1) The document discusses pricing options when volatility is uncertain and lies within a bounded range. It describes constructing a portfolio using a long call option and shorting the underlying asset to hedge against volatility risk. 2) It then discusses using static hedging to hedge options, like hedging a long digital call position using a short call spread. The goal is to find the optimal number of short calls to minimize residual risk, done by solving a partial differential equation. 3) Technical difficulties prevented finding the optimal hedge ratio using an Excel solver as planned. Instead, the document ends by listing references for further reading on uncertain volatility modeling and hedging techniques.

1. Uncertain Volatility with Static Hedge
Abukar Ali

2. Uncertain Volatility with Static Hedge
We begin with the Black-Scholes equation for pricing a vanilla option contract which is a
parabolic PDE with two variables, S and t and with constant parameters such as r, D, and sigma
(volatility). Instead of pricing the option with constant volatility, this assumption can be relaxed
and we can assume volatility to be parameter that lies within certain range.
Assume that volatility lies within the band
𝜎−
< 𝜎 < 𝜎+
Construct a portfolio consisting of the value of a long call option V(S,t), and hedge it by shorting
-∆ of the underlying asset. The value of the portfolio becomes
Π = 𝑉 − Δ𝑆 and 𝑑𝑠 = 𝑢𝑠𝑑𝑡 + 𝜎𝑆𝑑𝑋
The change in the value of the portfolio is
𝑑Π = (
𝜕𝑉
𝜕𝑡
+ 0.5𝜎2 𝜕2 𝑉
𝜕2 𝑆
) 𝑑𝑡 + (
𝜕𝑉
𝜕𝑆
− ∆) 𝑑𝑠, 𝑑Π = (
𝜕𝑉
𝜕𝑡
+ 0.5𝜎2 𝜕2 𝑉
𝜕2 𝑆
) 𝑑𝑡
Since volatility is unknown, we now assume long vanilla positions to take the lowest value of
the volatility range while short position takes the highest volatility value. The return of the
portfolio is set equal to the risk-free rate.
min
𝜎−<𝜎<𝜎+
(dΠ) = rΠdt
We set
min
𝜎−<𝜎<𝜎+
(
𝜕𝑉
𝜕𝑡
+ 0.5𝜎2 𝜕2 𝑉
𝜕2 𝑆
) 𝑑𝑡 = (𝑉 − 𝑆
𝜕𝑉
𝜕𝑆
) 𝑑𝑡
The value that the volatility takes depends sign of Gamma (Γ) and the worst-case value
satisfies1
𝜕𝑉−
𝜕𝑡
+ 0.5𝜎 (Γ)2 𝜕2 𝑉−
𝜕2 𝑆
+ 𝑟𝑆
𝜕𝑉−
𝜕𝑆
− 𝑟𝑉−
= 0 (2.1)
Where
Γ =
𝜕2 𝑉−
𝜕2 𝑆
and
𝜎(Γ) = 𝜎+
𝑖𝑓 Γ < 0
1
Equation 2.1 was originally derived by Avellaneda, Levy & Paras and Lyons. This is also the same as the
Hoggard – Whalley – Wilmott transaction cost model. See P.Wilmott on Quantitative Finance, 2 edition,
Chapter 48.

3. 𝜎(Γ) = 𝜎−
𝑖𝑓 Γ > 0
The worst-case scenario for a delta hedged long vanilla option is if volatility is low while the
opposite is true for a delta hedged short vanilla option.
The best option value, and hence the range of possible values can be found by solving
𝜕𝑉−
𝜕𝑡
+ 0.5𝜎 (Γ)2 𝜕2 𝑉−
𝜕2 𝑆
+ 𝑟𝑆
𝜕𝑉−
𝜕𝑆
− 𝑟𝑉−
= 0 (2.2)
Where
Γ =
𝜕2 𝑉−
𝜕2 𝑆
and
𝜎(Γ) = 𝜎+
𝑖𝑓 Γ < 0
𝜎(Γ) = 𝜎−
𝑖𝑓 Γ > 0
Static Hedging
Unlike delta hedging, static hedging does not require dynamic rebalancing of the hedged
portfolio. For example, suppose we have long position of a digital option which we would like
to hedge. To hedge the position, we can use call spread and discrete hedge the position from
time to time., However, instead of continues rebalancing, we can set up a one time hedge with
the same instrument as above (call spread) but with an optimum number of call spread to use
for the static hedge.
Consider the following example; suppose we want to buy a call option with strike price 𝐾1with
the following payoff:
𝑃 = 𝑀𝑎𝑥(𝑆 − 𝐾1, 0)
To hedge the above call position, we short another call with strike 𝐾2 and a
payoff 𝑃2 = 𝑀𝑎𝑥(𝑆 − 𝐾2, 0). The difference in the payoffs is then
𝐷𝑖𝑓𝑓 = 𝑀𝑎𝑥(𝑆 − 𝐾1) − 𝜆𝑀𝑎𝑥(𝑆 − 𝐾2, 0)
Were 𝜆 is the number of short positions we need to hedge the long call position. This portfolio
is said to be statically hedged and it has smaller payoff then the unhedged call.
To value the residual payoff (Diff) in the uncertain parameter framework, one needs to solve
𝜕𝑉−
𝜕𝑡
+ 0.5𝜎 (Γ)2 𝜕2 𝑉−
𝜕2 𝑆
+ 𝑟𝑆
𝜕𝑉−
𝜕𝑆
− 𝑟𝑉−
= 0
With the final condition
𝑉(𝑆, 𝑇) = 𝑀𝑎𝑥(𝑆 − 𝐾1) − 𝜆𝑀𝑎𝑥(𝑆 − 𝐾2, 0)

4. To find the optimum hedge, we solve for the optimum number of calls to short. So, given
market price for the original call, the statically hedged portfolio with the final condition and
with the optimum hedge should produce more tighter spread using the volatility scenario under
the static hedge and uncertain parameter framework.
We initially price up the digital call option and produce its payoff chart at expiry. To hedge the
digital call option we use a portfolio of long and short call options. As stated above, we price
up these options individually and show their payoff at expiry.
Figure 2.1 Digital Call
To hedge the digital call option with strike price of 100, we use initially a short call option with
a quantity of 0.05 and long call position with quantity of 0.05. Assume the calls have market
prices as given in table below
Table 2.1
Option
Type Strike Maturity Bid Ask Quantity
Target Option Digital Call 100 0.5 0 0 1
Hedging Option
1 Call 90 0.5 14.42 14.42 -0.05
Hedging Option
2 Call 110 0.5 4.22 4.22 0.05
The cost of setting up the hedge is
Hedging Cost = -0.05 x 14.42 + 0.05 x 4.22 = -0.51
We solve
𝜕𝑉−
𝜕𝑡
+ 0.5𝜎 (Γ)2
𝜕2
𝑉−
𝜕2 𝑆
+ 𝑟𝑆
𝜕𝑉−
𝜕𝑆
− 𝑟𝑉−
= 0
With the final condition
𝑉(𝑆, 𝑇) = 𝑀𝑎𝑥(𝑆 − 𝐾1) − 𝜆𝑀𝑎𝑥(𝑆 − 𝐾2, 0)
0
0.2
0.4
0.6
0.8
1
1.2
0
7.2
14.4
21.6
28.8
36
43.2
50.4
57.6
64.8
72
79.2
86.4
93.6
100.8
108
115.2
122.4
129.6
136.8
144
151.2
158.4
165.6
172.8
180

5. We deduct the result from the cost of setting up the hedge.
We repeat this process but this time we use the opposite scenario for the volatility and we
deduct the result from the cost of setting up the hedge.
The difference from both results should produce tighter price spread then what would have
been quoted for the original unhedged digital call option. To find the optimum number of calls
to sell and buy, we can use solver in excel.
Figure 2.2
Technical Problem.
Due to the copy office package currently running on my PC, the solver is NOT available and
downloading this add-in is also still causing some technical problems. See screen grab below.
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0
8.8
17.6
26.4
35.2
44
52.8
61.6
70.4
79.2
88
96.8
105.6
114.4
123.2
132
140.8
149.6
158.4
167.2
176
184.8
193.6
202.4
211.2
220

6. References
M. Avellaneda, A. Levy, A. Paras, “Pricing and hedging derivative securities in markets with uncertain
volatilities”, Journal of Applied Finance, Vol 1, 1995
Paul Wilmott, P.Wilmott on Quantitative Finance”, 2nd
Edition, Volume 3, Wiley.
CQF Lecture 2010/2011 – Heath, Jarrow and Morton Model
CQF Lecture 2010/ 2011 - Advanced Volatility Modelling in Complete Markets
P. Wilmott, A. Oztukel, “