# Risk Quant interview at the big four

#### Quasar Chunawala

Hi,

I recently attended a risk quant interview at Ernst and Young. It was for a risk model validation role typically for their Quantitative Advisory team under the umbrella of risk. This was a telephonic interview. I am disappointed with my performance, and I could have done better.
1. Tell us about yourself and a little bit about the quantitative finance projects that you have mentioned in your resume.

[Me:] I walked them through my resume, giving a brief overview of the work I did as a software programmer and why I switched fields to finance. I also talked about my reading habits, books that interest me, and some quantitative finance projects I've done on my own.

2. How do you derive the analytical expressions for the standard European call and put on equities?

[Me:] You can start in a discrete world and assume that the stock returns are binomially distributed, and the stock prices are binomial-like. One can derive an analytical expression for the expected payoff of the option discounted to the present. As the number of time-steps increases, in the continuous time limit, the binomial formula converges the Black-Scholes analytical expressions.

For a call,

$$C(S,t)=D_{f}S\Phi(d_{+})-XD_{d}\Phi(d_{-})$$

intuitively, the first term is expected value of the stock you are likely to receive and the second term is expected value of the exercise price paid.

The standard derivation of the Black Scholes PDE, proceeds by creating a special risk-free portfolio of going long option and shorting $$\Delta$$ units of the stock. The little change in the portfolio value $$d\Pi$$ in a small time period [dt] would be the sum of $$dC$$ and short $$-\Delta{dS}$$. Since, the option value is a function $$f$$ of the underlying $$x$$ and time $$t$$, the small change in it is given by the Ito's lemma. That way,

$$du(t,X_{t})=\frac{\partial{u}}{\partial{t}}(t,X_{t})dt+\frac{\partial{u}}{\partial{x}}(t,X_{t})dx+\frac{1}{2}\sigma^{2}{x^2}\frac{\partial{u}}{\partial{x^{2}}}(t,X_{t})dt$$

As the portfolio is risk-free, it must earn the risk-free rate of return.

After simplication, you get an expression for the Black Scholes PDE -

$$u_{t}+rxu_{x}+\frac{1}{2}\sigma^{2}x^{2}u_{xx}=ru$$

This can be reduced to:

$$u_{t}=\kappa{u_{xx}}$$

which is the standard heat equation.

3. Do you know of any other models where the volatility is not constant?

[Me:] I haven't implemented those by myself. But, I know that volatility is also modeled as stochastic, for example in the Heston model.

4. How do you approximate the Black Scholes PDE?

[Me:] For instruments, where no analytical solution can be derived, we numerically approximate it by constructing a space and time grid. Various finite difference schemes are used.

5. You've written about different products on your resume like Asian options, Barrier options... Do you know the delta profile of a barrier option?

[Me:] I know, the payoff of these products. But, I don't know the delta profile of a barrier.

6. A dice is rolled. What is the expection of the value on the dice face?

[Me:] The expected value is 7/2.

Mostly I'm self-learning. According to a famous adage, half knowledge is dangerous. So, I recently enrolled in a BSc. Maths course at a local university, so I would be learning & writing exams on real and complex analysis, Fourier series, ODEs, PDEs, geometry and probability. I would like to have a solid foundation in probability and then learn stochastic calculus.

I had few queries, which I was hoping, I would seek answers from friends here.
• Any interview tips? How do I get better at the game?

• How do you land an entry-level desk quant job? What's the best way to break in? Is it wise to work as a risk quant, acquire skills and then transition to a desk quant or rates/FX/Commodities model validation role in the future?

• Can I learn enough quantitative finance on my own(without going to an MFE school) to land a job, perhaps not immediately, a couple of years from now?
IMO, I could have fared better in the interview, if I had prepared atleast the very basics, such as delta profile of 1st gen exotics, what exactly is local volatility, what volatility models are used in the real world..

Last edited:

#### Quasar Chunawala

Hi,

I recently attended a risk quant interview at Ernst and Young. It was for a risk model validation role typically for their Quantitative Advisory team under the umbrella of risk. This was a telephonic interview. I am disappointed with my performance, and I could have done better.
1. Tell us about yourself and a little bit about the quantitative finance projects that you have mentioned in your resume.

[Me:] I walked them through my resume, giving a brief overview of the work I did as a software programmer and why I switched fields to finance. I also talked about my reading habits, books that interest me, and some quantitative finance projects I've done on my own.

2. How do you derive the analytical expressions for the standard European call and put on equities?

[Me:] You can start in a discrete world and assume that the stock returns are binomially distributed, and the stock prices are binomial-like. One can derive an analytical expression for the expected payoff of the option discounted to the present. As the number of time-steps increases, in the continuous time limit, the binomial formula converges the Black-Scholes analytical expressions.

For a call,

$$C(S,t)=D_{f}S\Phi(d_{+})-XD_{d}\Phi(d_{-})$$

intuitively, the first term is expected value of the stock you are likely to receive and the second term is expected value of the exercise price paid.

The standard derivation of the Black Scholes PDE, proceeds by creating a special risk-free portfolio of going long option and shorting $$\Delta$$ units of the stock. The little change in the portfolio value $$d\Pi$$ in a small time period [dt] would be the sum of $$dC$$ and short $$-\Delta{dS}$$. Since, the option value is a function $$f$$ of the underlying $$x$$ and time $$t$$, the small change in it is given by the Ito's lemma. That way,

$$du(t,X_{t})=\frac{\partial{u}}{\partial{t}}(t,X_{t})dt+\frac{\partial{u}}{\partial{x}}(t,X_{t})dx+\frac{1}{2}\sigma^{2}{x^2}\frac{\partial{u}}{\partial{x^{2}}}(t,X_{t})dt$$

As the portfolio is risk-free, it must earn the risk-free rate of return.

After simplication, you get an expression for the Black Scholes PDE -

$$u_{t}+rxu_{x}+\frac{1}{2}\sigma^{2}x^{2}u_{xx}=ru$$

This can be reduced to:

$$u_{t}=\kappa{u_{xx}}$$

which is the standard heat equation.

3. Do you know of any other models where the volatility is not constant?

[Me:] I haven't implemented those by myself. But, I know that volatility is also modeled as stochastic, for example in the Heston model.

4. How do you approximate the Black Scholes PDE?

[Me:] For instruments, where no analytical solution can be derived, we numerically approximate it by constructing a space and time grid. Various finite difference schemes are used.

5. You've written about different products on your resume like Asian options, Barrier options... Do you know the delta profile of a barrier option?

[Me:] I know, the payoff of these products. But, I don't know the delta profile of a barrier.

6. A dice is rolled. What is the expection of the value on the dice face?

[Me:] The expected value is 7/2.

Mostly I'm self-learning. According to a famous adage, half knowledge is dangerous. So, I recently enrolled in a BSc. Maths course at a local university, so I would be learning & writing exams on real and complex analysis, Fourier series, ODEs, PDEs, geometry and probability. I would like to have a solid foundation in probability and then learn stochastic calculus.

I had few queries, which I was hoping, I would seek answers from friends here.
• Any interview tips? How do I get better at the game?

• How do you land an entry-level desk quant job? What's the best way to break in? Is it wise to work as a risk quant, acquire skills and then transition to a desk quant or rates/FX/Commodities model validation role in the future?

• Can I learn enough quantitative finance on my own(without going to an MFE school) to land a job, perhaps not immediately, a couple of years from now?
IMO, I could have fared better in the interview, if I had prepared atleast the very basics, such as delta profile of 1st gen exotics, what exactly is local volatility, what volatility models are used in the real world..

Hi friends,

If anyone has any inputs, it'll be an awesome help!

Cheers,
Quasar.

@Quasar Chunawala your answers look great given the fact that you are a self-learner. I think I am putting a big morsel in my small mouth but I have a suggestion for your second answer. According to me the mention of using similarity reduction technique to solve that BS PDE further to get the closed form analytical solution (The famous BS formula as we know it) would have given the interviewer a jolt. Nevertheless, it was a great effort on your side.
Thanks,
Shreyash

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