) and (3.23), deduce that EP[
p (T ???? t)VT ] = 1 2 p 
Z 1 0 1 ???? EP[e( 1 2????
p 14 ????2)XT jFt]

32 d
5. By using the static replication formula (3.4), decompose the European option EP[e( 1 2????
p 14 ????2)XT jFt] on an innite sum of calls and puts.
6. Deduce the model-independent formula for the volatility swap.